arguments by analogy in critical thinking

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Analogy and Analogical Reasoning

An analogy is a comparison between two objects, or systems of objects, that highlights respects in which they are thought to be similar. Analogical reasoning is any type of thinking that relies upon an analogy. An analogical argument is an explicit representation of a form of analogical reasoning that cites accepted similarities between two systems to support the conclusion that some further similarity exists. In general (but not always), such arguments belong in the category of ampliative reasoning, since their conclusions do not follow with certainty but are only supported with varying degrees of strength. However, the proper characterization of analogical arguments is subject to debate (see §2.2 ).

Analogical reasoning is fundamental to human thought and, arguably, to some nonhuman animals as well. Historically, analogical reasoning has played an important, but sometimes mysterious, role in a wide range of problem-solving contexts. The explicit use of analogical arguments, since antiquity, has been a distinctive feature of scientific, philosophical and legal reasoning. This article focuses primarily on the nature, evaluation and justification of analogical arguments. Related topics include metaphor , models in science , and precedent and analogy in legal reasoning .

1. Introduction: the many roles of analogy

2.1 examples, 2.2 characterization, 2.3 plausibility, 2.4 analogical inference rules, 3.1 commonsense guidelines, 3.2 aristotle’s theory, 3.3 material criteria: hesse’s theory, 3.4 formal criteria: the structure-mapping theory, 3.5 other theories, 3.6 practice-based approaches, 4.1 deductive justification, 4.2 inductive justification, 4.3 a priori justification, 4.4 pragmatic justification, 5.1 analogy and confirmation, 5.2 conceptual change and theory development, online manuscript, related entries.

Analogies are widely recognized as playing an important heuristic role, as aids to discovery. They have been employed, in a wide variety of settings and with considerable success, to generate insight and to formulate possible solutions to problems. According to Joseph Priestley, a pioneer in chemistry and electricity,

analogy is our best guide in all philosophical investigations; and all discoveries, which were not made by mere accident, have been made by the help of it. (1769/1966: 14)

Priestley may be over-stating the case, but there is no doubt that analogies have suggested fruitful lines of inquiry in many fields. Because of their heuristic value, analogies and analogical reasoning have been a particular focus of AI research. Hájek (2018) examines analogy as a heuristic tool in philosophy.

Example 1 . Hydrodynamic analogies exploit mathematical similarities between the equations governing ideal fluid flow and torsional problems. To predict stresses in a planned structure, one can construct a fluid model, i.e., a system of pipes through which water passes (Timoshenko and Goodier 1970). Within the limits of idealization, such analogies allow us to make demonstrative inferences, for example, from a measured quantity in the fluid model to the analogous value in the torsional problem. In practice, there are numerous complications (Sterrett 2006).

At the other extreme, an analogical argument may provide very weak support for its conclusion, establishing no more than minimal plausibility. Consider:

Example 2 . Thomas Reid’s (1785) argument for the existence of life on other planets (Stebbing 1933; Mill 1843/1930; Robinson 1930; Copi 1961). Reid notes a number of similarities between Earth and the other planets in our solar system: all orbit and are illuminated by the sun; several have moons; all revolve on an axis. In consequence, he concludes, it is “not unreasonable to think, that those planets may, like our earth, be the habitation of various orders of living creatures” (1785: 24).

Such modesty is not uncommon. Often the point of an analogical argument is just to persuade people to take an idea seriously. For instance:

Example 3 . Darwin takes himself to be using an analogy between artificial and natural selection to argue for the plausibility of the latter:

Why may I not invent the hypothesis of Natural Selection (which from the analogy of domestic productions, and from what we know of the struggle of existence and of the variability of organic beings, is, in some very slight degree, in itself probable) and try whether this hypothesis of Natural Selection does not explain (as I think it does) a large number of facts…. ( Letter to Henslow , May 1860 in Darwin 1903)

Here it appears, by Darwin’s own admission, that his analogy is employed to show that the hypothesis is probable to some “slight degree” and thus merits further investigation. Some, however, reject this characterization of Darwin’s reasoning (Richards 1997; Gildenhuys 2004).

Sometimes analogical reasoning is the only available form of justification for a hypothesis. The method of ethnographic analogy is used to interpret

the nonobservable behaviour of the ancient inhabitants of an archaeological site (or ancient culture) based on the similarity of their artifacts to those used by living peoples. (Hunter and Whitten 1976: 147)

For example:

Example 4 . Shelley (1999, 2003) describes how ethnographic analogy was used to determine the probable significance of odd markings on the necks of Moche clay pots found in the Peruvian Andes. Contemporary potters in Peru use these marks (called sígnales ) to indicate ownership; the marks enable them to reclaim their work when several potters share a kiln or storage facility. Analogical reasoning may be the only avenue of inference to the past in such cases, though this point is subject to dispute (Gould and Watson 1982; Wylie 1982, 1985). Analogical reasoning may have similar significance for cosmological phenomena that are inaccessible due to limits on observation (Dardashti et al. 2017). See §5.1 for further discussion.

As philosophers and historians such as Kuhn (1996) have repeatedly pointed out, there is not always a clear separation between the two roles that we have identified, discovery and justification. Indeed, the two functions are blended in what we might call the programmatic (or paradigmatic ) role of analogy: over a period of time, an analogy can shape the development of a program of research. For example:

Example 5 . An ‘acoustical analogy’ was employed for many years by certain nineteenth-century physicists investigating spectral lines. Discrete spectra were thought to be

completely analogous to the acoustical situation, with atoms (and/or molecules) serving as oscillators originating or absorbing the vibrations in the manner of resonant tuning forks. (Maier 1981: 51)

Guided by this analogy, physicists looked for groups of spectral lines that exhibited frequency patterns characteristic of a harmonic oscillator. This analogy served not only to underwrite the plausibility of conjectures, but also to guide and limit discovery by pointing scientists in certain directions.

More generally, analogies can play an important programmatic role by guiding conceptual development (see §5.2 ). In some cases, a programmatic analogy culminates in the theoretical unification of two different areas of inquiry.

Example 6 . Descartes’s (1637/1954) correlation between geometry and algebra provided methods for systematically handling geometrical problems that had long been recognized as analogous. A very different relationship between analogy and discovery exists when a programmatic analogy breaks down, as was the ultimate fate of the acoustical analogy. That atomic spectra have an entirely different explanation became clear with the advent of quantum theory. In this case, novel discoveries emerged against background expectations shaped by the guiding analogy. There is a third possibility: an unproductive or misleading programmatic analogy may simply become entrenched and self-perpetuating as it leads us to “construct… data that conform to it” (Stepan 1996: 133). Arguably, the danger of this third possibility provides strong motivation for developing a critical account of analogical reasoning and analogical arguments.

Analogical cognition , which embraces all cognitive processes involved in discovering, constructing and using analogies, is broader than analogical reasoning (Hofstadter 2001; Hofstadter and Sander 2013). Understanding these processes is an important objective of current cognitive science research, and an objective that generates many questions. How do humans identify analogies? Do nonhuman animals use analogies in ways similar to humans? How do analogies and metaphors influence concept formation?

This entry, however, concentrates specifically on analogical arguments. Specifically, it focuses on three central epistemological questions:

• What criteria should we use to evaluate analogical arguments?
• What philosophical justification can be provided for analogical inferences?
• How do analogical arguments fit into a broader inferential context (i.e., how do we combine them with other forms of inference), especially theoretical confirmation?

Following a preliminary discussion of the basic structure of analogical arguments, the entry reviews selected attempts to provide answers to these three questions. To find such answers would constitute an important first step towards understanding the nature of analogical reasoning. To isolate these questions, however, is to make the non-trivial assumption that there can be a theory of analogical arguments —an assumption which, as we shall see, is attacked in different ways by both philosophers and cognitive scientists.

2. Analogical arguments

Analogical arguments vary greatly in subject matter, strength and logical structure. In order to appreciate this variety, it is helpful to increase our stock of examples. First, a geometric example:

Example 7 (Rectangles and boxes). Suppose that you have established that of all rectangles with a fixed perimeter, the square has maximum area. By analogy, you conjecture that of all boxes with a fixed surface area, the cube has maximum volume.

Two examples from the history of science:

Example 8 (Morphine and meperidine). In 1934, the pharmacologist Schaumann was testing synthetic compounds for their anti-spasmodic effect. These drugs had a chemical structure similar to morphine. He observed that one of the compounds— meperidine , also known as Demerol —had a physical effect on mice that was previously observed only with morphine: it induced an S-shaped tail curvature. By analogy, he conjectured that the drug might also share morphine’s narcotic effects. Testing on rats, rabbits, dogs and eventually humans showed that meperidine, like morphine, was an effective pain-killer (Lembeck 1989: 11; Reynolds and Randall 1975: 273).

Example 9 (Priestley on electrostatic force). In 1769, Priestley suggested that the absence of electrical influence inside a hollow charged spherical shell was evidence that charges attract and repel with an inverse square force. He supported his hypothesis by appealing to the analogous situation of zero gravitational force inside a hollow shell of uniform density.

Finally, an example from legal reasoning:

Example 10 (Duty of reasonable care). In a much-cited case ( Donoghue v. Stevenson 1932 AC 562), the United Kingdom House of Lords found the manufacturer of a bottle of ginger beer liable for damages to a consumer who became ill as a result of a dead snail in the bottle. The court argued that the manufacturer had a duty to take “reasonable care” in creating a product that could foreseeably result in harm to the consumer in the absence of such care, and where the consumer had no possibility of intermediate examination. The principle articulated in this famous case was extended, by analogy, to allow recovery for harm against an engineering firm whose negligent repair work caused the collapse of a lift ( Haseldine v. CA Daw & Son Ltd. 1941 2 KB 343). By contrast, the principle was not applicable to a case where a workman was injured by a defective crane, since the workman had opportunity to examine the crane and was even aware of the defects ( Farr v. Butters Brothers & Co. 1932 2 KB 606).

What, if anything, do all of these examples have in common? We begin with a simple, quasi-formal characterization. Similar formulations are found in elementary critical thinking texts (e.g., Copi and Cohen 2005) and in the literature on argumentation theory (e.g., Govier 1999, Guarini 2004, Walton and Hyra 2018). An analogical argument has the following form:

• \(S\) is similar to \(T\) in certain (known) respects.
• \(S\) has some further feature \(Q\).
• Therefore, \(T\) also has the feature \(Q\), or some feature \(Q^*\) similar to \(Q\).

(1) and (2) are premises. (3) is the conclusion of the argument. The argument form is ampliative ; the conclusion is not guaranteed to follow from the premises.

\(S\) and \(T\) are referred to as the source domain and target domain , respectively. A domain is a set of objects, properties, relations and functions, together with a set of accepted statements about those objects, properties, relations and functions. More formally, a domain consists of a set of objects and an interpreted set of statements about them. The statements need not belong to a first-order language, but to keep things simple, any formalizations employed here will be first-order. We use unstarred symbols \((a, P, R, f)\) to refer to items in the source domain and starred symbols \((a^*, P^*, R^*, f^*)\) to refer to corresponding items in the target domain. In Example 9 , the source domain items pertain to gravitation; the target items pertain to electrostatic attraction.

Formally, an analogy between \(S\) and \(T\) is a one-to-one mapping between objects, properties, relations and functions in \(S\) and those in \(T\). Not all of the items in \(S\) and \(T\) need to be placed in correspondence. Commonly, the analogy only identifies correspondences between a select set of items. In practice, we specify an analogy simply by indicating the most significant similarities (and sometimes differences).

We can improve on this preliminary characterization of the argument from analogy by introducing the tabular representation found in Hesse (1966). We place corresponding objects, properties, relations and propositions side-by-side in a table of two columns, one for each domain. For instance, Reid’s argument ( Example 2 ) can be represented as follows (using \(\Rightarrow\) for the analogical inference):

Hesse introduced useful terminology based on this tabular representation. The horizontal relations in an analogy are the relations of similarity (and difference) in the mapping between domains, while the vertical relations are those between the objects, relations and properties within each domain. The correspondence (similarity) between earth’s having a moon and Mars’ having moons is a horizontal relation; the causal relation between having a moon and supporting life is a vertical relation within the source domain (with the possibility of a distinct such relation existing in the target as well).

In an earlier discussion of analogy, Keynes (1921) introduced some terminology that is also helpful.

Positive analogy . Let \(P\) stand for a list of accepted propositions \(P_1 , \ldots ,P_n\) about the source domain \(S\). Suppose that the corresponding propositions \(P^*_1 , \ldots ,P^*_n\), abbreviated as \(P^*\), are all accepted as holding for the target domain \(T\), so that \(P\) and \(P^*\) represent accepted (or known) similarities. Then we refer to \(P\) as the positive analogy .

Negative analogy . Let \(A\) stand for a list of propositions \(A_1 , \ldots ,A_r\) accepted as holding in \(S\), and \(B^*\) for a list \(B_1^*, \ldots ,B_s^*\) of propositions holding in \(T\). Suppose that the analogous propositions \(A^* = A_1^*, \ldots ,A_r^*\) fail to hold in \(T\), and similarly the propositions \(B = B_1 , \ldots ,B_s\) fail to hold in \(S\), so that \(A, {\sim}A^*\) and \({\sim}B, B^*\) represent accepted (or known) differences. Then we refer to \(A\) and \(B\) as the negative analogy .

Neutral analogy . The neutral analogy consists of accepted propositions about \(S\) for which it is not known whether an analogue holds in \(T\).

Finally we have:

Hypothetical analogy . The hypothetical analogy is simply the proposition \(Q\) in the neutral analogy that is the focus of our attention.

These concepts allow us to provide a characterization for an individual analogical argument that is somewhat richer than the original one.

An analogical argument may thus be summarized:

It is plausible that \(Q^*\) holds in the target, because of certain known (or accepted) similarities with the source domain, despite certain known (or accepted) differences.

In order for this characterization to be meaningful, we need to say something about the meaning of ‘plausibly.’ To ensure broad applicability over analogical arguments that vary greatly in strength, we interpret plausibility rather liberally as meaning ‘with some degree of support’. In general, judgments of plausibility are made after a claim has been formulated, but prior to rigorous testing or proof. The next sub-section provides further discussion.

Note that this characterization is incomplete in a number of ways. The manner in which we list similarities and differences, the nature of the correspondences between domains: these things are left unspecified. Nor does this characterization accommodate reasoning with multiple analogies (i.e., multiple source domains), which is ubiquitous in legal reasoning and common elsewhere. To characterize the argument form more fully, however, is not possible without either taking a step towards a substantive theory of analogical reasoning or restricting attention to certain classes of analogical arguments.

Arguments by analogy are extensively discussed within argumentation theory. There is considerable debate about whether they constitute a species of deductive inference (Govier 1999; Waller 2001; Guarini 2004; Kraus 2015). Argumentation theorists also make use of tools such as speech act theory (Bermejo-Luque 2012), argumentation schemes and dialogue types (Macagno et al. 2017; Walton and Hyra 2018) to distinguish different types of analogical argument.

Arguments by analogy are also discussed in the vast literature on scientific models and model-based reasoning, following the lead of Hesse (1966). Bailer-Jones (2002) draws a helpful distinction between analogies and models. While “many models have their roots in an analogy” (2002: 113) and analogy “can act as a catalyst to aid modeling,” Bailer-Jones observes that “the aim of modeling has nothing intrinsically to do with analogy.” In brief, models are tools for prediction and explanation, whereas analogical arguments aim at establishing plausibility. An analogy is evaluated in terms of source-target similarity, while a model is evaluated on how successfully it “provides access to a phenomenon in that it interprets the available empirical data about the phenomenon.” If we broaden our perspective beyond analogical arguments , however, the connection between models and analogies is restored. Nersessian (2009), for instance, stresses the role of analog models in concept-formation and other cognitive processes.

To say that a hypothesis is plausible is to convey that it has epistemic support: we have some reason to believe it, even prior to testing. An assertion of plausibility within the context of an inquiry typically has pragmatic connotations as well: to say that a hypothesis is plausible suggests that we have some reason to investigate it further. For example, a mathematician working on a proof regards a conjecture as plausible if it “has some chances of success” (Polya 1954 (v. 2): 148). On both points, there is ambiguity as to whether an assertion of plausibility is categorical or a matter of degree. These observations point to the existence of two distinct conceptions of plausibility, probabilistic and modal , either of which may reflect the intended conclusion of an analogical argument.

On the probabilistic conception, plausibility is naturally identified with rational credence (rational subjective degree of belief) and is typically represented as a probability. A classic expression may be found in Mill’s analysis of the argument from analogy in A System of Logic :

There can be no doubt that every resemblance [not known to be irrelevant] affords some degree of probability, beyond what would otherwise exist, in favour of the conclusion. (Mill 1843/1930: 333)

In the terminology introduced in §2.2, Mill’s idea is that each element of the positive analogy boosts the probability of the conclusion. Contemporary ‘structure-mapping’ theories ( §3.4 ) employ a restricted version: each structural similarity between two domains contributes to the overall measure of similarity, and hence to the strength of the analogical argument.

On the alternative modal conception, ‘it is plausible that \(p\)’ is not a matter of degree. The meaning, roughly speaking, is that there are sufficient initial grounds for taking \(p\) seriously, i.e., for further investigation (subject to feasibility and interest). Informally: \(p\) passes an initial screening procedure. There is no assertion of degree. Instead, ‘It is plausible that’ may be regarded as an epistemic modal operator that aims to capture a notion, prima facie plausibility, that is somewhat stronger than ordinary epistemic possibility. The intent is to single out \(p\) from an undifferentiated mass of ideas that remain bare epistemic possibilities. To illustrate: in 1769, Priestley’s argument ( Example 9 ), if successful, would establish the prima facie plausibility of an inverse square law for electrostatic attraction. The set of epistemic possibilities—hypotheses about electrostatic attraction compatible with knowledge of the day—was much larger. Individual analogical arguments in mathematics (such as Example 7 ) are almost invariably directed towards prima facie plausibility.

The modal conception figures importantly in some discussions of analogical reasoning. The physicist N. R. Campbell (1957) writes:

But in order that a theory may be valuable it must … display an analogy. The propositions of the hypothesis must be analogous to some known laws…. (1957: 129)

Commenting on the role of analogy in Fourier’s theory of heat conduction, Campbell writes:

Some analogy is essential to it; for it is only this analogy which distinguishes the theory from the multitude of others… which might also be proposed to explain the same laws. (1957: 142)

The interesting notion here is that of a “valuable” theory. We may not agree with Campbell that the existence of analogy is “essential” for a novel theory to be “valuable.” But consider the weaker thesis that an acceptable analogy is sufficient to establish that a theory is “valuable”, or (to qualify still further) that an acceptable analogy provides defeasible grounds for taking the theory seriously. (Possible defeaters might include internal inconsistency, inconsistency with accepted theory, or the existence of a (clearly superior) rival analogical argument.) The point is that Campbell, following the lead of 19 th century philosopher-scientists such as Herschel and Whewell, thinks that analogies can establish this sort of prima facie plausibility. Snyder (2006) provides a detailed discussion of the latter two thinkers and their ideas about the role of analogies in science.

In general, analogical arguments may be directed at establishing either sort of plausibility for their conclusions; they can have a probabilistic use or a modal use. Examples 7 through 9 are best interpreted as supporting modal conclusions. In those arguments, an analogy is used to show that a conjecture is worth taking seriously. To insist on putting the conclusion in probabilistic terms distracts attention from the point of the argument. The conclusion might be modeled (by a Bayesian) as having a certain probability value because it is deemed prima facie plausible, but not vice versa. Example 2 , perhaps, might be regarded as directed primarily towards a probabilistic conclusion.

There should be connections between the two conceptions. Indeed, we might think that the same analogical argument can establish both prima facie plausibility and a degree of probability for a hypothesis. But it is difficult to translate between epistemic modal concepts and probabilities (Cohen 1980; Douven and Williamson 2006; Huber 2009; Spohn 2009, 2012). We cannot simply take the probabilistic notion as the primitive one. It seems wise to keep the two conceptions of plausibility separate.

Schema (4) is a template that represents all analogical arguments, good and bad. It is not an inference rule. Despite the confidence with which particular analogical arguments are advanced, nobody has ever formulated an acceptable rule, or set of rules, for valid analogical inferences. There is not even a plausible candidate. This situation is in marked contrast not only with deductive reasoning, but also with elementary forms of inductive reasoning, such as induction by enumeration.

Of course, it is difficult to show that no successful analogical inference rule will ever be proposed. But consider the following candidate, formulated using the concepts of schema (4) and taking us only a short step beyond that basic characterization.

Rule (5) is modeled on the straight rule for enumerative induction and inspired by Mill’s view of analogical inference, as described in §2.3. We use the generic phrase ‘degree of support’ in place of probability, since other factors besides the analogical argument may influence our probability assignment for \(Q^*\).

It is pretty clear that (5) is a non-starter. The main problem is that the rule justifies too much. The only substantive requirement introduced by (5) is that there be a nonempty positive analogy. Plainly, there are analogical arguments that satisfy this condition but establish no prima facie plausibility and no measure of support for their conclusions.

Here is a simple illustration. Achinstein (1964: 328) observes that there is a formal analogy between swans and line segments if we take the relation ‘has the same color as’ to correspond to ‘is congruent with’. Both relations are reflexive, symmetric, and transitive. Yet it would be absurd to find positive support from this analogy for the idea that we are likely to find congruent lines clustered in groups of two or more, just because swans of the same color are commonly found in groups. The positive analogy is antecedently known to be irrelevant to the hypothetical analogy. In such a case, the analogical inference should be utterly rejected. Yet rule (5) would wrongly assign non-zero degree of support.

To generalize the difficulty: not every similarity increases the probability of the conclusion and not every difference decreases it. Some similarities and differences are known to be (or accepted as being) utterly irrelevant and should have no influence whatsoever on our probability judgments. To be viable, rule (5) would need to be supplemented with considerations of relevance , which depend upon the subject matter, historical context and logical details particular to each analogical argument. To search for a simple rule of analogical inference thus appears futile.

Carnap and his followers (Carnap 1980; Kuipers 1988; Niiniluoto 1988; Maher 2000; Romeijn 2006) have formulated principles of analogy for inductive logic, using Carnapian \(\lambda \gamma\) rules. Generally, this body of work relates to “analogy by similarity”, rather than the type of analogical reasoning discussed here. Romeijn (2006) maintains that there is a relation between Carnap’s concept of analogy and analogical prediction. His approach is a hybrid of Carnap-style inductive rules and a Bayesian model. Such an approach would need to be generalized to handle the kinds of arguments described in §2.1 . It remains unclear that the Carnapian approach can provide a general rule for analogical inference.

Norton (2010, and 2018—see Other Internet Resources) has argued that the project of formalizing inductive reasoning in terms of one or more simple formal schemata is doomed. His criticisms seem especially apt when applied to analogical reasoning. He writes:

If analogical reasoning is required to conform only to a simple formal schema, the restriction is too permissive. Inferences are authorized that clearly should not pass muster… The natural response has been to develop more elaborate formal templates… The familiar difficulty is that these embellished schema never seem to be quite embellished enough; there always seems to be some part of the analysis that must be handled intuitively without guidance from strict formal rules. (2018: 1)

Norton takes the point one step further, in keeping with his “material theory” of inductive inference. He argues that there is no universal logical principle that “powers” analogical inference “by asserting that things that share some properties must share others.” Rather, each analogical inference is warranted by some local constellation of facts about the target system that he terms “the fact of analogy”. These local facts are to be determined and investigated on a case by case basis.

To embrace a purely formal approach to analogy and to abjure formalization entirely are two extremes in a spectrum of strategies. There are intermediate positions. Most recent analyses (both philosophical and computational) have been directed towards elucidating criteria and procedures, rather than formal rules, for reasoning by analogy. So long as these are not intended to provide a universal ‘logic’ of analogy, there is room for such criteria even if one accepts Norton’s basic point. The next section discusses some of these criteria and procedures.

3. Criteria for evaluating analogical arguments

Logicians and philosophers of science have identified ‘textbook-style’ general guidelines for evaluating analogical arguments (Mill 1843/1930; Keynes 1921; Robinson 1930; Stebbing 1933; Copi and Cohen 2005; Moore and Parker 1998; Woods, Irvine, and Walton 2004). Here are some of the most important ones:

These principles can be helpful, but are frequently too vague to provide much insight. How do we count similarities and differences in applying (G1) and (G2)? Why are the structural and causal analogies mentioned in (G5) and (G6) especially important, and which structural and causal features merit attention? More generally, in connection with the all-important (G7): how do we determine which similarities and differences are relevant to the conclusion? Furthermore, what are we to say about similarities and differences that have been omitted from an analogical argument but might still be relevant?

An additional problem is that the criteria can pull in different directions. To illustrate, consider Reid’s argument for life on other planets ( Example 2 ). Stebbing (1933) finds Reid’s argument “suggestive” and “not unplausible” because the conclusion is weak (G4), while Mill (1843/1930) appears to reject the argument on account of our vast ignorance of properties that might be relevant (G3).

There is a further problem that relates to the distinction just made (in §2.3 ) between two kinds of plausibility. Each of the above criteria apart from (G7) is expressed in terms of the strength of the argument, i.e., the degree of support for the conclusion. The criteria thus appear to presuppose the probabilistic interpretation of plausibility. The problem is that a great many analogical arguments aim to establish prima facie plausibility rather than any degree of probability. Most of the guidelines are not directly applicable to such arguments.

Aristotle sets the stage for all later theories of analogical reasoning. In his theoretical reflections on analogy and in his most judicious examples, we find a sober account that lays the foundation both for the commonsense guidelines noted above and for more sophisticated analyses.

Although Aristotle employs the term analogy ( analogia ) and discusses analogical predication , he never talks about analogical reasoning or analogical arguments per se . He does, however, identify two argument forms, the argument from example ( paradeigma ) and the argument from likeness ( homoiotes ), both closely related to what would we now recognize as an analogical argument.

The argument from example ( paradeigma ) is described in the Rhetoric and the Prior Analytics :

Enthymemes based upon example are those which proceed from one or more similar cases, arrive at a general proposition, and then argue deductively to a particular inference. ( Rhetoric 1402b15) Let \(A\) be evil, \(B\) making war against neighbours, \(C\) Athenians against Thebans, \(D\) Thebans against Phocians. If then we wish to prove that to fight with the Thebans is an evil, we must assume that to fight against neighbours is an evil. Conviction of this is obtained from similar cases, e.g., that the war against the Phocians was an evil to the Thebans. Since then to fight against neighbours is an evil, and to fight against the Thebans is to fight against neighbours, it is clear that to fight against the Thebans is an evil. ( Pr. An. 69a1)

Aristotle notes two differences between this argument form and induction (69a15ff.): it “does not draw its proof from all the particular cases” (i.e., it is not a “complete” induction), and it requires an additional (deductively valid) syllogism as the final step. The argument from example thus amounts to single-case induction followed by deductive inference. It has the following structure (using \(\supset\) for the conditional):

In the terminology of §2.2, \(P\) is the positive analogy and \(Q\) is the hypothetical analogy. In Aristotle’s example, \(S\) (the source) is war between Phocians and Thebans, \(T\) (the target) is war between Athenians and Thebans, \(P\) is war between neighbours, and \(Q\) is evil. The first inference (dashed arrow) is inductive; the second and third (solid arrows) are deductively valid.

The paradeigma has an interesting feature: it is amenable to an alternative analysis as a purely deductive argument form. Let us concentrate on Aristotle’s assertion, “we must assume that to fight against neighbours is an evil,” represented as \(\forall x(P(x) \supset Q(x))\). Instead of regarding this intermediate step as something reached by induction from a single case, we might instead regard it as a hidden presupposition. This transforms the paradeigma into a syllogistic argument with a missing or enthymematic premise, and our attention shifts to possible means for establishing that premise (with single-case induction as one such means). Construed in this way, Aristotle’s paradeigma argument foreshadows deductive analyses of analogical reasoning (see §4.1 ).

The argument from likeness ( homoiotes ) seems to be closer than the paradeigma to our contemporary understanding of analogical arguments. This argument form receives considerable attention in Topics I, 17 and 18 and again in VIII, 1. The most important passage is the following.

Try to secure admissions by means of likeness; for such admissions are plausible, and the universal involved is less patent; e.g. that as knowledge and ignorance of contraries is the same, so too perception of contraries is the same; or vice versa, that since the perception is the same, so is the knowledge also. This argument resembles induction, but is not the same thing; for in induction it is the universal whose admission is secured from the particulars, whereas in arguments from likeness, what is secured is not the universal under which all the like cases fall. ( Topics 156b10–17)

This passage occurs in a work that offers advice for framing dialectical arguments when confronting a somewhat skeptical interlocutor. In such situations, it is best not to make one’s argument depend upon securing agreement about any universal proposition. The argument from likeness is thus clearly distinct from the paradeigma , where the universal proposition plays an essential role as an intermediate step in the argument. The argument from likeness, though logically less straightforward than the paradeigma , is exactly the sort of analogical reasoning we want when we are unsure about underlying generalizations.

In Topics I 17, Aristotle states that any shared attribute contributes some degree of likeness. It is natural to ask when the degree of likeness between two things is sufficiently great to warrant inferring a further likeness. In other words, when does the argument from likeness succeed? Aristotle does not answer explicitly, but a clue is provided by the way he justifies particular arguments from likeness. As Lloyd (1966) has observed, Aristotle typically justifies such arguments by articulating a (sometimes vague) causal principle which governs the two phenomena being compared. For example, Aristotle explains the saltiness of the sea, by analogy with the saltiness of sweat, as a kind of residual earthy stuff exuded in natural processes such as heating. The common principle is this:

Everything that grows and is naturally generated always leaves a residue, like that of things burnt, consisting in this sort of earth. ( Mete 358a17)

From this method of justification, we might conjecture that Aristotle believes that the important similarities are those that enter into such general causal principles.

Summarizing, Aristotle’s theory provides us with four important and influential criteria for the evaluation of analogical arguments:

• The strength of an analogy depends upon the number of similarities.
• Similarity reduces to identical properties and relations.
• Good analogies derive from underlying common causes or general laws.
• A good analogical argument need not pre-suppose acquaintance with the underlying universal (generalization).

These four principles form the core of a common-sense model for evaluating analogical arguments (which is not to say that they are correct; indeed, the first three will shortly be called into question). The first, as we have seen, appears regularly in textbook discussions of analogy. The second is largely taken for granted, with important exceptions in computational models of analogy ( §3.4 ). Versions of the third are found in most sophisticated theories. The final point, which distinguishes the argument from likeness and the argument from example, is endorsed in many discussions of analogy (e.g., Quine and Ullian 1970).

A slight generalization of Aristotle’s first principle helps to prepare the way for discussion of later developments. As that principle suggests, Aristotle, in common with just about everyone else who has written about analogical reasoning, organizes his analysis of the argument form around overall similarity. In the terminology of section 2.2, horizontal relationships drive the reasoning: the greater the overall similarity of the two domains, the stronger the analogical argument . Hume makes the same point, though stated negatively, in his Dialogues Concerning Natural Religion :

Wherever you depart, in the least, from the similarity of the cases, you diminish proportionably the evidence; and may at last bring it to a very weak analogy, which is confessedly liable to error and uncertainty. (1779/1947: 144)

Most theories of analogy agree with Aristotle and Hume on this general point. Disagreement relates to the appropriate way of measuring overall similarity. Some theories assign greatest weight to material analogy , which refers to shared, and typically observable, features. Others give prominence to formal analogy , emphasizing high-level structural correspondence. The next two sub-sections discuss representative accounts that illustrate these two approaches.

Hesse (1966) offers a sharpened version of Aristotle’s theory, specifically focused on analogical arguments in the sciences. She formulates three requirements that an analogical argument must satisfy in order to be acceptable:

• Requirement of material analogy . The horizontal relations must include similarities between observable properties.
• Causal condition . The vertical relations must be causal relations “in some acceptable scientific sense” (1966: 87).
• No-essential-difference condition . The essential properties and causal relations of the source domain must not have been shown to be part of the negative analogy.

3.3.1 Requirement of material analogy

For Hesse, an acceptable analogical argument must include “observable similarities” between domains, which she refers to as material analogy . Material analogy is contrasted with formal analogy . Two domains are formally analogous if both are “interpretations of the same formal theory” (1966: 68). Nomic isomorphism (Hempel 1965) is a special case in which the physical laws governing two systems have identical mathematical form. Heat and fluid flow exhibit nomic isomorphism. A second example is the analogy between the flow of electric current in a wire and fluid in a pipe. Ohm’s law

states that voltage difference along a wire equals current times a constant resistance. This has the same mathematical form as Poiseuille’s law (for ideal fluids):

which states that the pressure difference along a pipe equals the volumetric flow rate times a constant. Both of these systems can be represented by a common equation. While formal analogy is linked to common mathematical structure, it should not be limited to nomic isomorphism (Bartha 2010: 209). The idea of formal analogy generalizes to cases where there is a common mathematical structure between models for two systems. Bartha offers an even more liberal definition (2010: 195): “Two features are formally similar if they occupy corresponding positions in formally analogous theories. For example, pitch in the theory of sound corresponds to color in the theory of light.”

By contrast, material analogy consists of what Hesse calls “observable” or “pre-theoretic” similarities. These are horizontal relationships of similarity between properties of objects in the source and the target. Similarities between echoes (sound) and reflection (light), for instance, were recognized long before we had any detailed theories about these phenomena. Hesse (1966, 1988) regards such similarities as metaphorical relationships between the two domains and labels them “pre-theoretic” because they draw on personal and cultural experience. We have both material and formal analogies between sound and light, and it is significant for Hesse that the former are independent of the latter.

There are good reasons not to accept Hesse’s requirement of material analogy, construed in this narrow way. First, it is apparent that formal analogies are the starting point in many important inferences. That is certainly the case in mathematics, a field in which material analogy, in Hesse’s sense, plays no role at all. Analogical arguments based on formal analogy have also been extremely influential in physics (Steiner 1989, 1998).

In Norton’s broad sense, however, ‘material analogy’ simply refers to similarities rooted in factual knowledge of the source and target domains. With reference to this broader meaning, Hesse proposes two additional material criteria.

3.3.2 Causal condition

Hesse requires that the hypothetical analogy, the feature transferred to the target domain, be causally related to the positive analogy. In her words, the essential requirement for a good argument from analogy is “a tendency to co-occurrence”, i.e., a causal relationship. She states the requirement as follows:

The vertical relations in the model [source] are causal relations in some acceptable scientific sense, where there are no compelling a priori reasons for denying that causal relations of the same kind may hold between terms of the explanandum [target]. (1966: 87)

The causal condition rules out analogical arguments where there is no causal knowledge of the source domain. It derives support from the observation that many analogies do appear to involve a transfer of causal knowledge.

The causal condition is on the right track, but is arguably too restrictive. For example, it rules out analogical arguments in mathematics. Even if we limit attention to the empirical sciences, persuasive analogical arguments may be founded upon strong statistical correlation in the absence of any known causal connection. Consider ( Example 11 ) Benjamin Franklin’s prediction, in 1749, that pointed metal rods would attract lightning, by analogy with the way they attracted the “electrical fluid” in the laboratory:

Electrical fluid agrees with lightning in these particulars: 1. Giving light. 2. Colour of the light. 3. Crooked direction. 4. Swift motion. 5. Being conducted by metals. 6. Crack or noise in exploding. 7. Subsisting in water or ice. 8. Rending bodies it passes through. 9. Destroying animals. 10. Melting metals. 11. Firing inflammable substances. 12. Sulphureous smell.—The electrical fluid is attracted by points.—We do not know whether this property is in lightning.—But since they agree in all the particulars wherein we can already compare them, is it not probable they agree likewise in this? Let the experiment be made. ( Benjamin Franklin’s Experiments , 334)

Franklin’s hypothesis was based on a long list of properties common to the target (lightning) and source (electrical fluid in the laboratory). There was no known causal connection between the twelve “particulars” and the thirteenth property, but there was a strong correlation. Analogical arguments may be plausible even where there are no known causal relations.

3.3.3 No-essential-difference condition

Hesse’s final requirement is that the “essential properties and causal relations of the [source] have not been shown to be part of the negative analogy” (1966: 91). Hesse does not provide a definition of “essential,” but suggests that a property or relation is essential if it is “causally closely related to the known positive analogy.” For instance, an analogy with fluid flow was extremely influential in developing the theory of heat conduction. Once it was discovered that heat was not conserved, however, the analogy became unacceptable (according to Hesse) because conservation was so central to the theory of fluid flow.

This requirement, though once again on the right track, seems too restrictive. It can lead to the rejection of a good analogical argument. Consider the analogy between a two-dimensional rectangle and a three-dimensional box ( Example 7 ). Broadening Hesse’s notion, it seems that there are many ‘essential’ differences between rectangles and boxes. This does not mean that we should reject every analogy between rectangles and boxes out of hand. The problem derives from the fact that Hesse’s condition is applied to the analogy relation independently of the use to which that relation is put. What counts as essential should vary with the analogical argument. Absent an inferential context, it is impossible to evaluate the importance or ‘essentiality’ of similarities and differences.

Despite these weaknesses, Hesse’s ‘material’ criteria constitute a significant advance in our understanding of analogical reasoning. The causal condition and the no-essential-difference condition incorporate local factors, as urged by Norton, into the assessment of analogical arguments. These conditions, singly or taken together, imply that an analogical argument can fail to generate any support for its conclusion, even when there is a non-empty positive analogy. Hesse offers no theory about the ‘degree’ of analogical support. That makes her account one of the few that is oriented towards the modal, rather than probabilistic, use of analogical arguments ( §2.3 ).

Many people take the concept of model-theoretic isomorphism to set the standard for thinking about similarity and its role in analogical reasoning. They propose formal criteria for evaluating analogies, based on overall structural or syntactical similarity. Let us refer to theories oriented around such criteria as structuralist .

A number of leading computational models of analogy are structuralist. They are implemented in computer programs that begin with (or sometimes build) representations of the source and target domains, and then construct possible analogy mappings. Analogical inferences emerge as a consequence of identifying the ‘best mapping.’ In terms of criteria for analogical reasoning, there are two main ideas. First, the goodness of an analogical argument is based on the goodness of the associated analogy mapping . Second, the goodness of the analogy mapping is given by a metric that indicates how closely it approximates isomorphism.

The most influential structuralist theory has been Gentner’s structure-mapping theory, implemented in a program called the structure-mapping engine (SME). In its original form (Gentner 1983), the theory assesses analogies on purely structural grounds. Gentner asserts:

Analogies are about relations, rather than simple features. No matter what kind of knowledge (causal models, plans, stories, etc.), it is the structural properties (i.e., the interrelationships between the facts) that determine the content of an analogy. (Falkenhainer, Forbus, and Gentner 1989/90: 3)

In order to clarify this thesis, Gentner introduces a distinction between properties , or monadic predicates, and relations , which have multiple arguments. She further distinguishes among different orders of relations and functions, defined inductively (in terms of the order of the relata or arguments). The best mapping is determined by systematicity : the extent to which it places higher-order relations, and items that are nested in higher-order relations, in correspondence. Gentner’s Systematicity Principle states:

A predicate that belongs to a mappable system of mutually interconnecting relationships is more likely to be imported into the target than is an isolated predicate. (1983: 163)

A systematic analogy (one that places high-order relations and their components in correspondence) is better than a less systematic analogy. Hence, an analogical inference has a degree of plausibility that increases monotonically with the degree of systematicity of the associated analogy mapping. Gentner’s fundamental criterion for evaluating candidate analogies (and analogical inferences) thus depends solely upon the syntax of the given representations and not at all upon their content.

Later versions of the structure-mapping theory incorporate refinements (Forbus, Ferguson, and Gentner 1994; Forbus 2001; Forbus et al. 2007; Forbus et al. 2008; Forbus et al 2017). For example, the earliest version of the theory is vulnerable to worries about hand-coded representations of source and target domains. Gentner and her colleagues have attempted to solve this problem in later work that generates LISP representations from natural language text (see Turney 2008 for a different approach).

The most important challenges for the structure-mapping approach relate to the Systematicity Principle itself. Does the value of an analogy derive entirely, or even chiefly, from systematicity? There appear to be two main difficulties with this view. First: it is not always appropriate to give priority to systematic, high-level relational matches. Material criteria, and notably what Gentner refers to as “superficial feature matches,” can be extremely important in some types of analogical reasoning, such as ethnographic analogies which are based, to a considerable degree, on surface resemblances between artifacts. Second and more significantly: systematicity seems to be at best a fallible marker for good analogies rather than the essence of good analogical reasoning.

Greater systematicity is neither necessary nor sufficient for a more plausible analogical inference. It is obvious that increased systematicity is not sufficient for increased plausibility. An implausible analogy can be represented in a form that exhibits a high degree of structural parallelism. High-order relations can come cheap, as we saw with Achinstein’s “swan” example ( §2.4 ).

More pointedly, increased systematicity is not necessary for greater plausibility. Indeed, in causal analogies, it may even weaken the inference. That is because systematicity takes no account of the type of causal relevance, positive or negative. (McKay 1993) notes that microbes have been found in frozen lakes in Antarctica; by analogy, simple life forms might exist on Mars. Freezing temperatures are preventive or counteracting causes; they are negatively relevant to the existence of life. The climate of Mars was probably more favorable to life 3.5 billion years ago than it is today, because temperatures were warmer. Yet the analogy between Antarctica and present-day Mars is more systematic than the analogy between Antarctica and ancient Mars. According to the Systematicity Principle , the analogy with Antarctica provides stronger support for life on Mars today than it does for life on ancient Mars.

The point of this example is that increased systematicity does not always increase plausibility, and reduced systematicity does not always decrease it (see Lee and Holyoak 2008). The more general point is that systematicity can be misleading, unless we take into account the nature of the relationships between various factors and the hypothetical analogy. Systematicity does not magically produce or explain the plausibility of an analogical argument. When we reason by analogy, we must determine which features of both domains are relevant and how they relate to the analogical conclusion. There is no short-cut via syntax.

Schlimm (2008) offers an entirely different critique of the structure-mapping theory from the perspective of analogical reasoning in mathematics—a domain where one might expect a formal approach such as structure mapping to perform well. Schlimm introduces a simple distinction: a domain is object-rich if the number of objects is greater than the number of relations (and properties), and relation-rich otherwise. Proponents of the structure-mapping theory typically focus on relation-rich examples (such as the analogy between the solar system and the atom). By contrast, analogies in mathematics typically involve domains with an enormous number of objects (like the real numbers), but relatively few relations and functions (addition, multiplication, less-than).

Schlimm provides an example of an analogical reasoning problem in group theory that involves a single relation in each domain. In this case, attaining maximal systematicity is trivial. The difficulty is that, compatible with maximal systematicity, there are different ways in which the objects might be placed in correspondence. The structure-mapping theory appears to yield the wrong inference. We might put the general point as follows: in object-rich domains, systematicity ceases to be a reliable guide to plausible analogical inference.

3.5.1 Connectionist models

During the past thirty-five years, cognitive scientists have conducted extensive research on analogy. Gentner’s SME is just one of many computational theories, implemented in programs that construct and use analogies. Three helpful anthologies that span this period are Helman 1988; Gentner, Holyoak, and Kokinov 2001; and Kokinov, Holyoak, and Gentner 2009.

One predominant objective of this research has been to model the cognitive processes involved in using analogies. Early models tended to be oriented towards “understanding the basic constraints that govern human analogical thinking” (Hummel and Holyoak 1997: 458). Recent connectionist models have been directed towards uncovering the psychological mechanisms that come into play when we use analogies: retrieval of a relevant source domain, analogical mapping across domains, and transfer of information and learning of new categories or schemas.

In some cases, such as the structure-mapping theory (§3.4), this research overlaps directly with the normative questions that are the focus of this entry; indeed, Gentner’s Systematicity Principle may be interpreted normatively. In other cases, we might view the projects as displacing those traditional normative questions with up-to-date, computational forms of naturalized epistemology . Two approaches are singled out here because both raise important challenges to the very idea of finding sharp answers to those questions, and both suggest that connectionist models offer a more fruitful approach to understanding analogical reasoning.

The first is the constraint-satisfaction model (also known as the multiconstraint theory ), developed by Holyoak and Thagard (1989, 1995). Like Gentner, Holyoak and Thagard regard the heart of analogical reasoning as analogy mapping , and they stress the importance of systematicity, which they refer to as a structural constraint. Unlike Gentner, they acknowledge two additional types of constraints. Pragmatic constraints take into account the goals and purposes of the agent, recognizing that “the purpose will guide selection” of relevant similarities. Semantic constraints represent estimates of the degree to which people regard source and target items as being alike, rather like Hesse’s “pre-theoretic” similarities.

The novelty of the multiconstraint theory is that these structural , semantic and pragmatic constraints are implemented not as rigid rules, but rather as ‘pressures’ supporting or inhibiting potential pairwise correspondences. The theory is implemented in a connectionist program called ACME (Analogical Constraint Mapping Engine), which assigns an initial activation value to each possible pairing between elements in the source and target domains (based on semantic and pragmatic constraints), and then runs through cycles that update the activation values based on overall coherence (structural constraints). The best global analogy mapping emerges under the pressure of these constraints. Subsequent connectionist models, such as Hummel and Holyoak’s LISA program (1997, 2003), have made significant advances and hold promise for offering a more complete theory of analogical reasoning.

The second example is Hofstadter and Mitchell’s Copycat program (Hofstadter 1995; Mitchell 1993). The program is “designed to discover insightful analogies, and to do so in a psychologically realistic way” (Hofstadter 1995: 205). Copycat operates in the domain of letter-strings. The program handles the following type of problem:

Suppose the letter-string abc were changed to abd ; how would you change the letter-string ijk in “the same way”?

Most people would answer ijl , since it is natural to think that abc was changed to abd by the “transformation rule”: replace the rightmost letter with its successor. Alternative answers are possible, but do not agree with most people’s sense of what counts as the natural analogy.

Hofstadter and Mitchell believe that analogy-making is in large part about the perception of novel patterns, and that such perception requires concepts with “fluid” boundaries. Genuine analogy-making involves “slippage” of concepts. The Copycat program combines a set of core concepts pertaining to letter-sequences ( successor , leftmost and so forth) with probabilistic “halos” that link distinct concepts dynamically. Orderly structures emerge out of random low-level processes and the program produces plausible solutions. Copycat thus shows that analogy-making can be modeled as a process akin to perception, even if the program employs mechanisms distinct from those in human perception.

The multiconstraint theory and Copycat share the idea that analogical cognition involves cognitive processes that operate below the level of abstract reasoning. Both computational models—to the extent that they are capable of performing successful analogical reasoning—challenge the idea that a successful model of analogical reasoning must take the form of a set of quasi-logical criteria. Efforts to develop a quasi-logical theory of analogical reasoning, it might be argued, have failed. In place of faulty inference schemes such as those described earlier ( §2.2 , §2.4 ), computational models substitute procedures that can be judged on their performance rather than on traditional philosophical standards.

In response to this argument, we should recognize the value of the connectionist models while acknowledging that we still need a theory that offers normative principles for evaluating analogical arguments. In the first place, even if the construction and recognition of analogies are largely a matter of perception, this does not eliminate the need for subsequent critical evaluation of analogical inferences. Second and more importantly, we need to look not just at the construction of analogy mappings but at the ways in which individual analogical arguments are debated in fields such as mathematics, physics, philosophy and the law. These high-level debates require reasoning that bears little resemblance to the computational processes of ACME or Copycat. (Ashley’s HYPO (Ashley 1990) is one example of a non-connectionist program that focuses on this aspect of analogical reasoning.) There is, accordingly, room for both computational and traditional philosophical models of analogical reasoning.

3.5.2 Articulation model

Most prominent theories of analogy, philosophical and computational, are based on overall similarity between source and target domains—defined in terms of some favoured subset of Hesse’s horizontal relations (see §2.2 ). Aristotle and Mill, whose approach is echoed in textbook discussions, suggest counting similarities. Hesse’s theory ( §3.3 ) favours “pre-theoretic” correspondences. The structure-mapping theory and its successors ( §3.4 ) look to systematicity, i.e., to correspondences involving complex, high-level networks of relations. In each of these approaches, the problem is twofold: overall similarity is not a reliable guide to plausibility, and it fails to explain the plausibility of any analogical argument.

Bartha’s articulation model (2010) proposes a different approach, beginning not with horizontal relations, but rather with a classification of analogical arguments on the basis of the vertical relations within each domain. The fundamental idea is that a good analogical argument must satisfy two conditions:

Prior Association . There must be a clear connection, in the source domain, between the known similarities (the positive analogy) and the further similarity that is projected to hold in the target domain (the hypothetical analogy). This relationship determines which features of the source are critical to the analogical inference.

Potential for Generalization . There must be reason to think that the same kind of connection could obtain in the target domain. More pointedly: there must be no critical disanalogy between the domains.

The first order of business is to make the prior association explicit. The standards of explicitness vary depending on the nature of this association (causal relation, mathematical proof, functional relationship, and so forth). The two general principles are fleshed out via a set of subordinate models that allow us to identify critical features and hence critical disanalogies.

To see how this works, consider Example 7 (Rectangles and boxes). In this analogical argument, the source domain is two-dimensional geometry: we know that of all rectangles with a fixed perimeter, the square has maximum area. The target domain is three-dimensional geometry: by analogy, we conjecture that of all boxes with a fixed surface area, the cube has maximum volume. This argument should be evaluated not by counting similarities, looking to pre-theoretic resemblances between rectangles and boxes, or constructing connectionist representations of the domains and computing a systematicity score for possible mappings. Instead, we should begin with a precise articulation of the prior association in the source domain, which amounts to a specific proof for the result about rectangles. We should then identify, relative to that proof, the critical features of the source domain: namely, the concepts and assumptions used in the proof. Finally, we should assess the potential for generalization: whether, in the three-dimensional setting, those critical features are known to lack analogues in the target domain. The articulation model is meant to reflect the conversations that can and do take place between an advocate and a critic of an analogical argument.

3.6.1 Norton’s material theory of analogy

As noted in §2.4 , Norton rejects analogical inference rules. But even if we agree with Norton on this point, we might still be interested in having an account that gives us guidelines for evaluating analogical arguments. How does Norton’s approach fare on this score?

According to Norton, each analogical argument is warranted by local facts that must be investigated and justified empirically. First, there is “the fact of the analogy”: in practice, a low-level uniformity that embraces both the source and target systems. Second, there are additional factual properties of the target system which, when taken together with the uniformity, warrant the analogical inference. Consider Galileo’s famous inference ( Example 12 ) that there are mountains on the moon (Galileo 1610). Through his newly invented telescope, Galileo observed points of light on the moon ahead of the advancing edge of sunlight. Noting that the same thing happens on earth when sunlight strikes the mountains, he concluded that there must be mountains on the moon and even provided a reasonable estimate of their height. In this example, Norton tells us, the fact of the analogy is that shadows and other optical phenomena are generated in the same way on the earth and on the moon; the additional fact about the target is the existence of points of light ahead of the advancing edge of sunlight on the moon.

What are the implications of Norton’s material theory when it comes to evaluating analogical arguments? The fact of the analogy is a local uniformity that powers the inference. Norton’s theory works well when such a uniformity is patent or naturally inferred. It doesn’t work well when the uniformity is itself the target (rather than the driver ) of the inference. That happens with explanatory analogies such as Example 5 (the Acoustical Analogy ), and mathematical analogies such as Example 7 ( Rectangles and Boxes ). Similarly, the theory doesn’t work well when the underlying uniformity is unclear, as in Example 2 ( Life on other Planets ), Example 4 ( Clay Pots ), and many other cases. In short, if Norton’s theory is accepted, then for most analogical arguments there are no useful evaluation criteria.

3.6.2 Field-specific criteria

For those who sympathize with Norton’s skepticism about universal inductive schemes and theories of analogical reasoning, yet recognize that his approach may be too local, an appealing strategy is to move up one level. We can aim for field-specific “working logics” (Toulmin 1958; Wylie and Chapman 2016; Reiss 2015). This approach has been adopted by philosophers of archaeology, evolutionary biology and other historical sciences (Wylie and Chapman 2016; Currie 2013; Currie 2016; Currie 2018). In place of schemas, we find ‘toolkits’, i.e., lists of criteria for evaluating analogical reasoning.

For example, Currie (2016) explores in detail the use of ethnographic analogy ( Example 13 ) between shamanistic motifs used by the contemporary San people and similar motifs in ancient rock art, found both among ancestors of the San (direct historical analogy) and in European rock art (indirect historical analogy). Analogical arguments support the hypothesis that in each of these cultures, rock art symbolizes hallucinogenic experiences. Currie examines criteria that focus on assumptions about stability of cultural traits and environment-culture relationships. Currie (2016, 2018) and Wylie (Wylie and Chapman 2016) also stress the importance of robustness reasoning that combines analogical arguments of moderate strength with other forms of evidence to yield strong conclusions.

Practice-based approaches can thus yield specific guidelines unlikely to be matched by any general theory of analogical reasoning. One caveat is worth mentioning. Field-specific criteria for ethnographic analogy are elicited against a background of decades of methodological controversy (Wylie and Chapman 2016). Critics and defenders of ethnographic analogy have appealed to general models of scientific method (e.g., hypothetico-deductive method or Bayesian confirmation). To advance the methodological debate, practice-based approaches must either make connections to these general models or explain why the lack of any such connection is unproblematic.

3.6.3 Formal analogies in physics

Close attention to analogical arguments in practice can also provide valuable challenges to general ideas about analogical inference. In an interesting discussion, Steiner (1989, 1998) suggests that many of the analogies that played a major role in early twentieth-century physics count as “Pythagorean.” The term is meant to connote mathematical mysticism: a “Pythagorean” analogy is a purely formal analogy, one founded on mathematical similarities that have no known physical basis at the time it is proposed. One example is Schrödinger’s use of analogy ( Example 14 ) to “guess” the form of the relativistic wave equation. In Steiner’s view, Schrödinger’s reasoning relies upon manipulations and substitutions based on purely mathematical analogies. Steiner argues that the success, and even the plausibility, of such analogies “evokes, or should evoke, puzzlement” (1989: 454). Both Hesse (1966) and Bartha (2010) reject the idea that a purely formal analogy, with no physical significance, can support a plausible analogical inference in physics. Thus, Steiner’s arguments provide a serious challenge.

Bartha (2010) suggests a response: we can decompose Steiner’s examples into two or more steps, and then establish that at least one step does, in fact, have a physical basis. Fraser (forthcoming), however, offers a counterexample that supports Steiner’s position. Complex analogies between classical statistical mechanics (CSM) and quantum field theory (QFT) have played a crucial role in the development and application of renormalization group (RG) methods in both theories ( Example 15 ). Fraser notes substantial physical disanalogies between CSM and QFT, and concludes that the reasoning is based entirely on formal analogies.

4. Philosophical foundations for analogical reasoning

What philosophical basis can be provided for reasoning by analogy? What justification can be given for the claim that analogical arguments deliver plausible conclusions? There have been several ideas for answering this question. One natural strategy assimilates analogical reasoning to some other well-understood argument pattern, a form of deductive or inductive reasoning ( §4.1 , §4.2 ). A few philosophers have explored the possibility of a priori justification ( §4.3 ). A pragmatic justification may be available for practical applications of analogy, notably in legal reasoning ( §4.4 ).

Any attempt to provide a general justification for analogical reasoning faces a basic dilemma. The demands of generality require a high-level formulation of the problem and hence an abstract characterization of analogical arguments, such as schema (4). On the other hand, as noted previously, many analogical arguments that conform to schema (4) are bad arguments. So a general justification of analogical reasoning cannot provide support for all arguments that conform to (4), on pain of proving too much. Instead, it must first specify a subset of putatively ‘good’ analogical arguments, and link the general justification to this specified subset. The problem of justification is linked to the problem of characterizing good analogical arguments . This difficulty afflicts some of the strategies described in this section.

Analogical reasoning may be cast in a deductive mold. If successful, this strategy neatly solves the problem of justification. A valid deductive argument is as good as it gets.

An early version of the deductivist approach is exemplified by Aristotle’s treatment of the argument from example ( §3.2 ), the paradeigma . On this analysis, an analogical argument between source domain \(S\) and target \(T\) begins with the assumption of positive analogy \(P(S)\) and \(P(T)\), as well as the additional information \(Q(S)\). It proceeds via the generalization \(\forall x(P(x) \supset Q(x))\) to the conclusion: \(Q(T)\). Provided we can treat that intermediate generalization as an independent premise, we have a deductively valid argument. Notice, though, that the existence of the generalization renders the analogy irrelevant. We can derive \(Q(T)\) from the generalization and \(P(T)\), without any knowledge of the source domain. The literature on analogy in argumentation theory ( §2.2 ) offers further perspectives on this type of analysis, and on the question of whether analogical arguments are properly characterized as deductive.

Some recent analyses follow Aristotle in treating analogical arguments as reliant upon extra (sometimes tacit) premises, typically drawn from background knowledge, that convert the inference into a deductively valid argument––but without making the source domain irrelevant. Davies and Russell introduce a version that relies upon what they call determination rules (Russell 1986; Davies and Russell 1987; Davies 1988). Suppose that \(Q\) and \(P_1 , \ldots ,P_m\) are variables, and we have background knowledge that the value of \(Q\) is determined by the values of \(P_1 , \ldots ,P_m\). In the simplest case, where \(m = 1\) and both \(P\) and \(Q\) are binary Boolean variables, this reduces to

i.e., whether or not \(P\) holds determines whether or not \(Q\) holds. More generally, the form of a determination rule is

i.e., \(Q\) is a function of \(P_1,\ldots\), \(P_m\). If we assume such a rule as part of our background knowledge, then an analogical argument with conclusion \(Q(T)\) is deductively valid. More precisely, and allowing for the case where \(Q\) is not a binary variable: if we have such a rule, and also premises stating that the source \(S\) agrees with the target \(T\) on all of the values \(P_i\), then we may validly infer that \(Q(T) = Q(S)\).

The “determination rule” analysis provides a clear and simple justification for analogical reasoning. Note that, in contrast to the Aristotelian analysis via the generalization \(\forall x(P(x) \supset Q(x))\), a determination rule does not trivialize the analogical argument. Only by combining the rule with information about the source domain can we derive the value of \(Q(T)\). To illustrate by adapting one of the examples given by Russell and Davies ( Example 16 ), let’s suppose that the value \((Q)\) of a used car (relative to a particular buyer) is determined by its year, make, mileage, condition, color and accident history (the variables \(P_i)\). It doesn’t matter if one or more of these factors are redundant or irrelevant. Provided two cars are indistinguishable on each of these points, they will have the same value. Knowledge of the source domain is necessary; we can’t derive the value of the second car from the determination rule alone. Weitzenfeld (1984) proposes a variant of this approach, advancing the slightly more general thesis that analogical arguments are deductive arguments with a missing (enthymematic) premise that amounts to a determination rule.

Do determination rules give us a solution to the problem of providing a justification for analogical arguments? In general: no. Analogies are commonly applied to problems such as Example 8 ( morphine and meperidine ), where we are not even aware of all relevant factors, let alone in possession of a determination rule. Medical researchers conduct drug tests on animals without knowing all attributes that might be relevant to the effects of the drug. Indeed, one of the main objectives of such testing is to guard against reactions unanticipated by theory. On the “determination rule” analysis, we must either limit the scope of such arguments to cases where we have a well-supported determination rule, or focus attention on formulating and justifying an appropriate determination rule. For cases such as animal testing, neither option seems realistic.

Recasting analogy as a deductive argument may help to bring out background assumptions, but it makes little headway with the problem of justification. That problem re-appears as the need to state and establish the plausibility of a determination rule, and that is at least as difficult as justifying the original analogical argument.

Some philosophers have attempted to portray, and justify, analogical reasoning in terms of some well-understood inductive argument pattern. There have been three moderately popular versions of this strategy. The first treats analogical reasoning as generalization from a single case. The second treats it as a kind of sampling argument. The third recognizes the argument from analogy as a distinctive form, but treats past successes as evidence for future success.

4.2.1 Single-case induction

Let’s reconsider Aristotle’s argument from example or paradeigma ( §3.2 ), but this time regard the generalization as justified via induction from a single case (the source domain). Can such a simple analysis of analogical arguments succeed? In general: no.

A single instance can sometimes lead to a justified generalization. Cartwright (1992) argues that we can sometimes generalize from a single careful experiment, “where we have sufficient control of the materials and our knowledge of the requisite background assumptions is secure” (51). Cartwright thinks that we can do this, for example, in experiments with compounds that have stable “Aristotelian natures.” In a similar spirit, Quine (1969) maintains that we can have instantial confirmation when dealing with natural kinds.

Even if we accept that there are such cases, the objection to understanding all analogical arguments as single-case induction is obvious: the view is simply too restrictive. Most analogical arguments will not meet the requisite conditions. We may not know that we are dealing with a natural kind or Aristotelian nature when we make the analogical argument. We may not know which properties are essential. An insistence on the ‘single-case induction’ analysis of analogical reasoning is likely to lead to skepticism (Agassi 1964, 1988).

Interpreting the argument from analogy as single-case induction is also counter-productive in another way. The simplistic analysis does nothing to advance the search for criteria that help us to distinguish between relevant and irrelevant similarities, and hence between good and bad analogical arguments.

4.2.2 Sampling arguments

On the sampling conception of analogical arguments, acknowledged similarities between two domains are treated as statistically relevant evidence for further similarities. The simplest version of the sampling argument is due to Mill (1843/1930). An argument from analogy, he writes, is “a competition between the known points of agreement and the known points of difference.” Agreement of \(A\) and \(B\) in 9 out of 10 properties implies a probability of 0.9 that \(B\) will possess any other property of \(A\): “we can reasonably expect resemblance in the same proportion” (367). His only restriction has to do with sample size: we must be relatively knowledgeable about both \(A\) and \(B\). Mill saw no difficulty in using analogical reasoning to infer characteristics of newly discovered species of plants or animals, given our extensive knowledge of botany and zoology. But if the extent of unascertained properties of \(A\) and \(B\) is large, similarity in a small sample would not be a reliable guide; hence, Mill’s dismissal of Reid’s argument about life on other planets ( Example 2 ).

The sampling argument is presented in more explicit mathematical form by Harrod (1956). The key idea is that the known properties of \(S\) (the source domain) may be considered a random sample of all \(S\)’s properties—random, that is, with respect to the attribute of also belonging to \(T\) (the target domain). If the majority of known properties that belong to \(S\) also belong to \(T\), then we should expect most other properties of \(S\) to belong to \(T\), for it is unlikely that we would have come to know just the common properties. In effect, Harrod proposes a binomial distribution, modeling ‘random selection’ of properties on random selection of balls from an urn.

There are grave difficulties with Harrod’s and Mill’s analyses. One obvious difficulty is the counting problem : the ‘population’ of properties is poorly defined. How are we to count similarities and differences? The ratio of shared to total known properties varies dramatically according to how we do this. A second serious difficulty is the problem of bias : we cannot justify the assumption that the sample of known features is random. In the case of the urn, the selection process is arranged so that the result of each choice is not influenced by the agent’s intentions or purposes, or by prior choices. By contrast, the presentation of an analogical argument is always partisan. Bias enters into the initial representation of similarities and differences: an advocate of the argument will highlight similarities, while a critic will play up differences. The paradigm of repeated selection from an urn seems totally inappropriate. Additional variations of the sampling approach have been developed (e.g., Russell 1988), but ultimately these versions also fail to solve either the counting problem or the problem of bias.

4.2.3 Argument from past success

Section 3.6 discussed Steiner’s view that appeal to ‘Pythagorean’ analogies in physics “evokes, or should evoke, puzzlement” (1989: 454). Liston (2000) offers a possible response: physicists are entitled to use Pythagorean analogies on the basis of induction from their past success:

[The scientist] can admit that no one knows how [Pythagorean] reasoning works and argue that the very fact that similar strategies have worked well in the past is already reason enough to continue pursuing them hoping for success in the present instance. (200)

Setting aside familiar worries about arguments from success, the real problem here is to determine what counts as a similar strategy. In essence, that amounts to isolating the features of successful Pythagorean analogies. As we have seen (§2.4), nobody has yet provided a satisfactory scheme that characterizes successful analogical arguments, let alone successful Pythagorean analogical arguments.

An a priori approach traces the validity of a pattern of analogical reasoning, or of a particular analogical argument, to some broad and fundamental principle. Three such approaches will be outlined here.

The first is due to Keynes (1921). Keynes appeals to his famous Principle of the Limitation of Independent Variety, which he articulates as follows:

Armed with this Principle and some additional assumptions, Keynes is able to show that in cases where there is no negative analogy , knowledge of the positive analogy increases the (logical) probability of the conclusion. If there is a non-trivial negative analogy, however, then the probability of the conclusion remains unchanged, as was pointed out by Hesse (1966). Those familiar with Carnap’s theory of logical probability will recognize that in setting up his framework, Keynes settled on a measure that permits no learning from experience.

Hesse offers a refinement of Keynes’s strategy, once again along Carnapian lines. In her (1974), she proposes what she calls the Clustering Postulate : the assumption that our epistemic probability function has a built-in bias towards generalization. The objections to such postulates of uniformity are well-known (see Salmon 1967), but even if we waive them, her argument fails. The main objection here—which also applies to Keynes—is that a purely syntactic axiom such as the Clustering Postulate fails to discriminate between analogical arguments that are good and those that are clearly without value (according to Hesse’s own material criteria, for example).

A different a priori strategy, proposed by Bartha (2010), limits the scope of justification to analogical arguments that satisfy tentative criteria for ‘good’ analogical reasoning. The criteria are those specified by the articulation model ( §3.5 ). In simplified form, they require the existence of non-trivial positive analogy and no known critical disanalogy. The scope of Bartha’s argument is also limited to analogical arguments directed at establishing prima facie plausibility, rather than degree of probability.

Bartha’s argument rests on a principle of symmetry reasoning articulated by van Fraassen (1989: 236): “problems which are essentially the same must receive essentially the same solution.” A modal extension of this principle runs roughly as follows: if problems might be essentially the same, then they might have essentially the same solution. There are two modalities here. Bartha argues that satisfaction of the criteria of the articulation model is sufficient to establish the modality in the antecedent, i.e., that the source and target domains ‘might be essentially the same’ in relevant respects. He further suggests that prima facie plausibility provides a reasonable reading of the modality in the consequent, i.e., that the problems in the two domains ‘might have essentially the same solution.’ To call a hypothesis prima facie plausible is to elevate it to the point where it merits investigation, since it might be correct.

The argument is vulnerable to two sorts of concerns. First, there are questions about the interpretation of the symmetry principle. Second, there is a residual worry that this justification, like all the others, proves too much. The articulation model may be too vague or too permissive.

Arguably, the most promising available defense of analogical reasoning may be found in its application to case law (see Precedent and Analogy in Legal Reasoning ). Judicial decisions are based on the verdicts and reasoning that have governed relevantly similar cases, according to the doctrine of stare decisis (Levi 1949; Llewellyn 1960; Cross and Harris 1991; Sunstein 1993). Individual decisions by a court are binding on that court and lower courts; judges are obligated to decide future cases ‘in the same way.’ That is, the reasoning applied in an individual decision, referred to as the ratio decidendi , must be applied to similar future cases (see Example 10 ). In practice, of course, the situation is extremely complex. No two cases are identical. The ratio must be understood in the context of the facts of the original case, and there is considerable room for debate about its generality and its applicability to future cases. If a consensus emerges that a past case was wrongly decided, later judgments will distinguish it from new cases, effectively restricting the scope of the ratio to the original case.

The practice of following precedent can be justified by two main practical considerations. First, and above all, the practice is conservative : it provides a relatively stable basis for replicable decisions. People need to be able to predict the actions of the courts and formulate plans accordingly. Stare decisis serves as a check against arbitrary judicial decisions. Second, the practice is still reasonably progressive : it allows for the gradual evolution of the law. Careful judges distinguish bad decisions; new values and a new consensus can emerge in a series of decisions over time.

In theory, then, stare decisis strikes a healthy balance between conservative and progressive social values. This justification is pragmatic. It pre-supposes a common set of social values, and links the use of analogical reasoning to optimal promotion of those values. Notice also that justification occurs at the level of the practice in general; individual analogical arguments sometimes go astray. A full examination of the nature and foundations for stare decisis is beyond the scope of this entry, but it is worth asking the question: might it be possible to generalize the justification for stare decisis ? Is a parallel pragmatic justification available for analogical arguments in general?

Bartha (2010) offers a preliminary attempt to provide such a justification by shifting from social values to epistemic values. The general idea is that reasoning by analogy is especially well suited to the attainment of a common set of epistemic goals or values. In simple terms, analogical reasoning—when it conforms to certain criteria—achieves an excellent (perhaps optimal) balance between the competing demands of stability and innovation. It supports both conservative epistemic values, such as simplicity and coherence with existing belief, and progressive epistemic values, such as fruitfulness and theoretical unification (McMullin (1993) provides a classic list).

5. Beyond analogical arguments

As emphasized earlier, analogical reasoning takes in a great deal more than analogical arguments. In this section, we examine two broad contexts in which analogical reasoning is important.

The first, still closely linked to analogical arguments, is the confirmation of scientific hypotheses. Confirmation is the process by which a scientific hypothesis receives inductive support on the basis of evidence (see evidence , confirmation , and Bayes’ Theorem ). Confirmation may also signify the logical relationship of inductive support that obtains between a hypothesis \(H\) and a proposition \(E\) that expresses the relevant evidence. Can analogical arguments play a role, either in the process or in the logical relationship? Arguably yes (to both), but this role has to be delineated carefully, and several obstacles remain in the way of a clear account.

The second context is conceptual and theoretical development in cutting-edge scientific research. Analogies are used to suggest possible extensions of theoretical concepts and ideas. The reasoning is linked to considerations of plausibility, but there is no straightforward analysis in terms of analogical arguments.

How is analogical reasoning related to the confirmation of scientific hypotheses? The examples and philosophical discussion from earlier sections suggest that a good analogical argument can indeed provide support for a hypothesis. But there are good reasons to doubt the claim that analogies provide actual confirmation.

In the first place, there is a logical difficulty. To appreciate this, let us concentrate on confirmation as a relationship between propositions. Christensen (1999: 441) offers a helpful general characterization:

Some propositions seem to help make it rational to believe other propositions. When our current confidence in \(E\) helps make rational our current confidence in \(H\), we say that \(E\) confirms \(H\).

In the Bayesian model, ‘confidence’ is represented in terms of subjective probability. A Bayesian agent starts with an assignment of subjective probabilities to a class of propositions. Confirmation is understood as a three-place relation:

\(E\) represents a proposition about accepted evidence, \(H\) stands for a hypothesis, \(K\) for background knowledge and \(Pr\) for the agent’s subjective probability function. To confirm \(H\) is to raise its conditional probability, relative to \(K\). The shift from prior probability \(Pr(H \mid K)\) to posterior probability \(Pr(H \mid E \cdot K)\) is referred to as conditionalization on \(E\). The relation between these two probabilities is typically given by Bayes’ Theorem (setting aside more complex forms of conditionalization):

For Bayesians, here is the logical difficulty: it seems that an analogical argument cannot provide confirmation. In the first place, it is not clear that we can encapsulate the information contained in an analogical argument in a single proposition, \(E\). Second, even if we can formulate a proposition \(E\) that expresses that information, it is typically not appropriate to treat it as evidence because the information contained in \(E\) is already part of the background, \(K\). This means that \(E \cdot K\) is equivalent to \(K\), and hence \(Pr(H \mid E \cdot K) = Pr(H \mid K)\). According to the Bayesian definition, we don’t have confirmation. (This is a version of the problem of old evidence; see confirmation .) Third, and perhaps most important, analogical arguments are often applied to novel hypotheses \(H\) for which the prior probability \(Pr(H \mid K)\) is not even defined. Again, the definition of confirmation in terms of Bayesian conditionalization seems inapplicable.

If analogies don’t provide inductive support via ordinary conditionalization, is there an alternative? Here we face a second difficulty, once again most easily stated within a Bayesian framework. Van Fraassen (1989) has a well-known objection to any belief-updating rule other than conditionalization. This objection applies to any rule that allows us to boost credences when there is no new evidence. The criticism, made vivid by the tale of Bayesian Peter, is that these ‘ampliative’ rules are vulnerable to a Dutch Book . Adopting any such rule would lead us to acknowledge as fair a system of bets that foreseeably leads to certain loss. Any rule of this type for analogical reasoning appears to be vulnerable to van Fraassen’s objection.

There appear to be at least three routes to avoiding these difficulties and finding a role for analogical arguments within Bayesian epistemology. First, there is what we might call minimal Bayesianism . Within the Bayesian framework, some writers (Jeffreys 1973; Salmon 1967, 1990; Shimony 1970) have argued that a ‘seriously proposed’ hypothesis must have a sufficiently high prior probability to allow it to become preferred as the result of observation. Salmon has suggested that analogical reasoning is one of the most important means of showing that a hypothesis is ‘serious’ in this sense. If analogical reasoning is directed primarily towards prior probability assignments, it can provide inductive support while remaining formally distinct from confirmation, avoiding the logical difficulties noted above. This approach is minimally Bayesian because it provides nothing more than an entry point into the Bayesian apparatus, and it only applies to novel hypotheses. An orthodox Bayesian, such as de Finetti (de Finetti and Savage 1972, de Finetti 1974), might have no problem in allowing that analogies play this role.

The second approach is liberal Bayesianism : we can change our prior probabilities in a non-rule-based fashion . Something along these lines is needed if analogical arguments are supposed to shift opinion about an already existing hypothesis without any new evidence. This is common in fields such as archaeology, as part of a strategy that Wylie refers to as “mobilizing old data as new evidence” (Wylie and Chapman 2016: 95). As Hawthorne (2012) notes, some Bayesians simply accept that both initial assignments and ongoing revision of prior probabilities (based on plausibility arguments) can be rational, but

the logic of Bayesian induction (as described here) has nothing to say about what values the prior plausibility assessments for hypotheses should have; and it places no restrictions on how they might change.

In other words, by not stating any rules for this type of probability revision, we avoid the difficulties noted by van Fraassen. This approach admits analogical reasoning into the Bayesian tent, but acknowledges a dark corner of the tent in which rationality operates without any clear rules.

Recently, a third approach has attracted interest: analogue confirmation or confirmation via analogue simulation . As described in (Dardashti et al. 2017), the idea is as follows:

Our key idea is that, in certain circumstances, predictions concerning inaccessible phenomena can be confirmed via an analogue simulation in a different system. (57)

Dardashti and his co-authors concentrate on a particular example ( Example 17 ): ‘dumb holes’ and other analogues to gravitational black holes (Unruh 1981; Unruh 2008). Unlike real black holes, some of these analogues can be (and indeed have been) implemented and studied in the lab. Given the exact formal analogy between our models for these systems and our models of black holes, and certain important additional assumptions, Dardashti et al. make the controversial claim that observations made about the analogues provide evidence about actual black holes. For instance, the observation of phenomena analogous to Hawking radiation in the analogue systems would provide confirmation for the existence of Hawking radiation in black holes. In a second paper (Dardashti et al. 2018, Other Internet Resources), the case for confirmation is developed within a Bayesian framework.

The appeal of a clearly articulated mechanism for analogue confirmation is obvious. It would provide a tool for exploring confirmation of inaccessible phenomena not just in cosmology, but also in historical sciences such as archaeology and evolutionary biology, and in areas of medical science where ethical constraints rule out experiments on human subjects. Furthermore, as noted by Dardashti et al., analogue confirmation relies on new evidence obtained from the analogue system, and is therefore not vulnerable to the logical difficulties noted above.

Although the concept of analogue confirmation is not entirely new (think of animal testing, as in Example 8 ), the claims of (Dardashti et al. 2017, 2018 [Other Internet Resources]) require evaluation. One immediate difficulty for the black hole example: if we think in terms of ordinary analogical arguments, there is no positive analogy because, to put it simply, we have no basis of known similarities between a ‘dumb hole’ and a black hole. As Crowther et al. (2018, Other Internet Resources) argue, “it is not known if the particular modelling framework used in the derivation of Hawking radiation actually describes black holes in the first place. ” This may not concern Dardashti et al., since they claim that analogue confirmation is distinct from ordinary analogical arguments. It may turn out that analogue confirmation is different for cases such as animal testing, where we have a basis of known similarities, and for cases where our only access to the target domain is via a theoretical model.

In §3.6 , we saw that practice-based studies of analogy provide insight into the criteria for evaluating analogical arguments. Such studies also point to dynamical or programmatic roles for analogies, which appear to require evaluative frameworks that go beyond those developed for analogical arguments.

Knuttila and Loettgers (2014) examine the role of analogical reasoning in synthetic biology, an interdisciplinary field that draws on physics, chemistry, biology, engineering and computational science. The main role for analogies in this field is not the construction of individual analogical arguments but rather the development of concepts such as “noise” and “feedback loops”. Such concepts undergo constant refinement, guided by both positive and negative analogies to their analogues in engineered and physical systems. Analogical reasoning here is “transient, heterogeneous, and programmatic” (87). Negative analogies, seen as problematic obstacles for individual analogical arguments, take on a prominent and constructive role when the focus is theoretical construction and concept refinement.

Similar observations apply to analogical reasoning in its application to another cutting-edge field: emergent gravity. In this area of physics, distinct theoretical approaches portray gravity as emerging from different microstructures (Linneman and Visser 2018). “Novel and robust” features not present at the micro-level emerge in the gravitational theory. Analogies with other emergent phenomena, such as hydrodynamics and thermodynamics, are exploited to shape these proposals. As with synthetic biology, analogical reasoning is not directed primarily towards the formulation and assessment of individual arguments. Rather, its role is to develop different theoretical models of gravity.

These studies explore fluid and creative applications of analogy to shape concepts on the front lines of scientific research. An adequate analysis would certainly take us beyond the analysis of individual analogical arguments, which have been the focus of our attention. Knuttila and Loettgers (2014) are led to reject the idea that the individual analogical argument is the “primary unit” in analogical reasoning, but this is a debatable conclusion. Linneman and Visser (2018), for instance, explicitly affirm the importance of assessing the case for different gravitational models through “exemplary analogical arguments”:

We have taken up the challenge of making explicit arguments in favour of an emergent gravity paradigm… That arguments can only be plausibility arguments at the heuristic level does not mean that they are immune to scrutiny and critical assessment tout court. The philosopher of physics’ job in the process of discovery of quantum gravity… should amount to providing exactly this kind of assessments. (Linneman and Visser 2018: 12)

Accordingly, Linneman and Visser formulate explicit analogical arguments for each model of emergent gravity, and assess them using familiar criteria for evaluating individual analogical arguments. Arguably, even the most ambitious heuristic objectives still depend upon considerations of plausibility that benefit by being expressed, and examined, in terms of analogical arguments.

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How to cite this entry . Preview the PDF version of this entry at the Friends of the SEP Society . Look up topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO). Enhanced bibliography for this entry at PhilPapers , with links to its database.

Other Internet Resources

• Crowther, K., N. Linnemann, and C. Wüthrich, 2018, “ What we cannot learn from analogue experiments ,” online at arXiv.org.
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• UCLA Reasoning Lab (UCLA)
• Dedre Gentner’s publications (Northwestern University)
• The Center for Research on Concepts and Cognition (Indiana University)

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Part Six: Evaluating Inductive Logic

Chapter Fifteen: Arguments from Analogy

Arguments that make their point by means of similarities are impostors, and, unless you are on your guard against them, will quite readily deceive you. —Plato

Analogies decide nothing, that is true, but they can make one feel more at home. —Sigmund Freud, New Introductory Lectures on Psychoanalysis

Correct Form for Arguments from Analogy

• The Total Evidence Condition (1): Relevant Similarities
• The Total Evidence Condition (2): Irrelevant Dissimilarities
• The Special Character of Arguments from Analogy

Arguments from analogy declare that because two items are the same in one respect they are the same in another. As Freud notes, they can make you feel at home—and for that reason they can be especially persuasive.

During World War I, the Socialist Party distributed leaflets to recent draftees, urging them to oppose the draft. The draft, they contended, violated the constitutional amendment against involuntary servitude. Oliver Wendell Holmes, chief justice of the Supreme Court, argued that they did not have the right to circulate the leaflets during wartime. The right to free speech, he asserted, “would not protect a man in falsely shouting fire in a theater and causing a panic.” Since in both cases “the words used . . . create a clear and present danger,” he concluded, the right to free speech did not protect the Socialists in expressing ideas that might harm the war effort. The argument begins with something familiar—of course we don’t have the right to falsely shout fire in a theater—and invites us to conclude the same about something less familiar—under certain circumstances we don’t even have the right to explain to others our interpretation of the U.S. Constitution. (Holmes is often quoted as calling it “a crowded theater.” He didn’t, though it is probably what he had in mind.)

Arguments from analogy are almost always enticing because, by their very nature, they use two of the quick-and-dirty shortcuts in reasoning described in Chapter 1. By beginning with the familiar, they exploit our dependence on the vividness shortcut; and by presenting similarities between the familiar and the unfamiliar, they take advantage of our dependence on the similarity shortcut. They are custom-made for the way our minds naturally operate. This makes us especially susceptible to them and heightens the importance of being able to evaluate them effectively.

15.1 Correct Form for Arguments from Analogy

Analogies are often used merely for rhetorical effect. Acel Moore of the Philadelphia Inquirer, for example, writes: “Writing editorials is a lot like wearing a navy blue suit and standing in a rainstorm on a cold day and wetting your pants; it may give you a warm feeling for a minute, but no one else is going to notice.” Moore doesn’t attempt to establish any conclusion based on the similarity—he simply makes note of it. Don’t jump to the conclusion that an analogy introduces an argument unless there really is—at least implicitly—a conclusion.

When there is an argument from analogy , as in the preceding free speech argument, it can typically be clarified according to the following form:

• A is F and G.

A and B, as always, are used here as name letters. They name the two analogs [1] —that is, the two things (or classes of things) that are said to be analogous. A, the basic analog , is the one that we are presumed to be more familiar with; in the free speech argument it is falsely shouting fire in a theater. B, the inferred analog , is the thing in question, the one that the argument draws a conclusion about; in the free speech argument it is expressing ideas that might harm the war effort.

We will continue to use F and G as property letters. F is the basic similarity , the property that the two analogs share, presumably without controversy. In the free speech argument, the basic similarity is that they create a clear and present danger. And G is the inferred similarity , the property that the inferred analog is purported to have on the grounds that the basic analog has it. Is not protected by the right to free speech is the inferred similarity in the free speech argument. Here is one good way to clarify the argument:

• Falsely shouting fire in a theater creates a clear and present danger and is not protected by the right to free speech.
• Expressing ideas that might harm the war effort creates a clear and present danger.
• ∴ Expressing ideas that might harm the war effort is not protected by the right to free speech..

Variations on this model are common. The basic or inferred analog, for example, will sometimes include more than one item, as in this example:

Manatees must be mammals, since whales and dolphins, like manatees, are sea creatures that give live birth, and whales and dolphins are definitely mammals.

In this case, the basic analog—the content of A —is whales and dolphins.

Likewise, either the basic similarity or the inferred similarity may include more than one property, as in this example:

Manatees must be mammals, since whales, like manatees, are sea creatures that give live birth and that nourish their young on the mother’s milk, and whales are definitely mammals.

In this example, the basic similarity—the content of F —is sea creatures that give live birth and nourish their young on the mother’s milk.

Clarifying an argument from analogy is usually a straightforward matter. It is easiest to begin by identifying the analogs—the two items that the arguer is comparing; insert the one that is not in question into the A position as the basic analog, and the one that is in question into the B position, as the inferred analog. Then insert the basic similarity—the property the two analogs uncontroversially share—into both premises as F. Finally, insert the inferred similarity into the first premise and the conclusion, as G.

Arguments from analogy are sometimes enthymemes. When there is an implicit statement, it is usually the second premise, the one that establishes the basic similarity. This is because arguers often assume, rightly, that the similarity between two analogs is so obvious that it goes without saying. Suppose I say to a friend of mine, whose son is about to enter first grade, “Since John behaves respectfully towards his parents, he will surely treat his teachers with respect.” The basic analog is John’s parents, the inferred analog is John’s teachers, and the inferred similarity is are treated with respect by John. But what is the basic similarity? We must identify a relevant trait that parents and teachers have in common, namely, that they are authority figures to John. Here is the clarified argument. (Brackets, as usual, indicate that premise 2 is implicit, but we also must supply to premise 1 the part about authority figures.)

• John’s parents are authority figures to him and are treated with respect by him.
• [John’s teachers will be authority figures to him.]
• ∴ John’s teachers will be treated with respect by him.

EXERCISES Chapter 15, set (a)

For each of these arguments from analogy, identify the basic analog, the inferred analog, the basic similarity, and the inferred similarity. Then clarify it in standard clarifying format.

Sample exercise. “Expressions of shock and sadness came from other coaches and administrators following the announcement by Tulane President Eamon Kelly that the school planned to drop its basketball program in the wake of the alleged gambling scheme and newly discovered NCAA violations. Coach Jim Killingsworth of TCU said: ‘I think they should deal with the problem, not do away with it. If they had something like that happen in the English department, would they do away with that? I feel like they should have tried to solve their problems.’” —Associated Press

Sample answer. Basic analog: English department. Inferred analog: basketball program. Basic similarity: is a college program (implicit). Inferred similarity: should not be eliminated if experiencing problems.

• The English department is a college program and should not be eliminated if it is experiencing problems.
• [The basketball program is a college program.]
• ∴ The basketball program should not be eliminated if it is experiencing problems.
• In a good marriage, partners often seek counseling to help them resolve their difficulties. You’re having trouble with your boss—why should a conflict in an employer–employee relationship be treated any differently?
• So you got tickets to the Metropolitan Opera’s production of the “Flying Dutchman”? You should try to smuggle in a flashlight and a good book. I made the mistake of going to Wagner’s “Parsifal”—that night was one of the most boring years of my life.
• Etiquette arbiter Emily Post contended that men need not remove their hats in elevators when there are women present. She reasoned that an elevator is a means of transportation, just like a streetcar, bus, subway, or train. The only difference is that an elevator travels vertically, rather than horizontally. A man is not expected to remove his hat in other vehicles, so there is no need for him to do so in an elevator.
• View expressed in a mid-20th century article by a professional sociologist: One attribute with which women are naturally and uniquely gifted is the care of children. Since the ill and infirm resemble children in many ways, being not merely physically weak and helpless but also psychologically dependent, it is fairly easy to conclude that women are also especially qualified to care for the sick.
• “Suppose you had a son, a fine writer who had brought national recognition for his college newspaper and a scholarship for himself. Suppose that, in his junior year, a big-city newspaper offered him a reporter’s job with a three-year guarantee at an unheard-of salary. Would you advise him to turn down the offer of a professional newspaper job? We know the answer. And we would not think twice before urging him, begging him, to hire on with the newspaper. After all, we’d say, the reason he was in college was to start to prepare himself for a decent career in the field of his choosing. So, why all the fulmination about a star athlete’s taking the chance to make himself a cool $5 million by doing for pay what he’s been doing for free (presumably) for three years?” —William Raspberry, Los Angeles Times
• “We feel instinctive sympathy for the defendant who pleads, ‘I tried to get a job and nobody would hire me. Only in desperation did I turn to robbery.’ Now consider the logically parallel defense: ‘I tried to seduce a woman legitimately and nobody would sleep with me. Only in desperation did I turn to rape.’ Nobody would buy that from a rapist, and nobody should buy it from a robber.” —Steven Landsburg, Forbes

15.2 The Total Evidence Condition (1): Relevant Similarities

If an argument from analogy can be loyally paraphrased in the form described above, then it satisfies the correct form condition. But for an inductive argument to be logically strong it must not only satisfy the correct form condition; it must also satisfy the total evidence condition. As with frequency arguments and inductive generalizations, there are two parts to the total evidence condition for arguments from analogy: the basic similarity must be relevant, and any dissimilarities must be irrelevant. If an argument does poorly on either one of these conditions, it should be judged no better than logically weak.

Although analogical arguments are sometimes accused of committing the fallacy of false analogy (or the fallacy of faulty analogy ), this fallacy is very much like the fallacy of hasty generalization. The existence of the named fallacy highlights the ease with which we can make mistakes in this sort of reasoning. But to accuse an argument from analogy of committing this fallacy says nothing about what has gone wrong with the argument. It is far better to explain more specifically how it is that some necessary condition for soundness has not been satisfied.

Total Evidence Condition for Arguments from Analogy

• The basic similarity must be relevant —it must count in favor of the inferred similarity.
• The dissimilarities must be irrelevant —any dissimilarity between the two analogs must not make the basic analog a better candidate for the inferred property.

The argument is logically weaker to the extent that it fails in either area.

15.2.1 The Relevance of the Basic Similarity

Begin your deliberations about the total evidence question by asking, Is the basic similarity relevant? The more relevant it is, the stronger the logic of the argument might be. When you consider this question, forget about the two analogs and simply consider to what extent the basic similarity counts in favor of the inferred similarity. A television advertising campaign by a dairy company shows old but cheerful citizens of the Republic of Georgia eating yogurt; they have eaten yogurt all their lives, we are told, and they are now well past the century mark—one woman is now 134! Eating yogurt, we are encouraged to believe, could do the same for us. The first step in evaluating how well this argument satisfies the total evidence condition is to ignore the two analogs (citizens of Georgia and us) and ask whether the basic similarity—eating yogurt—counts in favor of the inferred similarity—a long life. There is no special reason to think so, and the argument doesn’t help by providing one. So the logic of the argument is very weak.

More commonly an argument from analogy satisfies the condition at least to some degree. One large state university published the following story in its alumni magazine:

A preliminary appraisal of the results of a major assessment of faculty and graduate programs conducted by the Conference Board of Associated Research Councils placed our institution second in the nation among public research universities and in the top five overall. “It is gratifying to see our faculty receive this national recognition of their superior research and teaching,” said the Chancellor. Even though the study focused on graduate programs, he pointed out that the results could also be applied to the undergraduate program as well, since the two programs share the same faculty.

The university’s graduate program is the basic analog and its undergraduate program the inferred analog. The basic similarity is that the university’s excellent faculty staffs them. And the inferred similarity is that the academic programs are excellent. Is the basic similarity relevant? That is, does having an excellent faculty count toward the excellent academic programs? Of course it does. So this argument easily clears the first hurdle of the total evidence condition. But it is too soon to conclude that the argument is logically strong; there is still a second total evidence hurdle to clear.

15.2.2 Relevant Similarities and the Fallacy of Equivocation

Suppose I say, “Einstein was smart, and he was able to revolutionize physics. The physics teacher I had in high school is smart, too, so he should be able to revolutionize physics.” The basic similarity is relevant to the inferred similarity—smart is better than stupid when it comes to revolutionizing physics. But there is smart, and then there is smart. Surely my high school physics teacher is not as smart as Einstein. Doesn’t that weaken the argument? Let’s clarify it and see:

• Einstein was smart and was able to revolutionize physics.
• My high school physics teacher is smart.
• ∴ My high school physics teacher is able to revolutionize physics.

Smart shows up in both premises. To ask whether my high school physics teacher is as smart as Einstein is to ask, in effect, whether the word means the same thing in each case. It is a general expression. Recalling our coverage of generality in Chapter 5, this means that it is an expression that allows for degrees (examples were fine, bald, brown, living together, incompatible, wrong, and evil ). As we saw, generality is usually unproblematic. It becomes problematic, however, when the meaning of the expression shifts from one use to the next, and when the apparent success of the argument depends on that shift. In that case, the argument commits the fallacy of equivocation; the lesson from Chapter 5 is to eliminate the ambiguity.

Let’s eliminate the ambiguity by using the reasonable-premises approach in revising premise 2; in that case it is as follows:

2. My high school physics teacher is smart, though not as smart as Einstein.

While this is probably true, we now have a major problem with the logic of the argument—namely, it no longer satisfies the correct form condition, since the basic similarity, established in premise 1, is not asserted in premise 2. (The form is now something like this: 1. A is F and G; 2. B is sort of like F ; ∴ C . B is G. ) Let’s try revising it again, this time using the reasonable-logic approach. This gives us the following:

2. My high school physics teacher is just as smart as Einstein.

This nicely fixes the logical problem, but at the cost of what is pretty obviously a false premise. Either way, the argument is unsound.

The Oliver Wendell Holmes free speech argument, presented at the beginning of the chapter, provides a weightier example of the same problem. The basic similarity, creating a clear and present danger, certainly counts in favor of the inferred similarity of not being protected by the right to free speech. But is the danger caused by the wartime expression of potentially subversive ideas as clear and as present as the danger caused by the false shout of fire in a theater? If not, doesn’t this weaken the argument? Let’s take another look at Holmes’s clarified argument.

• ∴ Expressing ideas that might harm the war effort is not protected by the right to free speech.

The phrase clear and present danger, like the term smart in the Einstein example, is a general term that seems to apply to a greater degree in premise 1 than in premise 2. It is plausible to suppose that this shift contributes to the apparent success of the argument, and thus that the argument commits the fallacy of equivocation. So we should revise our paraphrase of premise 2 to eliminate the ambiguity. On the one hand, we could paraphrase it to say that those who scattered the leaflets created a clear and present danger, though less clear and present than falsely shouting fire in a theater. The premise would probably be true, but we would have created the same logical difficulty described in the Einstein argument—the basic similarity is not the same in each premise. On the other hand, we could paraphrase it to say that they created a clear and present danger that is just as clear and present as falsely shouting fire in a theater. We have now satisfied the correct form condition but probably have a false premise.

The problem is one to look for whenever you are clarifying an argument from analogy.

EXERCISES Chapter 15, set (b)

For each of the arguments in set (a), answer whether the basic similarity is relevant.

Sample exercise. See sample in set (a).

Sample answer. The basic similarity (that something is a college program) has some relevance to the inferred similarity (that it shouldn’t be eliminated if it is experiencing problems), but only to a limited extent. It is relevant only insofar as there is some weak presumption in any sort of institution that a program that has been set up was set up for a good reason.

15.3 The Total Evidence Condition (2): Irrelevant Dissimilarities

15.3.1 the irrelevance of the dissimilarities.

The second total evidence question is Are there relevant dissimilarities? Preferably they are irrelevant, for the more relevant the dissimilarities, the weaker the logic of the argument. When you consider this question, forget about the basic similarity and concentrate on the two analogs. There are always innumerable ways in which they are dissimilar, but most or all of them will be irrelevant. What matters is to what extent any dissimilarity makes the basic analog a better candidate for the inferred property.

Consider, for example, the free speech argument. There are many dissimilarities. One of the activities happens in a theater, for example, while the other could happen anywhere; but this is irrelevant, since there is no reason to think that things said in a theater are less deserving of protection by the right to free speech than things said anywhere else. Or, for example, one of them is spoken aloud, while the other could be written down; but again, this is irrelevant, for there is no general reason to think that the spoken word is more worthy of free speech protection than the written word.

Some of the dissimilarities, however, are relevant. In the theater case, what is expressed is intentionally deceptive, while in the leaflet case, what is expressed seems to have been utterly sincere. This, taken by itself, certainly makes the theater case a better candidate for exemption from free speech protection, and thus it counts as a relevant dissimilarity. Furthermore, in the theater case, the action is sure to have a harmful result; but in the leaflet case, there is no assurance that anyone will pay any attention or, if they do, that they will be influenced (in fact, it was established that no one had been persuaded by the leaflet). This, too, makes the theater case a better candidate for lack of protection by the right to free speech.

In short, even if we forget that the phrase clear and present danger may be equivocal, the argument does not score well on the second portion of the total evidence condition. Its logic can be judged, at best, as fairly weak. Brilliant jurist that he was, I should note that Oliver Wendell Holmes relied, as he should have, on a good deal more than just this argument in support of his conclusion.

Let’s now return to the academic excellence argument. Here is the clarification:

• The university’s graduate program is staffed by the university’s faculty and is academically excellent.
• The university’s undergraduate program is staffed by the university’s faculty.
• ∴ The university’s undergraduate program is academically excellent.

There are many dissimilarities between the graduate and undergraduate programs of any large state university. Graduate courses, for example, are usually assigned higher catalog numbers than are undergraduate courses. But this is irrelevant; catalog numbers are not like scores flashed by Olympic judges, with higher numbers going to better courses. Another difference is that in large state universities the graduate students tend to have much more exposure to the faculty than do the undergraduate students—their classes are much smaller and are more frequently taught by the regular faculty members. This is relevant, since student exposure to faculty can contribute powerfully to academic excellence. The conclusion may still be true. But even though this argument does well on the first condition, it performs badly on the second and so its logic must be considered weak.

EXERCISES Chapter 15, set (c)

For each of the arguments in set (a), do three things: ( i ) state an irrelevant dissimilarity, and explain, ( ii ) explain any relevant dissimilarities, and ( iii ) state your evaluation of the argument’s logic based on this and the previous exercise.

Sample answer. ( i ) The basketball program probably has a higher proportion of students on full scholarship than does the English department. This doesn’t seem relevant, since it doesn’t make English a better candidate for preservation in the face of difficulties. ( ii ) The most important dissimilarity is that the English department is not only an academic program, but also one that is central to the mission of the institution, while the basketball program is an athletic program and thus more peripheral to its mission. This means there is a far stronger impetus to work out English department difficulties before disbanding it. ( iii ) Though the argument is OK on the first part of the total evidence condition, it fails the second part and is logically very weak.

15.4 The Special Character of Arguments from Analogy

15.4.1 arguments from analogy as logical borrowers.

As you may have noticed, every example of an argument from analogy worked out in this chapter has been declared logically weak and thus unsound. This is not an aberration. Although not all arguments from analogy are unsound, they do establish their conclusions far less often than any other sort of argument. Plato, in the lead quotation for this chapter, calls them “impostors.” Analogical arguments, unlike any other arguments we look at in this book, have a built-in logical shortcoming.

Let’s take another look at the logical form of arguments from analogy:

• A (basic analog) is F (basic similarity) and G (inferred similarity).
• B (inferred analog) is F (basic similarity).
• ∴ B (inferred analog) is G (inferred similarity).

What is the source of logical strength for such an argument? Not the correct form condition; as with every other inductive argument, satisfying this condition merely qualifies the argument for any strength that might be conferred by the total evidence condition. Not the second part of the total evidence condition; the absence of relevant dissimilarities simply means there is no evidence to undermine whatever strength it has. [2]  This leaves the first part of the total evidence condition as the sole positive source of logical strength.

How does the first part of the total evidence condition provide logical strength? By virtue of the fact that the basic similarity counts in favor of the inferred similarity. But what does count in favor of mean here? The only meaning I know is a sound argument can be offered for it. So we can now see that logically strong analogical arguments derive their logical strength from another argument—the argument that can be offered from the inferred similarity to the basic similarity. We will call such an argument (an argument from F to G —see premise 1 of the form clarified above) a background argument . Stated simply: an analogical argument’s only logical strength is borrowed from a background argument.

Any other sort of argument can, in principle, lend its strength to an argument from analogy. For example, in the preceding chapter we looked briefly at the argument Every Japanese car I’ve ever owned has been well built, so that Toyota is probably well built. It could easily be clarified as an argument from analogy, clarified as follows:

• Every Japanese car I’ve ever owned has been a Japanese car and has been well built.
• [That Toyota is a Japanese car.]
• ∴ That Toyota is well built.

If the similarity is relevant in this case, it is because the background argument is a logically strong inductive generalization that goes from my experience of Japanese cars (the basic similarity) to the conclusion that Japanese cars in general are well built (the inferred similarity). The argument from analogy is logical only if this generalization works. So it borrows its logical strength from an inductive generalization.

The next passage, from Science News, provides a second example of borrowed logic in an argument from analogy.

The concept of “vintage year” took on a new meaning this week when two scientists presented the first chemical evidence that wine existed as far back as about 3500 bc. They had noticed a red stain while piecing together jars excavated from an Iranian site. They compared the stain with a similar stain in an ancient Egyptian vessel known to have contained wine. The researchers scraped the reddish residue from the jars and analyzed the samples with infrared spectroscopy. Residues from the Iranian and Egyptian jars looked alike and were full of tartaric acid, a chemical naturally abundant only in grapes. “Those crystals are a signature for wine,” says one researcher.

The argument can be clarified thus:

• The Egyptian jar had a certain red stain and contained wine.
• The Iranian jar had the same red stain.
• ∴ That Iranian jar contained wine.

In this case, if the similarity is relevant it is because the background argument is a sound explanatory argument (of a sort we will cover thoroughly in the next chapter) that establishes that the red stains (the basic similarity) have properties that are best explained as caused by wine (the inferred similarity). This argument’s logical strength is borrowed from an explanatory argument.

As a final example, arguments from analogy can even borrow their logical strength from deductive arguments. Consider the validity counterexamples of Chapter 10. In that chapter we started with an inverted—and invalid—Socrates argument:

• All men are mortal.
• Socrates is mortal.
• ∴ Socrates is a man.

We then offered as a validity counterexample this obviously invalid (because of true premises and false conclusion) Atlantic argument:

• All ponds are bodies of water.
• The Atlantic Ocean is a body of water.
• ∴ The Atlantic Ocean is a pond.

In this way we saw that the Socrates argument was invalid. Like any validity counterexample, the reasoning can be represented as an argument from analogy, clarified as follows:

• The Atlantic argument has a certain form and is invalid.
• The Socrates argument has the same form.
• ∴ The Socrates argument is invalid.

Here the relevance of the similarity depends on a deductive background argument; for the way to argue that a certain form (the basic similarity) is invalid (the inferred similarity) is by use of this valid affirming the antecedent argument, which has a self-evidently true first premise:

• If the form of an argument is such that it is possible for the premises to be true and the conclusion false, then the argument is invalid.
• This particular form is such that it is possible for the premises to be true and the conclusion false.
• ∴ The argument is invalid.

In this case, the logical strength of the analogical argument is borrowed from a sound deduction.

By its very nature, then, when an analogical argument works it works on borrowed logic. The two analogs mainly serve to get in the way by providing a basis for relevant dissimilarities. It is the background argument, which ignores the analogs and is concerned solely with the basic and inferred similarities, that serves as the argument’s motor. In the end, the background argument cannot itself be some other argument from analogy, since the background argument would depend on a background argument (and so on).

There are two practical lessons here. First, if you can see what the background argument is, bring it to the foreground when you clarify the argument, abandoning the analogical form. The Toyota argument, for example, would be much easier to evaluate properly if clarified as a complex argument composed of an inductive generalization and a frequency argument (as illustrated in Chapter 14); and the Iranian jar argument, likewise, if paraphrased as an explanatory argument. Second, if you cannot see what the background argument is, you should normally resist the temptation to judge it as logically strong until you better understand the background argument. As noted at the beginning of the chapter, analogical arguments are custom-made for the way our minds work, which makes them extraordinarily persuasive. But their inherent reliance on logical borrowing also makes them very good at concealing logical defects. When a persuasive car salesman won’t let you open the hood to inspect the motor, it may be prudent to shop elsewhere.

15.4.2 Arguments from Analogy as Psychological Lenders

From a logical point of view, analogical arguments are borrowers. But from a psychological point of view, they often put other arguments deeply into their debt. They can hint as well as hide.

Look, for example, at the Iranian jar argument. The analogy between the two stains is what suggested to the researchers that the jar had once contained wine. This set in motion a research effort in which samples scraped from both jars were examined by infrared spectroscopy, revealing crystals that were “a signature for wine.” One could perhaps say that this new evidence converts the initial analogical argument from a merely suggestive one into a logically strong one, by showing just how relevant the basic similarity (same red stain) is to the inferred similarity (that it contained wine). But it would be much clearer to simply say that the background argument displaces the argument from analogy. Analogical reasoning has lent a powerful psychological boost to the research program by producing the suggestive idea. Still, any logical strength it gains from that research program is borrowed from the background argument—that is, from the explanatory argument about crystals developed by the researchers. Clarity is increased if the initial analogy drops out of any account of the logical support for the conclusion—as long as it remains as a central feature of the history of the discovery. [3]

Analogical arguments can lend a valuable psychological boost to inquiry of every sort. Consider the free speech argument. Even if you are not persuaded by the proposed analogy between shouting fire and distributing leaflets, it is certainly suggestive. In particular, it suggests that you are wrong if you think that all expressions are protected. Further, it suggests a way of reasoning about which ones are not protected—namely, by thinking about the possible dangers caused by the speech in question. If that way of reasoning succeeds, the argument from analogy gets psychological credit for suggesting it, even if it gets no logical credit for supporting it.

Nineteenth-century philosopher John Stuart Mill aptly declared that good reasoners will consider any analogical argument as a “guidepost, pointing out the direction in which more rigorous investigations should be prosecuted.” Arguments from analogy brilliantly serve a necessary function in reasoning. We would be lost without good guideposts. But we should not confuse them with destinations.

EXERCISES Chapter 15, set (d)

Fully clarify and evaluate each of the arguments from analogy. In cases where you can see the background argument, you may clarify and evaluate either the analogical argument or the background argument.

• I’ve only seen one Hitchcock movie— Psycho. It was scary. Let’s try The Birds. I bet it will be scary too.
• To solve our drug problems, instead of outlawing drugs we must make them as safe and risk-free and—yes—as healthy as possible. It’s like sex. We recognize that people will continue to have sex for nonreproductive reasons, whatever the laws, and with that in mind we try to make sexual practices as safe as possible in order to minimize the spread of the sexually transmitted diseases.
• Question (investigator, to a university president): “Your administration will undertake reviews or investigations of members of your faculty without their being informed of the fact?”

A:  “I believe it’s very possible. I believe it happened in this case.”

Q:  “Do you consider that proper and appropriate?”

A:  “Personal opinion? Yes.”

Q: “Can you tell me why?”

A:  “I don’t know. Why not? I guess in an analogy, I don’t think J. Edgar Hoover, for example, ever advised everybody he was investigating that they were being investigated.”

Q:  “But he, J. Edgar Hoover, wasn’t running a university.”— Lingua Franca

• Breceda and lifeguards up and down the beach stressed the dangers of sleeping on the beach at night. “The people who get hurt are pretty much innocent,” Breceda said. “They take a walk on the beach at Puerto Vallarta at 3 a.m. and nothing happens, and so they assume it’s OK to do it here. But a whole different situation occurs here.” In addition to the dangers posed by muggers and rapists, people sleeping on the beach also could get run over by sweepers. — Los Angeles Times (Consider the argument attributed to the people who sleep on the beach.)
• “ Question: Surely society has a right to rid itself of a man like Ted Bundy? Answer : My main opposition to the death penalty is what it does to society. Our society kills people in cages. It is like going hunting in a zoo. In the cage they are not dangerous, but executing them is very dangerous—for us.” —I. Gray and M. Stanley, eds., A Punishment in Search of a Crime: Americans Speak Out Against the Death Penalty
• “At their August 1945 Potsdam meeting, Truman remarked to an aide, ‘Stalin is as near like Tom Pendergast as any man I know.’ Pendergast was a Missouri machine boss who helped get Truman elected to the Senate. For some superficial reason Truman concluded that, like Pendergast, Stalin was a man one could deal with, a man of his word. ‘It led Truman to believe that Stalin would hold free elections in Eastern Europe,’ says Deborah Larson, a UCLA political scientist.” — Associated Press
• Gerry Spence is serving as the pro bono defense attorney for an “environmental terrorist” who embedded metal plates in trees so that the bulldozers would be wrecked (and, potentially, the drivers injured). He is asked if “monkeywrenching” trees is ever justified. Spence’s sleight-of-hand answer reveals why he wins so many cases: “In most circumstances, breaking the law is improper. Now, suppose a tractor is about to run over a child. Is it improper to demolish the tractor? Suppose the tractor was going to run over something inanimate, a painting by Van Gogh that cost $32 million. Now, what about a tractor running down a tree? A 400-year-old original growth tree?” — Forbes
• “Thoughtful and right-minded men place their homage and consideration for woman upon an instinctive consciousness that her unmasculine qualities, whether called weaknesses, frailties, or what we will, are the sources of her characteristic and a special strength within the area of her legitimate endeavor. In actual war, it is the men who go to battle, enduring hardship and privation and suffering disease and death for the cause they follow. It is the mothers, wives, and maids betrothed, who neither following the camp nor fighting in battle, constitute at home an army of woman’s constancy and love whose yearning hearts make men brave and patriotic. So, in political warfare, it is perfectly fitting that actual strife and battle would be apportioned to men, and that the influence of woman, radiating from the homes of our land, should inspire to lofty aims and purposes those who struggle for the right.” —Grover Cleveland, Ladies Home Journal, 1905
• One philosopher, arguing that the rights of a rape victim to make decisions about her body can be more important than the right to life of a fetus, develops the following analogy: “Let me ask you to imagine this. You wake up in the morning and find yourself back to back in bed with an unconscious violinist. A famous unconscious violinist. He has been found to have a fatal kidney ailment, and the Society of Music Lovers has canvassed all the available medical records and found that you alone have the right blood type to help. They have therefore kidnapped you, and last night the violinist’s circulatory system was plugged into yours, so that your kidneys can be used to extract poisons from his blood as well as your own. The director of the hospital now tells you, ‘Look, we’re sorry the Society of Music Lovers did this to you—we would never have permitted it if we had known. But still, they did it, and the violinist now is plugged into you. To unplug you would be to kill him. But never mind, it’s only for nine months. By then he will have recovered from his ailment, and can safely be unplugged from you.’ Is it morally incumbent on you to accede to this situation?” —Judith Jarvis Thompson, Philosophy and Public Affairs
• “Look round the world. Contemplate the whole and every part of it. You will find it to be like one great machine, subdivided into an infinite number of lesser machines, which again admit of subdivisions to a degree beyond what human senses and faculties can trace and explain. All these various machines and their parts are adjusted to each other with an accuracy which ravishes into admiration all men who have ever contemplated them. From this we can see that the curious adapting of means to ends throughout all nature resembles exactly, though it much exceeds, the adapting of means to ends in the things made by human beings. Since, therefore, the effects resemble each other, we are led to infer, by all the rules of analogy, that the causes also resemble, and that there is an Author of Nature who is somewhat similar to the mind of man, though possessed of much larger faculties, proportioned to the grandeur of the work which he has executed. Therefore we prove at once the existence of God and his similarity to human mind and intelligence.” —David Hume, Dialogues Concerning Natural Religion

15.5 Summary of Chapter Fifteen

Arguments from analogy typically contend that because two items are the same in one respect, they are the same in another respect. The basic analog is compared to the inferred analog; because they have the basic similarity in common, it is concluded that the inferred analog also has the inferred similarity.

The total evidence condition has two parts. First, the basic similarity must be relevant—that is, it must count toward the presence of the inferred similarity. Second, there must not be any dissimilarities that are relevant—that is, any dissimilarity between the two analogs must not make the basic analog a better candidate for the inferred property. The argument is logically weaker to the extent that it fails in either of these two areas.

Their only positive logical strength comes from the background argument that establishes that the inferred similarity follows from the basic similarity; thus, whatever logical success analogical arguments have is borrowed. This makes it especially important to pay close attention to the first part of the total evidence condition. On the other hand, analogical arguments play an important psychological role in suggesting lines of reasoning, and so should be cultivated for that purpose.

15.6 Guidelines for Chapter Fifteen

• Structure arguments from analogy, when it would be loyal to do so, by identifying four things—the basic and inferred analogs and the basic and inferred similarities—then inserting each into its proper place in the form. Remember that the second premise, which declares the basic similarity, is often implicit.
• In considering whether an argument from analogy has satisfied the total evidence condition, first ask, Is the basic similarity relevant? To answer this question, look at the extent to which the basic similarity counts in favor of the inferred similarity.
• When the basic similarity is described by a general term, consider whether its meaning shifts from one use to the next. If it shifts enough to affect the soundness of the argument, revise your clarification to eliminate the ambiguity.
• In considering whether an argument from analogy has satisfied the total evidence condition, ask next, Are any of the dissimilarities relevant? To answer this question, look at the extent to which any dissimilarity makes the basic analog a better candidate than the inferred analog for the inferred property.
• When you can clearly see the background argument, clarify it rather than the argument from analogy. When you cannot see the background argument, you should normally reserve final judgment about the strength of the argument’s logic.

15.7 Glossary for Chapter Fifteen

Analogs —the two things (or classes of things) that are said to be similar in an argument from analogy.

Argument from analogy —an argument that asserts that because two items are the same in one respect, they are the same in another respect. They can be represented by this form:

Background argument —an argument that shows that the inferred similarity (of an analogical argument) follows from the basic similarity—that is, an argument that shows that the basic similarity is relevant.

Basic analog —in an argument from analogy, the item that we are presumably more familiar with, which is presumably known to have both the basic and the inferred similarities.

Basic similarity —in an argument from analogy, the property that the two analogs share, presumably without controversy.

Fallacy of false analogy —the mistake of using an argument from analogy in which the basic similarity is not relevant or in which there are relevant dissimilarities between the basic and inferred analogs. Because this term says nothing about what precisely has gone wrong with the argument, it is better to explain more specifically how it is that some necessary condition for soundness has not been satisfied. Also called the fallacy of faulty analogy.

Inferred analog —in an argument from analogy, the item in question, about which the argument is drawing its conclusion.

Inferred similarity —in an argument from analogy, the property that the inferred analog is alleged to have because the basic analog has it.

• The British usually spell it analogue . Historically, the term was analogon . ↵
• The second part of the total evidence condition for frequency arguments operates the same way. ↵
• To use terminology mentioned elsewhere in the text, it is important in the context of discovery, but not in the context of justification. ↵

An argument that asserts that because two items are the same in one respect, they are the same in another respect. They can be represented by this form:

1. A is F and G. 2. B is F. ∴ C . B is G.

The two things (or classes of things) that are said to be similar in an argument from analogy.

In an argument from analogy, the item that we are presumably more familiar with, which is presumably known to have both the basic and the inferred similarities.

In an argument from analogy, the item in question, about which the argument is drawing its conclusion.

In an argument from analogy, the property that the two analogs share, presumably without controversy.

In an argument from analogy, the property that the inferred analog is alleged to have because the basic analog has it.

The mistake of using an argument from analogy in which the basic similarity is not relevant or in which there are relevant dissimilarities between the basic and inferred analogs. Because this term says nothing about what precisely has gone wrong with the argument, it is better to explain more specifically how it is that some necessary condition for soundness has not been satisfied. Also called the fallacy of faulty analogy.

An argument that shows that the inferred similarity (of an analogical argument) follows from the basic similarity—that is, an argument that shows that the basic similarity is relevant.

A Guide to Good Reasoning: Cultivating Intellectual Virtues Copyright © 2020 by David Carl Wilson is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License , except where otherwise noted.

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Pursuing Truth: A Guide to Critical Thinking

Chapter 2 arguments.

The fundamental tool of the critical thinker is the argument. For a good example of what we are not talking about, consider a bit from a famous sketch by Monty Python’s Flying Circus : 3

2.1 Identifying Arguments

People often use “argument” to refer to a dispute or quarrel between people. In critical thinking, an argument is defined as

A set of statements, one of which is the conclusion and the others are the premises.

There are three important things to remember here:

• Arguments contain statements.
• They have a conclusion.
• They have at least one premise

Arguments contain statements, or declarative sentences. Statements, unlike questions or commands, have a truth value. Statements assert that the world is a particular way; questions do not. For example, if someone asked you what you did after dinner yesterday evening, you wouldn’t accuse them of lying. When the world is the way that the statement says that it is, we say that the statement is true. If the statement is not true, it is false.

One of the statements in the argument is called the conclusion. The conclusion is the statement that is intended to be proved. Consider the following argument:

Calculus II will be no harder than Calculus I. Susan did well in Calculus I. So, Susan should do well in Calculus II.

Here the conclusion is that Susan should do well in Calculus II. The other two sentences are premises. Premises are the reasons offered for believing that the conclusion is true.

2.1.1 Standard Form

Now, to make the argument easier to evaluate, we will put it into what is called “standard form.” To put an argument in standard form, write each premise on a separate, numbered line. Draw a line underneath the last premise, the write the conclusion underneath the line.

• Calculus II will be no harder than Calculus I.
• Susan did well in Calculus I.
• Susan should do well in Calculus II.

Now that we have the argument in standard form, we can talk about premise 1, premise 2, and all clearly be referring to the same thing.

2.1.2 Indicator Words

Unfortunately, when people present arguments, they rarely put them in standard form. So, we have to decide which statement is intended to be the conclusion, and which are the premises. Don’t make the mistake of assuming that the conclusion comes at the end. The conclusion is often at the beginning of the passage, but could even be in the middle. A better way to identify premises and conclusions is to look for indicator words. Indicator words are words that signal that statement following the indicator is a premise or conclusion. The example above used a common indicator word for a conclusion, ‘so.’ The other common conclusion indicator, as you can probably guess, is ‘therefore.’ This table lists the indicator words you might encounter.

Each argument will likely use only one indicator word or phrase. When the conlusion is at the end, it will generally be preceded by a conclusion indicator. Everything else, then, is a premise. When the conclusion comes at the beginning, the next sentence will usually be introduced by a premise indicator. All of the following sentences will also be premises.

For example, here’s our previous argument rewritten to use a premise indicator:

Susan should do well in Calculus II, because Calculus II will be no harder than Calculus I, and Susan did well in Calculus I.

Sometimes, an argument will contain no indicator words at all. In that case, the best thing to do is to determine which of the premises would logically follow from the others. If there is one, then it is the conclusion. Here is an example:

Spot is a mammal. All dogs are mammals, and Spot is a dog.

The first sentence logically follows from the others, so it is the conclusion. When using this method, we are forced to assume that the person giving the argument is rational and logical, which might not be true.

2.1.3 Non-Arguments

One thing that complicates our task of identifying arguments is that there are many passages that, although they look like arguments, are not arguments. The most common types are:

• Explanations
• Mere asssertions
• Conditional statements
• Loosely connected statements

Explanations can be tricky, because they often use one of our indicator words. Consider this passage:

Abraham Lincoln died because he was shot.

If this were an argument, then the conclusion would be that Abraham Lincoln died, since the other statement is introduced by a premise indicator. If this is an argument, though, it’s a strange one. Do you really think that someone would be trying to prove that Abraham Lincoln died? Surely everyone knows that he is dead. On the other hand, there might be people who don’t know how he died. This passage does not attempt to prove that something is true, but instead attempts to explain why it is true. To determine if a passage is an explanation or an argument, first find the statement that looks like the conclusion. Next, ask yourself if everyone likely already believes that statement to be true. If the answer to that question is yes, then the passage is an explanation.

Mere assertions are obviously not arguments. If a professor tells you simply that you will not get an A in her course this semester, she has not given you an argument. This is because she hasn’t given you any reasons to believe that the statement is true. If there are no premises, then there is no argument.

Conditional statements are sentences that have the form “If…, then….” A conditional statement asserts that if something is true, then something else would be true also. For example, imagine you are told, “If you have the winning lottery ticket, then you will win ten million dollars.” What is being claimed to be true, that you have the winning lottery ticket, or that you will win ten million dollars? Neither. The only thing claimed is the entire conditional. Conditionals can be premises, and they can be conclusions. They can be parts of arguments, but that cannot, on their own, be arguments themselves.

Finally, consider this passage:

I woke up this morning, then took a shower and got dressed. After breakfast, I worked on chapter 2 of the critical thinking text. I then took a break and drank some more coffee….

This might be a description of my day, but it’s not an argument. There’s nothing in the passage that plays the role of a premise or a conclusion. The passage doesn’t attempt to prove anything. Remember that arguments need a conclusion, there must be something that is the statement to be proved. Lacking that, it simply isn’t an argument, no matter how much it looks like one.

2.2 Evaluating Arguments

The first step in evaluating an argument is to determine what kind of argument it is. We initially categorize arguments as either deductive or inductive, defined roughly in terms of their goals. In deductive arguments, the truth of the premises is intended to absolutely establish the truth of the conclusion. For inductive arguments, the truth of the premises is only intended to establish the probable truth of the conclusion. We’ll focus on deductive arguments first, then examine inductive arguments in later chapters.

Once we have established that an argument is deductive, we then ask if it is valid. To say that an argument is valid is to claim that there is a very special logical relationship between the premises and the conclusion, such that if the premises are true, then the conclusion must also be true. Another way to state this is

An argument is valid if and only if it is impossible for the premises to be true and the conclusion false.

An argument is invalid if and only if it is not valid.

Note that claiming that an argument is valid is not the same as claiming that it has a true conclusion, nor is it to claim that the argument has true premises. Claiming that an argument is valid is claiming nothing more that the premises, if they were true , would be enough to make the conclusion true. For example, is the following argument valid or not?

• If pigs fly, then an increase in the minimum wage will be approved next term.
• An increase in the minimum wage will be approved next term.

The argument is indeed valid. If the two premises were true, then the conclusion would have to be true also. What about this argument?

• All dogs are mammals
• Spot is a mammal.
• Spot is a dog.

In this case, both of the premises are true and the conclusion is true. The question to ask, though, is whether the premises absolutely guarantee that the conclusion is true. The answer here is no. The two premises could be true and the conclusion false if Spot were a cat, whale, etc.

Neither of these arguments are good. The second fails because it is invalid. The two premises don’t prove that the conclusion is true. The first argument is valid, however. So, the premises would prove that the conclusion is true, if those premises were themselves true. Unfortunately, (or fortunately, I guess, considering what would be dropping from the sky) pigs don’t fly.

These examples give us two important ways that deductive arguments can fail. The can fail because they are invalid, or because they have at least one false premise. Of course, these are not mutually exclusive, an argument can be both invalid and have a false premise.

If the argument is valid, and has all true premises, then it is a sound argument. Sound arguments always have true conclusions.

A deductively valid argument with all true premises.

Inductive arguments are never valid, since the premises only establish the probable truth of the conclusion. So, we evaluate inductive arguments according to their strength. A strong inductive argument is one in which the truth of the premises really do make the conclusion probably true. An argument is weak if the truth of the premises fail to establish the probable truth of the conclusion.

There is a significant difference between valid/invalid and strong/weak. If an argument is not valid, then it is invalid. The two categories are mutually exclusive and exhaustive. There can be no such thing as an argument being more valid than another valid argument. Validity is all or nothing. Inductive strength, however, is on a continuum. A strong inductive argument can be made stronger with the addition of another premise. More evidence can raise the probability of the conclusion. A valid argument cannot be made more valid with an additional premise. Why not? If the argument is valid, then the premises were enough to absolutely guarantee the truth of the conclusion. Adding another premise won’t give any more guarantee of truth than was already there. If it could, then the guarantee wasn’t absolute before, and the original argument wasn’t valid in the first place.

2.3 Counterexamples

One way to prove an argument to be invalid is to use a counterexample. A counterexample is a consistent story in which the premises are true and the conclusion false. Consider the argument above:

By pointing out that Spot could have been a cat, I have told a story in which the premises are true, but the conclusion is false.

Here’s another one:

• If it is raining, then the sidewalks are wet.
• The sidewalks are wet.
• It is raining.

The sprinklers might have been on. If so, then the sidewalks would be wet, even if it weren’t raining.

Counterexamples can be very useful for demonstrating invalidity. Keep in mind, though, that validity can never be proved with the counterexample method. If the argument is valid, then it will be impossible to give a counterexample to it. If you can’t come up with a counterexample, however, that does not prove the argument to be valid. It may only mean that you’re not creative enough.

• An argument is a set of statements; one is the conclusion, the rest are premises.
• The conclusion is the statement that the argument is trying to prove.
• The premises are the reasons offered for believing the conclusion to be true.
• Explanations, conditional sentences, and mere assertions are not arguments.
• Deductive reasoning attempts to absolutely guarantee the truth of the conclusion.
• Inductive reasoning attempts to show that the conclusion is probably true.
• In a valid argument, it is impossible for the premises to be true and the conclusion false.
• In an invalid argument, it is possible for the premises to be true and the conclusion false.
• A sound argument is valid and has all true premises.
• An inductively strong argument is one in which the truth of the premises makes the the truth of the conclusion probable.
• An inductively weak argument is one in which the truth of the premises do not make the conclusion probably true.
• A counterexample is a consistent story in which the premises of an argument are true and the conclusion is false. Counterexamples can be used to prove that arguments are deductively invalid.

( Cleese and Chapman 1980 ) . ↩︎

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An analogy is when you point out the similarities between two different things.

If I say “the mind is like a computer: it takes certain inputs, processes them, and then spits out results”, I’m drawing an analogy – I’m saying that these two different things – the mind and a computer – are actually alike in some way. But I’m not giving an argument – I’m not drawing any conclusion from my analogy. I’m just drawing parallels, maybe trying to make you think about the mind in a different way, illustrating something about how the mind works.

An argument by analogy is when you point out the similarities between two or more things and then draw some conclusion.

Suppose there is a particular car that I’m deciding whether or not to buy. It’s a ten-year-old Honda Civic with 75,000 kms on the clock, it has a little rust but not much, and it’s only had one owner and she drove it carefully and got it serviced regularly.

And suppose I reason like this:

My last car was a Honda Civic, and when I bought it it was 10 years old had 75,000 kms on the clock, it has a little rust but not much, it had had only one previous owner, and that owner had looked after it well. And my last car served me well – I drove it for 5 years and it gave me no trouble and needed very little in the way of repairs. So this car, being so similar to my last, will probably likewise serve me well – I should buy it.

P1. Car A was a Honda Civic, ten years old, 75,000 kms, only a little rust, well looked after, and it served me well for 5 years each and required very little in the way of repairs. P2. Car B is a Honda Civic, ten years old, 75,000 kms, only a little rust, well looked after.                                          C. Car B will last 5 years and hardly need any repairs.

General structure of arguments by analogy:

A has characteristics W, X, Y and Z B has characteristics W, X, and Y                                           B will have characteristic Z as well.

Things to consider in evaluating an argument by analogy:

• How similar are the things being compared?
• Are the similarities relevant?  (e.g. if the similarities mentioned were all things like colour, which is surely irrelevant to how well the car goes, it would not be a good argument.)
• Are there any relevant differences between the things being compared?
• How many similar cases are we dealing with?  (e.g. If I had had 3 cars which were similar in all of these respects and had all served me well, the argument would be stronger.)

Analogy Copyright © 2024 by Stephanie Gibbons and Justine Kingsbury is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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Analogical Reasoning

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Argument by analogy ; Case-based reasoning ; Metaphorical thinking

Analogical Reasoning and Its Uses

Analogical reasoning or argument by analogy can be defined as a specific way of thinking, based on the idea that because two or more things are similar in some respects, they are probably also similar in some further respect. Integrating various human-level reasoning mechanisms, arguing by analogical thinking, use analogies by transferring knowledge from one particular entity (the analogue or source) to another one (the target). Furthermore, it refers to the linguistic form, which corresponds to the process of relating the source and the target. As specific form of inference or reasoning, analogies draw conclusions by applying heuristics to propositions or observations as well as by interpolating logical steps or patterns. Analogies focus on relating specific particularities in two or more cases or things to form the basis for a conclusion involving an additional...

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23 Arguments VIII: Analyzing Analogies

Okay, on to analogies. Let’s again start with a more precise definition.

• An analogical argument or argument by analogy is an argument in which known similarities between one thing and another (or multiple others) is/are used to infer that some additional similarity is likely to hold between them.

(Side note: An analog is something that is like something else; an analogy a drawing of a comparison. Given that, why do you think we talk about “analog” technologies, particularly in recording?)

The general form of an analogical argument :

• A and B are known to be similar in respects p, q, r, … 
• A is also known to be x .

Therefore, B will likely be x as well.

Elements of analysis of an analogical argument:

Individuals Compared: A, B

Known Similarities (of A and B): p, q, r, …

Note: there is no set number of similarities that are required. The list “ p, q, r,… ” is just a placeholder. There could be only one similarity, or there could be many more than three.

Additional Fact Known about A that is predicted to hold of B as well: x

Note: the  conclusion  of the argument will claim that x holds or is likely to hold of B.

Examples of Analogical Arguments with Elements Analyzed

1.     1. During the Great Depression there was high unemployment, low wages, and low national productivity, all of which ended due to the increase in government spending for World War II.

2. Today there is high unemployment, low wages, and low national productivity.

Conclusion: high unemployment, low wages, and low national productivity will end if there is an increase in government spending comparable to that during WWII.

(Note: This is an argument some economists will make in times of recession, as for example back in 2008.)

Individuals Compared : (A) the economy of the Great Depression, (B) the economy today

Known Similarities of A and B : (p) high unemployment, (q) low wages, (r) low national productivity

Additional Fact (x) Known about A that is predicted to hold of B as well : it ended/will end because of an increase in government spending

2.       1. A watch is something has many parts which are interrelated in complex yet ordered (i.e., non-random) ways, and which gets that order and complexity from an intelligent  being who designed it.

2. The universe has many parts which are interrelated in complex yet ordered ways.

Conclusion: the universe must have gotten its complexity from an intelligent being who designed it.

(Note:  this is a famous argument for the existence of God known as the Argument from Design.)

Individuals Compared : (A) a watch, (B) the universe

Known Similarities of A and B : (p) many parts, (q) parts are related in complex ways, (r) relations among parts is ordered

Additional Fact (x) Known about A that is predicted to hold of B as well : its order and complexity comes from an intelligent designer

3.      1. Skeletor, a drug designed to speed the healing of broken bones, proved very effective in clinical trials on men who were between 30 and 50 years of age who had broken legs.

2. I am a man between 30 and 50 years of age, and I have a broken leg.

Conclusion: Skeletor will be very effective in helping heal my broken leg.

( Note : Skeletor is an entirely fictitious drug; the fact that I thought of using the He-man villain as a pharmaceutical name shows why I think I should really be in marketing.)

Individuals Compared : (A) men in clinical trial, (B) me

Known Similarities of A and B : (p) male, (q) between 30 and 50, (r) have broken legs

Additional Fact (x) Known about A that is predicted to hold of B as well : very effective in healing broken legs

4.      1. Rush Limbaugh smokes cigars, has a lot of money, and is politically conservative.

2. Robert DeNiro smokes cigars and has a lot of money.

Conclusion: Therefore, he is probably politically conservative too.

Individuals Compared : (A) Rush Limbaugh, (B) Robert DeNiro

Known Similarities of A and B : (p) smoke cigars, (q) have a lot of money

Additional Fact (x) Known about A that is predicted to hold of B as well : is politically conservative

What we now want to know is, when is an analogical argument  strong ?

Criteria for Strength of an Analogical Argument

• … there must be enough similarities between the things being compared;
• … the similarities must be relevant to the drawing of the conclusion;
• … there must not be relevant dissimilarities that are overlooked or suppressed.

Based on these, we get the definition of another  fallacy : an analogical argument that is weak (for whatever reason) is said to be guilty of the fallacy of faulty analogy .

With that in mind, let’s ask, are the above examples above strong? As with generalizations, you may not easily be able to answer one way or another. What you should be able to do, however, is specify what you would need to know in order to answer that question .

Above Examples of Analogical Arguments Analyzed for Strength

• There are multiple similarities which appear to be relevant between the historical example and the current one referred to; only economists and/or economic historians will be in a position to judge whether there are enough similarities and whether there are any neglected but relevant dissimilarities . So those of us who lack the relevant expertise can only say: maybe this is a strong argument, but without more information, we can’t be sure. It’s not  obviously  weak, however.
• There are multiple similarities which appear to be relevant. There are, however, some obvious dissimilarities that are not mentioned, for instance, that we have direct knowledge of how watches are made by their designers, that the universe is vastly larger than a watch, and that watches have obvious purposes (for telling time). Such dissimilarities lead many philosophers to conclude that this is a weak argument (though it has its supporters as well). (If you find this argument interesting and want to think more about it, take another philosophy class!)
• There are multiple similarities which appear to be relevant. While it’s possible that there could be relevant dissimilarities, there aren’t any obvious ones. This is, in fact, the sort of reasoning we implicitly use any time we choose to take a medication (at least for its intended purpose). We rely on being sufficiently similar to those it has already been proven effective in treating – that’s why we think it will work on us too. So this is a strong argument.
• Two similarities between the men are noted, but only one of them (wealth) is in any clear way related to political leanings. So, it is pretty clearly weak: not enough relevant similarities, and no consideration or relevant dissimilarities. It is thus a faulty analogy .

Important Note : it can be difficult to analyze analogies (more so than generalizations), because much more depends on your background knowledge of the topic. It’s this that will help you know whether what’s mentioned is relevant, or whether there are important things being left out. When you are doing your exercises, keep that in mind, and do your best to think imaginatively about the situations described. You should always be able to specify the known and predicted similarities, because those will be stated. But, more often than not, you may have to say that you don’t know enough to assess whether there are enough relevant similarities or unstated dissimilarities.

Distinguishing Generalizations from Analogies

Sometimes it can be hard to tell if you’re dealing with an analogy or a generalization. Both involve two things (target and sample groups; A and B being compared) and an argument that considers them in relation to one another. So how do you know which is which?

The two  main things to look for : once you’ve got the two things/groups and you’re trying to decide what kind of argument it is, ask,

( 1 ) is one included in the other as a smaller subset of it?

( 2 ) Is the thing the conclusion is about a group or a single individual?

In most cases, once you answer these questions you’ll know what you’re dealing with. If the answer to 1 is ‘yes,’ and the conclusion is about the larger group, then it’s a generalization. A generalization  never makes a conclusion about a single individual, and it  does  involve starting with a small group and moving to a claim about a larger group that includes the small one.

If the answer to 2 is ‘single individual,’ it’s an analogy. Analogies sometimes draw conclusions about groups (but not on the basis of what’s true of a subset of the group), but they often draw conclusions about individual things or people.

If the answer to 2 is ‘group,’ then you need to ask question 1 again and determine whether two distinct groups are being compared, or whether a conclusion about a large group is being drawn on the basis of a smaller one.

Practice is how you learn, so your exercises are where you will really get a handle on all of this.

Phil-P102 Critical Thinking and Applied Ethics Copyright © 2020 by R. Matthew Shockey is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License , except where otherwise noted.

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A Workbook For Arguments, Part 3: Analogies, Sources, Causal Claims, and Theoretical Virtues… Oh My!

In this series, I walk through the critical thinking rules  that are sections of the book  A Workbook for Arguments: A Complete Course in Critical Thinking . The information presented in this series derives largely from the aforementioned source. Nevertheless, paraphrases and original examples are mixed into the posts alongside the content derived from the book.

Arguments by analogy, rather than multiplying examples to support a generalization, argue from one specific example to another, reasoning that because the two examples are alike in many ways (or some particular way(s)), they are also alike in one further specific way.

For example, here is how a doctor might argue that everyone should have a regular physical checkup:

People take in their car for servicing and checkups every few months without complaint. Why shouldn’t they take similar care of their bodies?

This argument suggests that getting a regular physical checkup is like taking your car in for regular servicing. Cars need that kind of attention — otherwise, major problems may develop. Aren’t our bodies like that too? The argument amounts to:

People should take their cars in for regular service and checkups (otherwise major problems may develop). People’s bodies are like cars (because human bodies, too, are complex systems that can develop problems if not regularly checked up). Therefore, people should take themselves in for regular “service” and checkups too.

Notice the italicized word “like” in the second premise. When an argument stresses the likeness between two cases, it is very probably an argument from analogy. Here is another striking example.

An interesting switch was pulled in Rome yesterday by Adam Nordwell, an American Chippewa chief. As he descended from his plane dressed in full tribal regalia, Nordwell announced in the name of the American Indian people that he was taking possession of Italy “by right of discovery” in the same way that Christopher Columbus did in America. “I proclaim this day the day of the discovery of Italy,” said Nordwell. “What right did Columbus have to discover America when it had already been inhabited for thousands of years? The same right I now have to come to Italy and proclaim the discovery of your country.”

Nordwell is suggesting that his “discovery” of Italy is like Columbus’ “discovery” of America in at least one important way: both Nordwell and Columbus claimed a country that already had been inhabited by its own people for centuries. Thus, Nordwell insists that he has as much “right” to claim Italy as Columbus had to claim America. But, of course, Nordwell has no right at all to claim Italy. It follows, then, that Columbus had no right to claim America.

Nordwell has no right to claim Italy for another people, let alone “by right of discovery” (because Italy has been inhabited by its own people for centuries). Columbus’ claim to America “by right of discovery” is like Nordwell’s claim to Italy (America, too, had been inhabited by its own people for centuries). Therefore, Columbus had no right to claim America for another people, let alone “by right of discovery.”

How do we evaluate arguments by analogy? The first premise of an argument by analogy makes a claim about the example used as an analogy. Remember Rule 3: make sure this premise is true. It’s true that cars need regular service and checkups to keep major problems from developing, and it’s also true that Adam Nordwell could not legitimately claim Italy for the Chippewa.

The second premise in arguments by analogy claims that the example in the first premise is like the case about which the argument draws a conclusion. Evaluating this premise is harder, and needs a rule of its own.

Rule 12: Analogies require relevantly similar examples

Arguments by analogy do not require that the example used as an analogy be exactly like the example in the conclusion. Our bodies are not just like cars, after all. We are flesh and bone, not metal; we have neither wheels nor seats nor windshield wipers. We have brains (well, some of us do…). Analogies require relevant similarities. The precise composition of cars is irrelevant to the doctor’s point. The argument is about the upkeep of complex systems.

One relevant difference between our bodies and cars is that our bodies do not need regular “service” in the way our cars do. Cars regularly need oil changes, new pumps or transmissions, and the like. Replacing body parts or fluids is much rarer: think organ transplants or blood transfusions. On the other hand, it’s true that we need regular checkups — otherwise problems can develop undetected — and older and strenuously used bodies, like older and higher mileage cars, likely need checkups more often. So the doctor’s analogy is partly successful. The “service” part is somewhat weak, but the checkup part is persuasive.

Likewise, twentieth-century Italy is not like fifteenth-century America in every respect. Italy is known to every twentieth-century school child, whereas America was unknown to much of the world in the fifteenth century. Nordwell is not an explorer, and a commercial jet is not the Santa Maria.

Importantly, though, these differences are not relevant to Nordwell’s analogy. Nordwell simply means to remind us that it is senseless to claim a country already inhabited by its own people. Whether that land is known to the world’s schoolchildren, or how the “discoverer” arrived there, is irrelevant. The more appropriate reaction might have been to try to establish diplomatic relations, as we would try to do today if somehow the land and people of Italy had just been discovered. That’s Nordwell’s point, and taken in that way his analogy makes a good argument.

In conclusion, when evaluating and analyzing an analogy, ask the following:

In what ways are the two things similar? Are these similarities relevant or irrelevant to the desired conclusion?

In what ways are the two things different? Are these differences relevant or irrelevant to the desired conclusion?

The key is that X and Y must be relevantly alike in order for an argument by analogy using X and Y to succeed.

Sources and Causes

In the final section of this Workbook For Arguments, let’s look at critical thinking in relation to sources and causation.

For sources, be sure that your sources are informed, impartial, independent, and cross-checked.

With respect to causes, it is important to know that nearly all causal claims start with correlations. All of the following may be the case:

E1 may be correlated with E2 because E1 causes E2.

E1 may be correlated with E2 because E2 causes E1.

E1 may be correlated with E2 because E1 and E2 are both caused by or correlated with some third thing, E3.

Finally, the correlation between E1 and E2 might just be a coincidence, with no causal connection between them.

Another crucially important aspect of causal claims is the notion of multiple causation. In many cases, physical, psychological, sociological, cultural, biological, and historical phenomena have many different causes that each contribute to the overall effect. For instance, to say the sole cause of the Civil War was slavery is an oversimplification. While disputes concerning the legality and morality of slavery were the predominant cause of the Civil War, a variety of other causal factors were at play as well. The concept of multiple causation is particularly relevant in this case. Such factors include the election of Abraham Lincoln, Bleeding Kansas, increased sectionalism and regional tension, differences in economy and society, economic considerations involving trade, and much more.

Additionally, causes and effects may interpenetrate. For instance, open-mindedness leads to reading more, and reading more leads to open-mindedness.

Questions to ask when evaluating and analyzing causal claims are as follows:

(1) Is a correlation really present?

(2) What are potential explanations of the correlation?

(3) Which explanation of the correlation is the best?

(4) If an explanation or cause is identified, ask: by what mechanism does this occur? In other words, how does this occur?

Consider the following causal claim:

Eating dinner together with a family leads to greater emotional health in teenagers.

• It could be that greater emotional health actually increases the likelihood and willingness of teens to eat with their families (or to encourage familial bonding).
• It could be that greater emotional health actually leads to eating with family more, and then eating with family more, in turn, contributes further to emotional well-being (a loop of cause-effect-cause).
• It could be that a third causal factor leads to both greater emotional well-being in teens and an increase in family dinners, such as a loving, stable, and supportive family (and familial life).

Evaluating Theories and Hypotheses

To evaluate a theory or hypothesis for its explanatory/theoretical virtues, use the following tools.

Simplicity : How many entities are postulated? How many types of entities are postulated? What is the nature of the entities? Are there arbitrary, unprincipled, or brute propositions or entities to which the theory or hypothesis is committed?

Also look at the amount and complexity of the claims made by the theory.

Simplicity is principally judged by ontological commitment: theories that cover the data with fewer entities postulated are (in general) more likely to be true.

As stated in another post, evaluating theories and hypotheses also requires taking into account: intrinsic probability, coherence, and explanatory scope.

Allow me to unpack these a little bit further. The greatest hypothesis or explanation is the one with the overall greatest balance of intrinsic probability and accuracy.

By “intrinsic probability” of a hypothesis, I mean the probability independent of the evidence we have for or against it. The intrinsic probability of a hypothesis is determined entirely by its modesty and coherence.

By “accuracy” of a hypothesis, I mean the degree to which a hypothesis’ predictions correspond to reality. We measure accuracy by looking at “evidence.”

By “evidence” I mean something which makes a hypothesis more probable than it would have been otherwise. Let me give you an example. Imagine you have two jars of red and blue jelly beans. In the first jar, 90% of the jellybeans are blue and the rest are red. In the second jar, 90% of the jellybeans are red and the rest are blue.

Now imagine you are handed a jelly bean from one of the jars, but you don’t know which jar it came from. If it’s a blue bean, that’s evidence it came from the first jar, not the second. The blue bean doesn’t disprove that it came from the second jar because the second jar also has blue beans. Nevertheless, it’s more likely that it came from the first because there are more blue jelly beans in the first than in the second. Similarly, if it’s a red bean, that’s evidence it came from the second jar.

Mathematicians have a formula called Bayes’ Theorem that can be used to specify the relationship between intrinsic probability, accuracy, and the overall or final probability of a hypothesis. It follows from Bayes’ Theorem that a hypothesis is probably true just in case it has a greater overall balance of intrinsic probability and explanatory power than do its alternatives collectively.

Intrinsic probability is determined by modesty and coherence. By “modesty,” I mean a measure of how much the hypothesis asserts. The more a hypothesis claims, the more ways there are for it to be false and so (before we start looking at evidence) the less likely it is to be true.

By “coherence,” I mean a measure of how well the parts of a hypothesis fit together. If the different parts count against each other, the hypothesis is less coherent and less likely to be true.

Bayesian probability assigning is all about looking at competing hypotheses and asking whether we would expect the data given the hypothesis — in which case the data is evidence for the hypothesis — or whether we would be very surprised by the data given the hypothesis — in which case the data is evidence against the hypothesis.

With all that out of the way, the series is now finished. I really hope you enjoyed it and found it both helpful and enlightening!

Author: Joe

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Analogical Argument — Definition and Examples

Analogical reasoning definition.

Analogical reasoning, also known as analogical argument or argument by analogy, suggests that if two or more things are similar in one way, they are probably similar in other ways.

Using analogies can explain or clarify an object or idea through comparison. Analogical reasoning uses analogies to persuade or make an argument.

There are typically two to three premises in an analogical argument, which ultimately identify the analogy. Those premises are then used to come to some conclusion. The analogical argument would then be structured as follows:

Analogical thinking

Analogical thinking is when someone uses information from one area to help with problem-solving in another area. Through this type of critical thinking, several everyday items made their way into production thanks to technology developed by NASA.

Companies took the technology developed for space exploration and found ways to transform it, creating real-world products widely used today. Items such as memory foam, baby formula, thermometers, invisible braces, and even the Super Soaker exist today due to analogical problem solving.

Argument by analogy examples

The first step in creating an analogical argument is to determine how the two things are similar. Then determine if those similarities support the conclusion. Consider these examples:

Analogical induction

Argument by use of analogy can be made through inductive reasoning, meaning it creates an assumption based on the identified similarities. While the premises are typically accurate, the conclusions based on the similarities may or may not be correct. The following uses analogical induction and produces a potentially inaccurate conclusion:

Deductive analogy

Analogical arguments can also be made through deductive reasoning. Through this approach, the premises lead to a correct conclusion. The following uses analogical deduction:

False analogy

Analogical arguments must be checked to make sure the conclusion is accurate. Sometimes the argument creates a false analogy where the similarities between the two things lead to an invalid conclusion. Consider the following:

Middle Way Society

An ethical approach to a better life, by integrating desires and avoiding dogmatic extremes, critical thinking 12: analogies.

Analogies are comparisons made in an argument to help prove a point. You’re arguing about one thing and you put it in the terms of another, to help people to see it in a different light. For example:

Getting into your car to drive a few hundred metres to the corner shop is as ridiculous as hopping that distance: both are clumsy, grossly inefficient, and enough to make you a laughing stock.

There are obviously some parallels here, but that doesn’t mean that the analogy is particularly successful. One reason for its lack of success may be that we tend to view inefficiency in using fuel rather differently from inefficiency in using our own bodily energy. Hopping a few hundred metres might just be seen as a good, though rather bizarre, form of useful exercise, whereas driving that distance wastes fuel – which we can more easily measure. The ridiculousness of hopping might also be exactly what makes it positive fun for some, whereas driving a car a few hundred metres would only be ‘ridiculous’ in the sense of drawing condemnation from the ecologically-minded. What looked like similarities at first turn out to be rather stretched and thin.

A well-judged analogy can be really helpful. It can help people to ‘think outside the box’ of the cognitive models they’re in the habit of using, and bring in the imagination to allow them to consider their experience in a more open way. However, it’s also very easy to dismiss a poorly-applied analogy. The problem is that there will always be dissimilarities as well as similarities between the two things being compared, so it is very easy just to latch onto the dissimilarities and use them as an excuse to dismiss the argument, if you’re a bit resistant to it in the first place. But a Middle Way approach involves trying to reach a balanced appreciation both of the similarities and the dissimilarities.

So, when you come across an analogy, it helps to think clearly about what the analogy is being used to support, and what sorts of relevant similarities and dissimilarities there are. The analogy may also need to be seen in a wider context, as there may be counter-arguments based on strong dissimilarities that just aren’t being considered. Here’s what I hope is a useful checklist:

• What is the analogy trying to show?
• Is the analogy relevant to what it is trying to show?
• What are the relevant similarities?
• What are the relevant dissimilarities?
• Are the assumptions being made about the things being compared correct?
• Are there other important dissimilarities that are not being taken into account?

Here are a couple more examples to illustrate the application of some of these questions:

Politicians in Britain today are just like African dictators, only out to get what they can from the country and squirrel it away in their offshore bank accounts. We will never get straight politicians.

This analogy is weak because the assumptions being made about British politicians are factually dubious. There may be a few cases of corruption, but these are nowhere near the scale of certain well-known corrupt African dictators (such as Mobutu in Congo). Of course, African dictators themselves are also rather varied, and some may not be particularly corrupt.

Jess has red hair and likes reading like her sister. She’ll probably become an English teacher like her sister.

Here the analogy is between Jess and her sister, but the fact of her having red hair is of no relevance to the probability of her becoming an English teacher. The fact that she likes reading is relevant, but is not strong enough by itself to support the conclusion, as lots of people who like reading do not become English teachers.

Assess the strength of these analogical arguments:

1. Cars should be restricted just as guns are, because they are lethal weapons just like guns. Cars kill and injure people just as much as guns do.

2. Motorists who kill people through reckless driving should be given a life sentence just like a murderer. The outcome is the same: a dead person.

3. More people are killed by horse-riding each year than by taking ecstasy. Ecstasy is thus less dangerous than horse-riding, and it is inconsistent to maintain horse-riding as a legal activity whilst banning ecstasy.

4. The practice of arranged marriage (as practised, for example, in Asian and Islamic cultures) is necessary to take into account young people’s lack of experience when they choose a partner. We need someone else to make this choice for us when we are inexperienced. This has been effectively admitted in Western culture when people use dating agencies and dating websites to select a partner for them, so it is hypocritical for people who use these services to criticise arranged marriage.

 Index of previous blogs in the Critical Thinking course

Picture: Nude man hopping on right foot (Edward Muybridge studies in locomotion)

3 thoughts on “ Critical Thinking 12: Analogies ”

1. A dissimilar analogy becase the analogy tries to show that cars and guns are weapons, cars are not purchased with the intention of using them as weapons. Cars and guns have no relevant similarity. the assumption that they are the same is incorrect 2. This is probably a similar analogy, it does sow there is a relevant similarity between the two outcomes from dangerous driving that maims and use of guns which murder, but it is not really a useful or relevant comparison. 3. This analogy attempts to show that horse riding and ecstacy produce the same results, but it is a dissimilar analogy. Ban the drug if it is harmful but to ban a sport such as horse riding because harm may sometimes result is a dissimilar analogy, taking ecstacy and sport are not relevant similarities 4. Here is a similar analogy, to critise an arranged marriage is hypocritical, having a partner chosen for a person or finding a partner via a dating agency can be viewed as being similar, unless marriage is forced on the couple in the arranged marriage, it is not forced from matching by a dating agency, then there is a dissimilarity between the two methods, if both partners agree with the decision made for them that would be analogous to the latter method of finding a partner. The matter of force has to be taken into account when deciding if it is similar or dissimilar.

1. This analogy has a certain strength if you feel that the deaths and injuries caused by cars is too high in proportion to the advantages they provide. Admittedly the analogy weakens when you consider that guns are specifically designed to cause injury and death and although there are a lot more road deaths a year, say in the UK, than gun deaths, there are a lot more cars. I would nevertheless feel it would be appropriate to have some further restrictions for cars such as raising the age when one can drive to 21 and having longer driving bans for drink/reckless driving.

1. This analogy goes too far as there is a significant difference in intending to kill someone and doing it by accident. However, that doesn’t mean to say that people are not responsible to a lesser or greater extent according to the circumstances of the death they have caused. This needs to be taken into account in any judicial proceedings.

3. I refer to your answer to this analogy put forward by David Knutt after being sacked by the government as drugs Zsar , which makes sense to me. “He makes a good point with the analogy, but it does have some weaknesses that he is rather reluctant to acknowledge. The main problem is that statistics about fatalities and serious injuries are not the only indicator of danger – in the case of ecstasy there could be danger of addiction or of longer-term mental or physical health effects, neither of which are the case with horse-riding.”

4. This is a fairly weak analogy as although it highlights the similarities in the two approaches in regards to affording certain advantages that go with being selective, it doesn’t take into such factors as choice, attraction, coercion, vested interests etc.

Here are my answers 1. As both Barry and Norma recognised, the problem here is that the differing intentions behind cars and guns are not taken into account. Cars also have many useful effects apart from their side-effect of occasionally killing people. So quite a weak analogy.

2. The similarity in outcome here, again, might distract us from a potentially great difference in intention. However, there can be different degrees of recklessness, from slight carelessness to a state in which you don’t care who you kill, and also different degrees of responsibility for that carelessness. At the extreme end you could make a case for treating the worst dangerous drivers like murderers, but the argument doesn’t take into account this incrementality.

3. This is, indeed, a summary of David Knutt’s argument, though with a slightly altered conclusion (Knutt was arguing about government policy). The analogy would be much stronger if it was only explicitly concerned with danger of death, rather than potentially with other types of danger.

4. I agree with Barry here. There is a similarity between arranged marriage and using a dating website in the sense that you let someone else do your matching for you. However, the circumstances are rather different. On a dating website both parties chose to use it, and are not under any pressure to accept the outcome (or even to take it at all seriously if they don’t find a good match). In contrast, traditional arranged marriages are loaded with traditional expectations, and the basis of the parents’ choice is likely to be much more socially restricted than that on a dating website. Arranged marriage should not be confused with forced marriage, but there are also degrees of social pressure on the young people concerned.

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[U08] Analogical arguments

Module: Values

• U01. Three types of values
• U02. Intrinsic values
• U03. Facts and moral values
• U04. Religion & morality
• U05. Relativism & absolutism
• U06. Reflective equilibrium
• U07. A quiz about moral relativism
• U09. Utility and rights
• U10. Types of rights

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Everyone of us holds many beliefs. By and large, we hope that our beliefs are not irrational or unreasonable. And the more strongly we hold a set of beliefs, the stronger we hope that they are not irrational. Consistency is the minimum requirement of rationality. If we have a set of beliefs which are at odds with each other, rationality demands us to revise them so that the whole set becomes internally consistent. Most of us hold some beliefs on moral/ethical issues (e.g., such as whether suicide is morally wrong, or whether someone should be denied of the opportunity of education just because of his/her sex or skin color). Consistency plays one of the most important roles in moral reasoning (indeed reasoning of any kind). So if we do hold some moral beliefs, we should at least make sure that those beliefs are consistent with each other (putting aside for the moment the further question whether those beliefs are also true).

Analogical arguments are very often used in moral reasoning (i.e., reasoning about moral issues), and they often help us think about the consistency of our moral outlook (i.e., the set of moral beliefs we hold). The basic idea of analogical arguments in moral reasoning is: treat similar case similarly. For example, suppose we are already convinced that action X (e.g., slapping a human baby for fun) is morally wrong. Also suppose we are convinced that action Y (e.g., lashing a dog for fun) is similar to X in many aspects relevant to deciding whether or not the action is morally wrong (e.g., both actions are infliction of pain on innocent beings, both are voluntary, and both are motivated by trivial self-interests, namely fun). Then, to be consistent, we should also think that Y is morally wrong to a similar extent to which X is wrong.

In short, an analogical argument in ethics employs a presumably uncontroversial case X, to which we are supposed to agree to give a certain verdict, and a controversial case Y, to which we disagree about what verdict to give. The purpose of an analogical argument is to settle the controversy of Y, by comparing it with Y (which may be either a real case or a fictional case). What an analogical argument does is (1) to show that X and Y are analogous to each other (i.e., similar in many relevant aspects), and then (2) to point out that for the sake of consistency, the controversial case Y should receive a verdict similar to the one given to the uncontroversial case X. Analogical arguments in ethics often take the following basic form:

P1. Action X is morally wrong/permissible/unblamable/right/virtuous/etc.. [Presumably uncontroversial verdict for X] P2. Actions X and Y are analogous to each other. [Analogy between X and Y] C. Action Y is morally wrong/permissible/unblamable/right/virtuous/etc.. [From P1 & P2] Or P1. State-of-affairs X is morally good/bad/neutral. [Presumably uncontroversial verdict for X] P2. States-of-affairs X and Y are analogous to each other. [Analogy between X and Y] C. State-of-affairs Y is morally good/bad/neutral. [From P1 & P2] Or P1. Object X has/lacks intrinsic (i.e., non-instrumental) value/disvalue. [Presumably uncontroversial verdict for X] P2. Objects X and Y are analogous to each other. [Analogy between X and Y] C. Object Y has/lacks intrinsic value/disvalue. [From P1 & P2]

We may call P1 the evaluative premise because it employs evaluative terms, such as "morally wrong/permissible/unblamable/right/virtuous", "morally good/bad", and "intrinsic value/disvalue". We may call P2 the analogy-stating premise because it states an analogy without trying to evaluative what are being compared by the analogy. Finally, C is the evaluative conclusion of the analogical argument.

Analogical arguments are often complex arguments, in which there are sub-arguments for P1 and P2. So in order to evaluate a complex analogical arguments, we will need to evaluate all its sub-arguments. Before looking at some concrete examples, we should bear in mind that there are at least two ways in which an analogical argument can be challenged.

(1) An analogical argument can be challenged most directly by questioning the analogy-stating premise P2 - i.e., questioning whether the analogy drawn between X and Y is a good one. If P2 is unacceptable, then the argument fails to give good reasons for accepting conclusion C. The criteria for assessing an analogy includes (a) whether the aspects of the two cases being compared are relevant to the questions being asked about them (e.g., whether they are morally wrong), (b) whether the number of relevant similarities between the two cases are enough so that the conclusion drawn for the one case should be similar to that for the other case, and (c) whether there are relevant differences between the two cases which outweigh the relevant similarities between them so that the conclusion drawn for the one case should be different from that for the other.

(2) Even if premise P2 is acceptable, the argument can still be challenged by questioning the evaluative premise P1 - i.e., questioning the evaluative verdict for the presumably uncontroversial case. If it could be shown that P1 is actually doubtful, then the C is also doubtful.

§1. An example

Let us now look at a concrete example of an analogical argument in ethics (in this case, animal ethics).

Consider the follows passage:

The relationship between small children and their parents is just like the relationship between dogs and their owners. When a dog, and likewise a small child, wants something badly and does not get it, he or she will repeatedly make demands in ways that often irritate those who take care of them. For instance, when a dog wants a walk but the owner is busy doing something else, the dog will bark and bark and bark nonstop so that the owner gets the message and the dog eventually gets a walk. Likewise, when a small child wants a toy but the parent refuses to buy, the child will repeat his of her demands nonstop by whinging, crying and yelling so the parent gets the message and the child eventually gets the toy. The worse thing in both cases is that dogs, and likewise small children, are not creatures that adults can reason with: they simply don't listen or understand. That is why dog owners are often irritated by and get angry with their dogs in just the same way that parents are often irritated by and get angry with their small children. Furthermore, dogs and likewise small children cannot really tell what is right from what is wrong. It is dog owners and parents who are morally (as well as legally) responsible for the bad behaviour of their dogs and small children, respectively, if that causes damage to the interests of other people. Finally, the worst thing that could happen to a dog, and likewise a small child, is for it to be ignored: it is better to be cruel to it than to be totally indifferent to it as if it doesn't exist. For the reasons given above, it is morally permissible for dog owners to beat their dogs if that is necessary for disciplining them. Therefore, likewise, it is morally permissible for parents to beat their small children if that is necessary for disciplining them.

The above is a complex analogical argument, the basic structure of which is as follows.

P1. It is morally permissible for dog owners to beat their dogs if that is necessary for disciplining them. [Evaluative premise, presumably acceptable] P2. The relationship between dogs and their owners is just like the relationship between small children and their parents. [Analogy-stating premise] C. It is morally permissible for parents to beat their small children if that is necessary for disciplining them. [From P1 & P2]

Is the above analogical argument good? You need to do at least two things in order to answer the question.

(a) Decide whether the evaluative premise P1 is acceptable. In order to do that, it will be help to write out and evaluate the sub-argument for P1.

(b) Do the same with the analogy-stating premise P2.

Important Reflections : The demand of consistency tells us nothing about which ones of our inconsistent beliefs should be rid of. So an analogy (e.g., P2) can be used to argue for a claim (e.g., C) as well as for the opposite claim (e.g., not-C). For example, someone, who disapprove of parents beating their small children, can actually employ the same analogy to condemn dog owners beating their dogs in the following way:

P1*. It is morally impermissible (i.e., morally wrong) for parents to beat their small children even if that is necessary for disciplining them. [Evaluative premise, not-C] P2*. The relationship between dogs and their owners is just like the relationship between small children and their parents. [Analogy-stating premise, P2] C*. It is morally impermissible for dog owners to beat their dogs even if that is necessary for disciplining them. [not-P1, from P1* & P2*]

Suppose the analogy (i.e., P2 or P2*) is correct. Consistency only demands us to hold either (a) that beating a small child is morally as permissible as beating a dog or, on the other side of the same coin, (b) that beating a dog is morally as impermissible as beating a small child.

In other words, we should hold either both P1 and C on the one hand or both P1* ( not-C) and C* ( not-P1) on the other hand. But the demand of consistency does not tell us which option to take. So what do we do? Answer: we compare the acceptability of evaluative premise P1 and the acceptability of evaluative premise P1* - i.e., we write out and evaluate the sub-arguments for P1 and P1* respectively. If P1 is more acceptable than P1*, then so is conclusion C more acceptable than conclusion C*, and vice versa.

In the light of the above considerations of the logic of analogical arguments in ethics, one useful tip is as follows:

If you want to construct a good analogical argument, then you need to (1) give a good sub-argument for your analogy-stating premise, and (2) make sure that your evaluative premise is more acceptable than the negation of your (evaluative) conclusion - otherwise, your opponents would have good reasons to reject your conclusion, which, by virtue of consistent, will further lead to the rejection of your evaluative premise.

In short, an analogy can be used to settle a controversial case in ethics, but, on the other side of the same coin, it can also be used to unsettle a case which is originally thought to be uncontroversial. Considerations independent of the analogy (e.g., sub-arguments for P1 and not-C, respectively) are needed in deciding whether the analogy helps to settle or unsettles.

§2. Racism/Sexism & Speciesism (An Argument for Animal Liberation)

Consider the following passage:

Racism is the view that the interests (e.g., health, pleasure, desire/preference-satisfaction) of members of one race (e.g., white people) are morally more significant than the interests of members of other races (e.g., colour people) just because the former are the interests of people belonging to a particular racial group whereas the latter are not, so that it is morally justified to promoted/protected the former in expense of the latter whenever the two are in conflict each other. An example of racism is that white babies receive better medical treatments than black babies from government hospitals simply because the former belong to, but the latter do not belong to, the white race. Sexism is the view that the interests members of one sex (e.g., male human beings) are morally more significant than the interests of members of another sex (e.g., female human beings) just because the former are the interests of people belonging to a particular sex whereas the latter are not, so that it is morally justified to promoted/protected the former in expense of the latter whenever the two are in conflict each other. An example of sexism is that boys receive, but girls do not receive, school education simply because the former are male whereas the latter are not. Speciesism, which has the same structure as racism and sexism, is the view that the interests of members of one species (e.g., Homo sapiens or, in other words, human beings) are morally more significant than the interests of members of other species (e.g., non-human beings) just because the former are the interests of beings belonging to a particular species whereas the latter are not, so that it is morally justified to promoted/protected the former in expense of the latter whenever the two are in conflict each other. An example is speciesism is that nonhuman animals (e.g., mice, rabbits, dogs, chimpanzees, ...etc.) are used in medical experiments (which cause them immense suffering) but human beings (e.g., human infants) are not used in such experiments simply because the former are do not, whereas the latter do, belong to the species of Homo sapiens. Now, racism and sexism are both morally unjustified. For how important an interest is (e.g., the interest in receiving medical treatment) does not depend on the skin-colour or the sex of the person to whom the interest belong. Black babies have just the same interest in receiving good medical treatment as white babies. Girls have just the same interest in receiving education as boys. Likewise, a dog has just the same interest in avoiding pain (of same intensity and duration) as a human infant. Pain is pain no matter whose pain it is. So the interest in avoiding pain is same for everyone who is capable of suffering pain. In short, species-membership is just as arbitrary a discriminating factor as skin colour and sex in deciding how important an interest is. Hence, (to be consistent, we should conclusion that) speciesism is as morally unjustified as racism and sexism.

P1. Racism (or sexism) is morally unjustified. [Evaluative premise, presumably acceptable] P2. Speciesism is analogous to racism (or sexism). [Analogy-stating premise] C. Speciesism is morally unjustified. [From P1 & P2]

P1*. Speciesism is morally justified. [Evaluative premise, not-C] P2*. Speciesism is analogous to racism (or sexism). [Analogy-stating premise, P2] C*. Racism (or sexism) is morally justified. [not-P1, from P1* & P2*]

Suppose the analogy (between speciesism on the one hand and racism/sexism on the other hand, i.e., P2 or P2*) is correct, consistency only demands us to hold either (a) that speciesism is morally as unjustified as racism/sexism or, on the other side of the same coin, (b) that speciesism is morally as justified as racism/sexism. In other words, we hold either both P1 and C on the one hand or both P1* ( not-C) and C* ( not-P1) on the other hand. But the demand of consistency does not tell us which option to take.

(a) State all the sub-arguments that you can think of for P1 and P1* respectively in the above two analogical arguments.

(b) Do you think the analogy-stating premise P2 (or P2*) above is true? Give reasons for your answer.

(c) Do you think speciesism is morally unjustified? Give reasons for your answer.

§3. The Violinist & The Fetus (An Argument Defending Abortion)

We grant that the fetus is a person from the moment of conception. How does the argument [against abortion] go from here? Something like this, I take it. Every person has a right to life. So the fetus has a right to life. No doubt the mother has a right to decide what shall happen in and to her body; everyone would grant that. But surely a person's right to life is stronger and more stringent than the mother's right to decide what happens in and to her body , and so outweighs it. So the fetus may not be killed; and abortion may not be performed. But now let me ask you to imagine this. You wake up in the morning and find yourself back to back in bed with an unconscious famous violinist. He has been found to have a fatal kidney ailment, and the Society of Music Lovers has [...] kidnapped you, and last night the violinist's circulatory system was plugged into yours so that your kidneys could be used to extract poisons from his blood as well as your own. The director of the hospital now tells you: "Look, we're sorry the Society of Lovers Lovers did this to you - we would never have permitted it if we had known. But still, they did it and the violinist now is plugged into you. To unplug you would be to kill him. But never mind, it's only for nine months. By then he will have recovered from his ailment and can safely be unplugged from you." Is it morally incumbent on you to accede to this situation? No doubt it would be very nice of you if you did, a great kindness. But do you have to accede to it? [... The story continues:] There you are, in bed with the violinist, and the director of the hospital says to you: "It's all most distressing, and I deeply sympathize, but you see this is putting an additional strain on your kidneys, and you'll be dead within the month. But you have to stay where you are all the same, because unplugging you would be directly killing an innocent violinist, and that's murder, and that's impermissible." If anything in the world is true, it is that you do not commit murder, you do not do what is impermissible, if you reach around to your back and unplug yourself from that violinist to save your life. (From J. J. Thomson, "A Defense of Abortion", Philosophy and Public Affairs 1, no. 1 (1971)).

The above analogical argument has the following form:

P1. It is morally permissible for agent A to perform action X under circumstances C. P2. Agent B performing action Y under circumstances D is analogous to agent A performing action X under circumstances C. C. It is morally permissible for agent B to perform action Y under circumstances D. [From P1 & P2]

In the above analogical argument: Action X is unplugging the person (who has been plugged to the violinist) from the violinist. Agent A is the person who has been plugged to the violinist. Circumstances C is (1) that the person did not volunteer to be plugged to the violinist - he or she was kidnapped, and (2) the person will die unless he or she is unplugged from the violinist.

Answer the following questions:

• What are action Y, agent B, and circumstances D?
• Rewrite the argument according to your answers to (b).
• Do you think premise P1 is true? Give reasons for your answer.
• Do you think premise P2 is true? Give reasons for your answer.
• Do you think conclusion C is true? Give reasons for your answer.

Imagine the following situation: Some tapeworms are intelligent creatures having individual personalities just like you and me (i.e., they are persons), and raw fish (sashimi) is well know as a carrier of the eggs of those intelligent tapeworms. You have eaten some raw fish carrying those tapeworm eggs, which will develop into huge intelligent tapeworms inside your body in several month. There is a drug which, if you take it, can kill the eggs so that they will not develop in your body.

(a) Do you think it is morally permissible for you to take the drug? Give reasons for your answer.

(b) If your answers for (a) is yes, then do you also think that the use of the morning-after pill is morally permissible? Give reasons for your answer.

(c) Use the fictional case above to formulate an analogical argument supporting the use of morning-after pill.

§4. Animal Sexual Behaviour & Human Sexual Behaviour (An Argument Defending Human Sexual Infidelity)

Consider the follows passage: We cannot blame men or women for sexual infidelity. Recent research has shown that animals cheat sexually all the time. Why should the situation be any different with humans? Female animals look for males who show signs of biological fitness, and will mate with such a male even if they are already bonded with another male. The same is also true of male animals. Just so it is with humans. As we should not blame animals for sexual infidelity, neither should we blame human beings for sexual infidelity.

(a) Rewrite the above argument so that it contains an evaluative premise, an analogy-stating premise, and an evaluative conclusion.

(b) Do you think the analogy-stating premise is true? Give reasons for your answer.

(c) Do you think the evaluative premise is true? Give reasons for your answer.

(d) Do you think the evaluative conclusion is acceptable? Give reasons for your answer.

(e) If you find any fault in the above argument other than those that you have pointed out in your answers to (b), (c), and (d), please say what it is.

§5. An argument from Aristotle

Consider the follows passage from Aristotle:

It is clear that the rule of the soul over the body, and of the mind and the rational element over the passionate, is natural and expedient; whereas the equality of the two or the rule of the inferior is always hurtful. the same holds good for animals in relation to men; for tame animals have a better nature than wild, and all tame animals are better off when they are ruled by man; for then they are preserved. Again, the male is by nature superior, and the female inferior; and the one rules, and the other is ruled; this principle of necessity extends to all mankind. Where then there is such a difference as that between soul and body, or between men and animals (as is the case of those business it is to use their body, and who can do nothing better), the lower sort are by nature slaves, and it is better for them as for all inferiors that they should be under the rule of a master. For he who can be, and therefore is, another's and he who participates in rational principle enough to apprehend, but not to have such a principle, is a slave by nature. Whereas the lower animals cannot even apprehend such a principle; they obey their instinct. And indeed the use made of slaves and of tame animals is not very different; for both with their bodies minister to the needs of life. [...] It is clear, then, that some men are by nature free and others slaves, and that for these latter slavery is both expedient and right. (Aristotle, Politics, book 1, chapters 4-5 (emphases added))

The above passage from Aristotle actually contains three analogical arguments for three conclusions, namely (C1) it is morally justified for human beings (especially the male ones) to dominate nonhuman animals, (C2) it is morally justified for men to dominate women, and (C3) it is morally justified for those who are more rational to be masters and those who are less rational to be slaves.

If C3 is true and if there is a standard and a corresponding test for rationality, then people who fail the test should all be the slaves of those who pass the test. Next, if C2 is true, then no woman should be allowed to hold a position (e.g., in business institutes, in the government, or whatever) which is more senior than some position held by a man. Finally, if C1 is true, then is nothing wrong human beings to use nonhuman animals in ways that are conducive to human interests (e.g., confining them in zoos, doing experiments on them, or hunting them for recreation).

(a). Which group of beings in the world do you think is most benefited by Aristotle's three conclusions taken together?

(b) Rewrite Aristotle's argument for C3 so that it contains an evaluative premise, an analogy-stating premise, and C3 as the evaluative conclusion.

(c) Rewrite Aristotle's argument for C2 so that it contains an evaluative premise, an analogy-stating premise, and C2 as the evaluative conclusion.

(d) Rewrite Aristotle's argument for C1 so that it contains an evaluative premise, an analogy-stating premise, and C1 as the evaluative conclusion Do you think the analogy-stating premise is true? Give reasons for your answer.

(e) Evaluate the rewritten argument in (b).

(f) Evaluate the rewritten argument in (c).

(g) Evaluate the rewritten argument in (d).

§6. Artifacts & Restored Natural Entities (An Argument Devaluing Restored Natural Entities)

A natural area (e.g., a rainforest) may be damaged by human activity or natural disaster (e.g., open-pit mining or natural fire). A damaged natural area may be repaired or restored (e.g., by reintroducing to the area plant and animal species which were original there) so that it appears quite similar to how it looks like before the damage. We call this resulting area a "restored natural area". Now, consider the following passage:

When humans modify a natural area they create an artifact, a product of human labour and human design. This restored natural area might resemble a wild and unmodified natural system, but it is, in actuality, a product of human thought [i.e., human intention], the result of human desire and interests. The ethical importance of the distinction between artifacts and natural entities is thus derived from the anthropocentric nature of artifacts, their ontological reliance on human interests, plans and projects. In contrast to natural entities, artifacts, as human instruments, are always a means to the furtherance of some human end. [...] If the categorical imperative is applied to a treatment of artifacts and natural entities, we find a crucial difference: artifacts must be treated as means, for their existence and value only exist in a dependent relationship with human aims and goals; but natural entities, existing apart from human projects, can be considered as ends-in-themselves. (From E. Katz, Nature As Subject: Human Obligation and Natural Community, Lanham, MD.: Rowman & Littlefield.)

The analogical argument above can be reformulated as follows.

P1. Artifact have instrumental value but lack intrinsic value. P2. Restored natural areas analogous to artifacts. C. Restored natural areas have instrumental value but lack intrinsic value.

(a) Do you think the evaluative premise P1 is acceptable? Give reasons for your answer.

(b) Do you think the analogy-stating premise P2 is acceptable? Give reasons for your answer.

(c) Do you think the evaluative conclusion is acceptable? Give reasons for your answer.

Hint: You may consult: Yeuk-Sze Lo, "Natural and Artifactual: Restored Nature as Subjects", Environmental Ethics 21(1999): 247-66.

Written by Norva Lo

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50 Examples Of Analogies For Critical Thinking

By forcing students to distill one relationship in order to understand another, it’s almost impossible to solve analogies without understanding.

What Are The Best Examples Of Analogies For Critical Thinking?

by Terry Heick

In our guide to teaching with analogies , we offered ideas, definitions, categories, and examples of analogies.

This post is a more specific version of that article where we focus specifically on types and examples of analogies rather than looking at teaching with analogies more broadly. Below, we offer more than 20 different types of analogies and examples of type of analogy as well–which results in nearly 100 examples of analogies overall.

Note that because an analogy is simply a pattern established by the nature of a relationship between two ‘things,’ there are an infinite number of kinds of analogies. You could, for example, set up an analogy by pairing two objects only loosely connected–brick and road, for example: a brick is to a road as…

Of course, analogies are best solved by creating a sentence that accurately captures the ‘truest and best’ essence of the relationship of the first two items in the analogy. So in the above brick/road example, you might say that ‘bricks used to be used to create roads,’ at which point all kinds of possibilities emerge: Bricks used to be used to create roads as glass used to be used to create bottles, yielding the analogy:

Bricks : Road :: Glass : Bottle

You could also use this in a specific content area–Social Studies, for example:

Bricks : Road :: Pamphlets : Propaganda

Language Arts?

Bricks : Roads :: Couplets : Sonnets? Maybe, but this leaves out the critical ‘used to be…’ bit.

You get the idea. By forcing students to distill one relationship in order to understand another, it’s almost impossible to accurately solve analogies without at least some kind of understanding–unless you use multiple-choice, in which case a lucky guess could do the trick.

Now, that’s a purposely far-fetched example. In most teaching and learning circumstances like courses and classrooms, analogies are used in common forms that are more or less obvious: part to whole, cause and effect, synonym and antonym, etc. This makes them less subjective and creative and easier to score on a multiple-choice question and can reduce the subjectivity of actually nailing down the uncertain relationship between ‘bricks’ and ‘roads.’ It becomes much easier when you use something with a more clear relationship, like ‘sapling is to tree as zygote is to…’

Of course, this misses the genius of analogies: asking students to see–and sometimes even create–the relationship between things rather than ‘choosing’ the ‘type’ of analogy. Analogies are brilliant teaching and learning tools that we use all of the time in everyday life to explain something by explaining something else .

(If you’d like to read more about this idea, I discussed it in some in ‘ Why Questions Are More Important Than Answers .’) And it’s at this point that it feels like exploring how to use analogies for critical thinking might be more interesting than merely offering types of analogies, but for the sake of packaging and time, I’ll finish this post and re-address the ‘analogies for critical thinking’ bit later.

Why Analogies Are Valuable For Learning

In the guide to teaching with analogies shown above, I explained that, “Academic analogies are useful for teaching and learning because they require students to analyze a thing (or things), and then transfer that analysis that analysis to another thing. This kind of transfer requires at least some kind of conceptual grasp–understanding.”

I went on to offer that “This makes them useful for assessment, but they can also be used as an effective learning strategy as well. As students create incorrect analogies, analyze the relationships their analogies are suggesting, and then correct them accordingly, students are grappling with ideas, monitoring and revising their thinking, and otherwise actively consider the often complex relationships between disparate things.”

In fact, we’ve begun using analogies in our TeachThought University courses. They’re genius little tools to both cause and measure understanding. And while there are some common types of analogies that you (and students) will see most commonly, (antonyms, categories, part to whole, cause and effect, etc.), the truth is that unless two objects or ideas represent an entirely unique circumstance that doesn’t exist anywhere else in the universe, there should always be an analogous pairing or counterpart somewhere. They even function strongly as psychology-based critical thinking strategies .

Put another way, there are nearly an infinite number of analogies and an uncountable number of types of analogies. Consider the following:

Father : Pops :: Henry VIII : ______?

Tissue: Kleenex :: ______ : ______?

You could call these ‘slang’ analogies but the latter isn’t really slang. You could say also call them ‘more commonly known as’ analogies or even synonyms but that’s entirely the essence of the relationship either. This is a unique relationship–as so many are. You get the point: That there are an impossible number of things and relationships so there aren’t a set number of ‘types of analogies.’

The question for you, as a teacher, is which are the most helpful for you to cause and measure understanding with students? So for now, we’ve included the most common types of analogies and then added in some less common but still useful types of analogies. We’ve tried to make some simple and some more complex just to demonstrate the range and value of analogies in critical thinking.

Some, I’ve added commentary to. Others, I just included the examples. The general pattern I’ve used is to start with a simple example and then create a more complex analogy.

Note, there may be some disagreement about some of the ‘answers’ here–either from you as a reader or your students. That’s good! If your students are arguing that democracies aren’t actually the ‘opposite’ of a dictatorship, that means they likely at least vaguely grasp each and arguing about the similarities and differences!

What more can you ask for in introducing or reviewing content?

50+ Examples Of Analogies For Critical Thinking

1. Synonym Analogies

Funny : Humorous :: Hardworking : Diligent

Lead : Guide :: Drawing : Illustration

Mom : Mother :: Dog : _______

Beginner : Novice :: Law : ______

2. Antonym Analogies

Night : Day :: Right : Left

Wet : Dry : Hot : Cold

Open: Closed :: Free : ______

Empiricism : ______ :: Small : Big

3. Part/Whole Analogies

Electron: Molecule :: Country : Continent

Toe : Foot :: Finger : Hand

Stars : Galaxy :: Molecules : Object

Data : Scientific Process :: Thesis Statement : ______

4. Cause/Effect Analogies

Spin : Dizzy :: Jump : Elevate

Honesty: Trust :: Light : Plant Growth

Itch : Scratch :: Virus : Cold

Read : Learn :: Try : Improve

Rise of Social Messaging : Demise of Email :: __________: French Revolution

Writing Process : Idea Organization :: Eye Contact : ______

5. Thing/Function Analogies

Broom : Sweep :: Paintbrush : Paint

Freezer: Freeze :: Paper Towel : Wipe

Martin Luther King Jr.’s “I Have A Dream” : Civil Rights :: ________ : LBGTQ rights

Gravity : Pull :: Conflict : ______

6. Thing/Characteristic Analogies

Democracy : Equality :: Monarchy : One Ruler

Water : Wet :: Concrete : Hard

Mountain : Tall :: Desert : Dry

Sugar : Sweet :: Cheetah : Fast

Water : Wet :: Circles : ______

Wall : Bricks :: Sonnet : Lines

Mountain : Tall :: Globalization : _____

7. Thing/Context Analogies (similar to Part/Whole and other categories of analogies)

Cello : Symphony :: Creek : Forest

Plane : Hangar :: Surfboard : Beach

Field : Farm :: Audience : Arena

Conflict : Story :: Emoji : Texting

8. Example/Type Analogies

Orange : Fruit :: Hydrogen : Element

Kangaroo : Marsupial : Dog :: Mammal

Ford Mustang : Muscle Car :: Subaru BRZ : Sports Car

Deontology : Ethics: Cubism : Art

______ : Immigration Policy :: iPhone : Smartphone

9. Category/Subcategory Analogies

Phylum : Kingdom :: Aisle : Department

Sonnet: Petrarchan Sonnet :: Rose : Red Rose

Shape : Quadrilateral :: _______ : Orbit

10. Object/Classification Analogies

Bowl : Dish :: Sword : Weapon

Cat : Feline :: Dog : Canine

Porsche 911 : Sports Car :: Alligator : Reptile

Rain : Precipitation :: ______ : Rhyme Scheme

12. Fact/Opinion Analogies

Wet : Soaked :: 7 Seconds : Fast

6′ 4″ : Tall :: Awake : Smart

It’s 93 degrees : It’s Hot :: ________ :

13. Step/Process Analogies

Evaporation : Water Cycle :: ______ : Evolution

Test Theory : Scientific Process :: Stir : Make Chocolate Milk

Revision : Writing Process :: ______ :

______ : Sentence Diagramming

14. Problem/Solution Analogies

Tape : Paper Tear :: Knee Scrape : Bandaid

Alliteracy : Habits :: Lack of Cardiovascular Endurance : Lack of Exercise

Climate Change : Reduce Greenhouse Gases :: ________ : Poverty

15. Symbol/Referent Analogies

Peace Sign : Hippies :: Red Cross: Medical Professional

To make that a bit more complex, consider Peace Sign : Vietnam :: _____ : ______ where it could be seen that rather just “The Peace Sign characterized Hippies as…” you instead of “The Peace Sign was seen as a counter-symbol to Vietnam as…”, and so on. The following example is equally complex:

Guillotine: French Revolution :: Faulkner’s use of setting in A Rose for Emily : ________

To answer that, you’d have to know whether or not it was commonly considered for the guillotine to ‘represent’ the Fresh Revolution and then further, exactly how Faulkner used setting in ‘A Rose for Emily.’

16. Producer/Product Analogies

Sheep : Wool :: Milk : Cow

As with others, the first analogy is simple:

Cow : Milk :: Beehive : Honey

The second sets up at simple (Producer/Product) but the second part asks the student to think (and know) more:

Cow: Milk :: Industrialism : _____

Clearly, these can be subjective but if you use this to your advantage (in a debate or discussion, or by asking the student to defend their choices, for example) that’s a good thing. You can also use a multiple-choice format to reduce some of this subjectivity if you need thinks nice and tidy in a lesson or assessment.

17. Noun/Adjective Analogies

Lemon : Yellow :: Snow : White

Flamingo : Pink :: Rhinoceros : Grey

Cardinal : Red :: Irony : ______

18. Task/Subtask Analogies

Kick : Soccer :: Dribble : Basketball

Plan : Prioritize :: Lead : Communicate

Drive : Steer :: Live : Breathe

19. Kinds Of Measurement Analogies

Vegetable Harvest : Bushels :: Liquid : Gallon

Geometric Shape : Degrees :: Marine Distance : Nautical Miles

City : Blocks :: Farms : Acres

20. Finish the Set or Sequence Analogies

Salt : Pepper :: Peas : Carrots

2 : 8 :: 5 : 20

21. Strength & Weakness Analogies

Lighthouse : Brightness :: Flashlight : Portability

Abundant Supply : Solar Energy :: Low Cost : Coal

Potential Profits : Capitalism :: ______ : Artificial Intelligence

Plastic : Pollution :: Greed : ______

22. Spatial Relationship (e.g., Geography) Analogies

South America : North America :: Ireland : ______

Floor : Ceiling :: Conclusion : Introduction

Peanut Butter : Bread :: Chapters : Book Covers

23. Increasing or Decreasing Intensity Analogies

Cool : Cold :: Warm : Hot

Aggressive : Fierce :: Amused : Elated

Instability : Turmoil :: Change : Revolution

Speed of Sound : Speed of Light :: ______ : Gammar Ray Bursts

24. Thing/Group Analogies (similar to Part/Whole Analogies)

Fish : School :: Lion : Pride

Flock : Birds :: Pack : Wolves

People : Community :: Tree : Forest

25. Rhyme Analogies

Jump : Bump :: Wire : Fire

Ship : Blip :: Stop : Lop

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Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).

IARML@IJCAI’2024: Workshop on the Interactions between Analogical Reasoning and Machine Learning, at IJCAI’2024, August, 2024, Jeju, South Korea

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[1]Equal contribution. \cortext [1]Corresponding authors.

Enhancing Analogical Reasoning in the Abstraction and Reasoning Corpus via Model-Based RL

This paper demonstrates that model-based reinforcement learning (model-based RL) is a suitable approach for the task of analogical reasoning. We hypothesize that model-based RL can solve analogical reasoning tasks more efficiently through the creation of internal models. To test this, we compared DreamerV3, a model-based RL method, with Proximal Policy Optimization, a model-free RL method, on the Abstraction and Reasoning Corpus (ARC) tasks. Our results indicate that model-based RL not only outperforms model-free RL in learning and generalizing from single tasks but also shows significant advantages in reasoning across similar tasks.

1 Introduction

The Abstraction and Reasoning Corpus (ARC) 1 1 1 ARC dataset URL: https://github.com/fchollet/ARC-AGI is known as a benchmark for evaluating abstraction and reasoning abilities  [ 1 ] . Examples of ARC tasks can be seen in Fig.  1 . To solve an ARC task, one needs the ability to abstract a common rule among given demos and apply this rule to a new test input grid to infer the corresponding grid  [ 2 ] . Additionally, to solve untrained tasks, one must have the ability to discover various knowledge during the learning process and selectively apply this knowledge in the task-solving process  [ 2 ] .

This study interprets the two abilities required by ARC not as separate skills but as one concept of analogical reasoning. Analogical reasoning refers to the process of understanding new situations or solving problems based on existing knowledge that can be applied to different contexts or problems  [ 3 ] . First, the process of solving an ARC task can be seen as solving a new problem (test input grid) based on the transformations from input grids to output grids in each demo (existing knowledge), which exactly fits the definition described above. Second, for a model trained on various ARC tasks to solve a new ARC task, it must understand the demos of the new ARC task (new situations) based on the solution processes (existing knowledge) applicable to the training tasks (different problems) and solve the task, which also aligns with the concept of analogical reasoning. Third, since the evaluation tasks in ARC are typically more complex and challenging than the training tasks  [ 4 ] , the ability to infer new problem-solving processes based on learned knowledge through analogical reasoning can enhance the performance on ARC.

In this study, noting the critical importance of analogical reasoning capabilities in solving ARC, we conducted reinforcement learning (RL) for training ARC tasks. Learning ARC with RL can offer the following potential benefits from an analogical reasoning perspective. First, RL can enhance its learning strategies through reward-based learning. In the process of learning ARC tasks, RL agents learn to maximize rewards across various tasks, thereby developing analogical reasoning capabilities. Second, RL can learn policies that effectively operate in new, yet similar environments based on experience. During the learning of various ARC tasks, RL agents can learn to apply rules learned in one scenario to another, a key element of analogical reasoning. Third, RL allows agents to produce their own data, enabling effective learning even without high-quality given data. In tasks like ARC, where data may be limited, RL agents can quickly infer patterns from a few demos and learn to apply these to the new input.

This study compared model-based RL and model-free RL to evaluate the efficiency of acquiring analogical reasoning abilities. The key difference between these two methodologies lies in the presence or absence of an internal environmental model. Model-based RL builds an internal model of the environment based on the agent’s experiences and learns policies through predictable scenarios, which has the potential to significantly improve decision-making in complex situations. On the other hand, model-free RL learns to optimize rewards through direct interaction with the environment, allowing for the development of effective action policies without constructing an environmental model.

This study hypothesized that the internal model of model-based RL could aid in analogical reasoning. To test this hypothesis, we observed the learning processes of ARC tasks using DreamerV3  [ 5 ] , a representative model-based RL algorithm, and Proximal Policy Optimization (PPO)  [ 6 ] , a general model-free RL algorithm, demonstrating the potential for RL agents to acquire analogical reasoning. Additionally, we evaluated the performance and learning efficiency of both algorithms by fine-tuning them on tasks similar to the pre-trained tasks and comparing the analogical reasoning capabilities they acquired.

2 Model-Based RL and Model-Free RL

RL has two main categories: model-based RL and model-free RL. Before conducting experiments to compare these categories from the perspective of abstraction reasoning, this paper briefly reviews the characteristics of each category and compares the algorithms that will be used in this experiment.

2.1 Model-Based RL

Model-based RL builds an internal model of the environment to predict various possibilities and establish more efficient policies based on these predictions. In problems requiring abstract thinking and pattern recognition like ARC, the model-based RL approach provides the ability to generalize to new situations that the agent has not experienced through its internal model. Representative algorithms utilizing the model-based RL approach include Dyna-Q  [ 7 ] , I2A  [ 8 ] , MBMF  [ 9 ] , and World Model  [ 10 ] . Also, AlphaZero  [ 11 ] , which successfully surpassed human performance in board games such as Go, Chess, and Shogi based on Monte Carlo Tree Search (MCTS), is the most well-known. Recently, DreamerV3  [ 12 ] , based on the World Model, has been the subject of extensive research.

A key feature of DreamerV3 is its ability to extract important features from input data and convert these into a latent representation, which is then used to learn a predictive model of the environment’s dynamics. This model functions by taking the current state and actions as inputs and predicting the next state’s representation and rewards. This approach allows for the simulation of future scenarios and the evaluation of various scenarios. In summary, DreamerV3 operates by repeating the following three processes: 1) collect data based on a random policy or the current policy, 2) update the agent’s model based on the collected data, and 3) generate and evaluate virtual future scenarios using the updated model to improve the policy. This process excels in multi-step tasks like mining diamonds in Minecraft. We chose DreamerV3 for its superior handling of ARC, compared to other model-based approaches.

2.2 Model-Free RL

Model-free RL operates directly through interactions with the external environment, basing its action policies on reward information. This methodology utilizes only the agent’s experiences without prior modeling of the environment, making it relatively simple to implement and fast to execute. These characteristics could constrain the flexible reasoning and rapid adaptation required in ARC. Representative algorithms of model-free RL include DQN  [ 13 ] , PPO  [ 14 ] , and SAC  [ 15 ] .

Among these, the Proximal Policy Optimization (PPO) algorithm is one of the most commonly used algorithms in RL, based on the policy gradient method. PPO is designed to achieve stable and efficient learning even in complex environments. Its process can be summarized similarly to DreamerV3 in the following two steps: 1) perform actions in the environment based on the current policy to collect data, and 2) incrementally improve the policy using the policy gradient technique based on the collected data.

3 Experiments

We conducted an experiment comparing the ARC task learning of DreamerV3 and PPO, which are representative algorithms of model-based RL and model-free RL, respectively, to demonstrate the differences in analogical reasoning capabilities between these methodologies. Through this experiment, we aimed to answer the following research questions (RQs):

Can Model-Based RL Learn a Single Task? ( 𝒜 → 𝒜 → 𝒜 𝒜 \mathcal{A}\rightarrow\mathcal{A} caligraphic_A → caligraphic_A )

Can Model-Based RL Reason about Tasks Similar to Pre-Trained Task? ( 𝒜 → 𝒜 ′ → 𝒜 superscript 𝒜 ′ \mathcal{A}\rightarrow\mathcal{A^{\prime}} caligraphic_A → caligraphic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )

Can Model-Based RL Reason about Sub-Tasks of Pre-Trained Task? ( 𝒜 ⁢ ℬ → 𝒜 → 𝒜 ℬ 𝒜 \mathcal{AB}\rightarrow\mathcal{A} caligraphic_A caligraphic_B → caligraphic_A )

• 𝒜 ℬ 𝒜 \mathcal{A,B}\rightarrow\mathcal{A} caligraphic_A , caligraphic_B → caligraphic_A )
• 𝒜 ℬ 𝒜 ℬ \mathcal{A,B}\rightarrow\mathcal{AB} caligraphic_A , caligraphic_B → caligraphic_A caligraphic_B )

We scope our paper to provide answers on RQ1 and RQ2. By addressing RQ1 and RQ2 through experiments, we aim to demonstrate the abilities of model-based RL in efficiently learning and applying knowledge across similar tasks. The following descriptions are the common setting of every experiment.

The vast action space of ARC is one of the biggest obstacles to the RL  [ 2 ] . According to the ARCLE framework  [ 16 ] , action is defined as a combination of operation and selection; where operation means the type of action to be performed, and selection determines grids to which these actions are applied. We force strict restrictions on these operations and selections to weaken the challenge posed by the vast action space. In experiments, RL agents can select only five operations: Rotate90, Rotate270, FlipH, FlipV, and Submit  [ 16 ] , and selections are always fixed as the entire grid. These limitations influenced the selection of tasks.

We found tasks from among the 400 training tasks that could be solved with the restricted operations and selections of actions, and 7 tasks satisfied this condition. Among these tasks, we selected 4 tasks for our experiment as shown in Fig.  1 . First, we selected tasks that need diagonal flipping on a 3 × 3 3 3 3\times 3 3 × 3 grid or N × N 𝑁 𝑁 N\times N italic_N × italic_N grids. These tasks need the right combination of actions (rotate and flip) before submission. Thus, we thought these tasks were good for measuring analogical reasoning ability. Next, we chose tasks that need to rotate a 3 × 3 3 3 3\times 3 3 × 3 grid and flip horizontally on N × N 𝑁 𝑁 N\times N italic_N × italic_N grids. These tasks need only one action except submit, so analogical reasoning ability are not necessary for agents.

We designed the reward based on a sparse reward such that an agent gets a reward of 1000 if it does the submit action and the state matches the correct grid. Additionally, even if the submit action is not executed, reaching the correct grid grants an extra reward of 1. Although sparse rewards can present challenges in learning tasks with a wide search space, we overcame these difficulties through simple tasks and restricted action spaces. Furthermore, by giving a small reward for just finding the correct grid, the RL agent can learn the importance of achieving the goal state, and by offering a large reward for the submit action, we prevented the agent from repeatedly reaching the correct grid without submitting in a single episode.

In all experiments, we measured the data efficiency of learning through the accuracy relative to the number of environment steps used during training. We decided to use environment steps to ensure consistency with the experimental settings of DreamerV3  [ 12 ] , as we wanted to monitor data efficiency throughout the learning process. Additionally, we employed the pass@3 method used in ARC to measure accuracy  [ 2 ] . The term pass@3 grants three submission opportunities per episode, and an episode is considered successful if the correct answer is found within three attempts. If three incorrect submissions are made or the predetermined episode length (50) is exceeded, the attempt is judged unsuccessful.

All tasks used in the experiments were trained using the RL environment for ARC, ARCLE  [ 16 ] . Each task was originally composed of a few demos and one test input; however, there was a concern that this could lead to overfitting during the learning process. Therefore, we augmented the training with 1,000 demos and used an additional 100 test inputs for evaluating the model. Furthermore, when training a single task, the agent was trained over 100,000 environment steps, whereas for fine-tuning a new task on a pre-trained model, the agent was trained for 50,000 environment steps.

3.1 Can Model-Based RL Learn a Single Task? ( 𝒜 → 𝒜 → 𝒜 𝒜 \mathcal{A}\rightarrow\mathcal{A} caligraphic_A → caligraphic_A )

The first experiment focused on a single ARC task requiring analogical reasoning, where the objective was to learn the common rules from given demo pairs and apply these rules effectively to a test input. This experiment involved training on four ARC tasks, each illustrated in Fig.  1 , using the DreamerV3 and PPO algorithms. Tasks requiring multiple actions to solve, such as Diagonal Flipping, demand analogical reasoning, whereas tasks that can be solved with a single action, such as CCW Rotation and Horizontal Flipping, are less complex. DreamerV3, a model-based RL algorithm, exhibited superior performance in Diagonal Flip tasks, which involve more complex reasoning. In contrast, PPO, recognized for its stable learning capabilities, performed comparably or even better in simpler tasks.

It is noteworthy that DreamerV3 achieved 100% performance on uniformly sized 3 × 3 3 3 3\times 3 3 × 3 tasks but only managed about 40% on tasks with varying sizes. Conversely, PPO’s performance remained consistent regardless of grid size and was influenced solely by task difficulty. This suggests a performance drop in DreamerV3 during the encoding processes.

A common interval of zero performance was observed in DreamerV3’s training across various tasks, followed by rapid performance improvement. This pattern suggests that DreamerV3 is not merely learning simple problem patterns but is developing a deeper understanding of concepts essential for analogical reasoning. Initially, DreamerV3 demonstrated about 40% effectiveness due to learning straightforward patterns. Later, the agent recognized that this approach was insufficient for fully solving tasks, prompting it to explore and learn from various trials. Consequently, DreamerV3 displayed significant performance improvements in later learning stages, outperforming earlier results. These observations imply that DreamerV3’s latent representations capture conceptual content that enhances learning efficiency and illustrates the potential for analogical reasoning within model-based RL frameworks.

3.2 Can Model-Based RL Reason Tasks Similar to Pre-Trained Task? ( 𝒜 → 𝒜 ′ → 𝒜 superscript 𝒜 ′ \mathcal{A}\rightarrow\mathcal{A^{\prime}} caligraphic_A → caligraphic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )

The second experiment evaluated the learning efficiency of an agent when facing a task similar to one it was previously trained on. For instance, in Fig.  3(a) , we observed the performance of diagonal flip tasks across various grid sizes for agents pre-trained on a 3 × 3 3 3 3\times 3 3 × 3 diagonal flip task. Conversely, in Fig.  3(b) , we reversed the roles of the pre-trained and adaptation tasks to evaluate both agents’ performances.

In both scenarios, PPO consistently displayed a performance level of around 20%. This outcome was anticipated, considering that the agent’s performance on pre-trained tasks had similarly been around 20%, as evidenced in previous experiments (Fig.  2(a) and Fig.  2(b) ). The insufficient performance of the pre-trained model likely inhibited any positive influence on the adaptation process, resulting in no notable performance gains compared to the non-pre-trained scenario.

An interesting finding from the DreamerV3 results was the direct impact of the pre-trained model’s performance on adaptation. For example, while the model without pre-training achieved about 40% performance on the same task, the pre-trained model approached 100%, mirroring the 3 × 3 3 3 3\times 3 3 × 3 Diagonal Flip task’s success previously encountered. This pattern was further supported by the performance contrasts observed between Fig.  2(a) and Fig.  3(b) . Here, a task that once achieved 100% performance dropped to around 40% after pre-training, reflecting the outcomes seen with the N × N 𝑁 𝑁 N\times N italic_N × italic_N Diagonal Flip tasks. These outcomes illustrate that DreamerV3 tailored its problem-solving strategies based on insights drawn from pre-trained tasks, suggesting that better results could have been obtained if adaptation in Fig.  3(b) had also been initiated from a higher-performing pre-trained model.

Towards the latter part of training in Fig.  3(a) , a sudden decline in performance was noted. While the exact cause of this drop cannot be definitively established, we hypothesize that it might be due to the same type of interval observed in previous results (Fig.  2 ). Although the experiment’s step limit prevented precise verification, it seems probable that the DreamerV3 agent was undergoing an interval similar to earlier tests, attempting to learn new concepts. Further experimentation beyond 50,000 steps and additional theoretical analysis are required to explore this phenomenon more deeply, which could further substantiate DreamerV3’s capability for analogical reasoning.

4 Discussion

Restriction of action space.

Our study examined the analogical reasoning abilities of reinforcement learning (RL) algorithms and their ability to apply learned concepts. In the ARCLE, an agent’s actions consist of operation and selection, requiring the policy to adeptly handle both components. However, the challenge of learning the selection process significantly increases the complexity of the tasks. To mitigate this, we simplified the action space by providing selections through an oracle in this experiment. This adjustment allowed the agents to focus more on mastering operations without the added difficulty of selection, streamlining the learning process. Future research should explore the possibility of incorporating the selection process more autonomously within the reasoning tasks. This would test the agent’s ability to handle the complete process of decision-making, potentially leading to more analogical reasoning ability for agents.

Restriction of Tasks

This paper focused on learning simple tasks involving just one or two operations, such as diagonal flipping tasks, a Rotating task, and a horizontal flipping task. Future research should explore additional challenges such as the Rotate and the Flip task, which could provide valuable insights into the model’s ability to generalize and reason about sub-components of learned tasks. It is also crucial to assess the efficiency of learning multiple related actions (multi-tasking) simultaneously. To overcome this challenge, previously learned knowledge must be retained while new knowledge is acquired. In other words, it is essential to be robust against catastrophic forgetting. Should such an agent be developed, two types of experiments will be essential: 1) The agent should be trained on a large and diverse set of tasks to verify the absence of catastrophic forgetting. 2) The agent should be trained on a wide range of tasks with varying difficulties, followed by an evaluation of its capability to adapt to untrained tasks.

Applying Meta-Learning

In addressing the ARC tasks, the ability to apply learned concepts to entirely new contexts is essential. Meta-learning, or ‘learning to learn,’ can be considered to adapt quickly to new tasks. Future studies should focus on developing meta-learning frameworks that can efficiently abstract underlying task structures, allowing for rapid generalization across varying problem settings. A deeper exploration into meta-learning could utilize frameworks such as Model-Agnostic Meta-Learning (MAML)  [ 17 ] which has shown promise in various domains. Implementing and refining meta-learning techniques like MAML could lead to breakthroughs in developing AI models that can learn not only untrained tasks more efficiently but also their knowledge to solve untrained tasks.

Applying Transfer Learning

Transfer learning can significantly enhance the model’s ability to utilize knowledge acquired in one context and apply it to different yet related tasks. This process often involves the use of deep neural networks, specifically their ability to approximate and adapt policies through layers that capture generalizable features, which are crucial for the successful transfer of knowledge across tasks within the same domain or between different domains   [ 18 ] . Conducting such studies would provide deeper insights into the flexible application of acquired knowledge, a critical aspect of analogical reasoning. These investigations will pave the way for creating adaptable and efficient learning systems that thrive in dynamically changing environments.

Theoretical Analysis of Model-Based RL

Finally, our findings indicate that model-based RL can facilitate more efficient learning in analogical reasoning tasks compared to model-free approaches. The inherent capabilities of model-based methods to infer and generalize from limited data were particularly beneficial. However, our study also highlights the need for further investigation into the specific mechanisms through which these models store and retrieve task-specific patterns. A deeper understanding of these processes could inform the development of more robust model-based systems, enhancing their capability to handle a wider array of complex reasoning tasks.

5 Related Works

The winner of the ARC Challenge  [ 19 ] utilized a total of 142 domain-specific languages (DSLs) to combine various transformed images  [ 20 ] . The ARC Challenge later expanded into ARCathon  [ 21 ] , and the ARCathon 2022 winner’s approach involved exploring using 166 DSLs that included more complex and diverse features  [ 22 ] . This attempted to challenge the broad generalization of DSLs; however, it also introduced limitations due to the inclusion of very infrequently used features, making the DSLs too complex. Compared to humans can solve about 80% of ARC evaluation tasks  [ 23 ] , DSL-based search algorithms have shown performances around 30-40%  [ 4 , 21 ] . However, these approaches, being fundamentally simple search-based, have inherent weaknesses in complex tasks and are prone to overfitting to specific tasks defined by artificially designed DSLs, making them difficult to apply or generalize to other tasks. This fundamental limitation may render them unsuitable for analogical reasoning.

Subsequent studies have attempted to solve tasks using neural networks. Some research efforts were based on program synthesis, progressively recognizing increasingly complex patterns, while other studies have tried to learn the complexities of ARC through the computational abilities of Large Language Models (LLMs). However, studies without DSLs have shown relatively low performance. Even the most recent studies based on program synthesis  [ 24 ] or LLMs  [ 25 ] have only achieved performances of 6.5–6.75% on untrained tasks  [ 26 ] . Recently, there was interesting research utilizing human-solving processes  [ 27 ] , but this study did not contain the performance on untrained tasks. The low performance of these studies is attributed to the following reasons: In the case of program synthesis, the method of combining all possessed knowledge to acquire new knowledge leads to poor learning efficiency  [ 28 ] , and LLMs are known to be weak in incremental reasoning  [ 29 ] . These limitations can be particularly harmful in evaluation tasks, which are known to be comparatively more complex and difficult than training tasks  [ 4 ] . Additionally, the research utilizing human-solving processes has a risk to untrained tasks due to their reliance on offline learning from given solutions  [ 30 ] .

Ensemble-based research that combines the previous approaches has also been announced  [ 4 , 26 ] . These methods were expected to compensate for the weaknesses of existing approaches through the ensemble. An ensemble-based study combining program synthesis and LLMs showed a performance of 14.75%  [ 26 ] , and the approach of the ARCathon 2023 winner exhibited about 33% performance, significantly surpassing previous studies. However, the performance of ensemble-based research that combined DSL, program synthesis, and LLMs was only 40.25%  [ 4 ] , which is just 0.25% higher than the performance of a baseline that used only DSL. This result means that just one more task out of 400 evaluation tasks was solved. Ultimately, the experimental results of these ensemble-based studies further highlighted the limitation that the presence of program synthesis and LLMs does not significantly impact performance compared to DSL.

Meanwhile, there are some attempts to apply reinforcement learning (RL) algorithms in a limited way based on program synthesis and LLMs methods  [ 26 , 31 ] . The limited application of RL algorithms can be attributed to factors such as the absence of appropriate rewards, high-dimensional complex states, and extensive search spaces, but fundamentally, the absence of an environment for ARC was the main reason. However, an environment for training ARC tasks  [ 16 ] has been developed recently. Furthermore, this research demonstrated successful training of an ARC task using PPO, suggesting the potential applicability of various RL models.

6 Conclusion

In this paper, we compared analogical reasoning abilities between model-based and model-free RL, using a subset of ARC to demonstrate the effectiveness of model-based RL. This approach has shown superior learning efficiency and adaptability in tasks compared to model-free methods like PPO. Specifically, DreamerV3 outperformed in tasks similar to those it had previously solved, displaying a remarkable ability to generalize across various task dimensions.

The observed performance dips followed by rapid recoveries during DreamerV3’s training are thought to represent periods of conceptual consolidation. These intervals align with theoretical expectations of analogical reasoning in AI systems and highlight DreamerV3’s potential to manage complex cognitive tasks. Future research should further explore these learning dynamics to better understand the mechanisms that enable such advanced reasoning. This will help enhance RL agents capable of analogical reasoning and adaptation in dynamic environments, reducing the gap between the reasoning ability of humans and RL agents.

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