definition of sufficient condition in critical thinking

Contents Introduction Definition of "necessary condition" Definition of "sufficient condition" Necessary conditions that are not jointly sufficient Sufficient conditions that are not necessary The concept of converse relations "Is a necessary condition for" and "is a sufficient condition for" are converse relations Four possible combinations --> Practice exercise #1 Practice exercise #2 --> Different kinds (or modes) of necessary condition 1. Introduction

2. definition of "necessary condition", 3. definition of "sufficient condition", 4. necessary conditions that are not sufficient example 4.1 - a set of conditions that are individually necessary without being jointly sufficient. thomas white, the author of a recent textbook in philosophy, attempted to use as his example the specifying of the necessary and sufficient conditions for hearing music from a walkman®. here is the list of necessary conditions that white offered (irrelevant conditions have been here removed, and the list has been renumbered): the walkman is in good working order. the batteries are good. [  note 1  ] the earphones are plugged in. the tape has music on it and is in good working condition. you operate the controls correctly. -- discovering philosophy , by thomas i. white, p. 25 (englewood cliffs, nj: prentice hall), 1991. white then goes on to make a too-hasty claim: "... taken all together they are sufficient  " ( ibid. , p. 25). unfortunately, for his illustrative purposes, the list is by no means sufficient. here are just a few of the many additional necessary conditions that my own students, in previous years, have offered: the listener must not be deaf. the ambient (surrounding) sound must not drown out the earphones. the listener must be wearing the earphones, or must be close enough to them, to hear the music. there must be nothing blocking the sound in the listener's ears. the tape must be inserted correctly; the door of the walkman must be closed; and the tape must not be at the end of the reel (more specifically, it must be positioned so that some of the parts of the tape which contain recorded music will pass over the playback heads). the earphones are in good working order. the listener does not die in the time between operating the controls correctly and the music's emerging from the earphones. even this expanded list is not complete. i imagine that will little effort, you yourself can come up with still further necessary conditions. indeed, there does not seem to be any practical limit to the number of necessary conditions. conclusion: sometimes (as in the case of hearing music from a walkman), it is (far) easier to specify necessary conditions than sufficient ones. 5. sufficient conditions that are not necessary example 5.1 - a set of conditions that are (jointly) sufficient without being individually necessary. a sufficient condition for travelling from calgary to vancouver would be your taking an uneventful trip as a passenger on a regularly scheduled air flight. but while that method of getting from the one city to the other would suffice, it is by no means necessary. there are all sorts of other conditions that would also suffice for your getting from calgary to vancouver. you could take via rail; or you could travel by car; or you could hike; or you could ride a bicycle; or you could travel on horseback; or ..., ... example 5.2 - a(nother) set of conditions that are (jointly) sufficient without being individually necessary. if you'll forgive the morbid example, think of all the ways a person might die: having his/her head chopped off; being at `ground-zero' when a nuclear bomb is detonated; tearing a gaping hole in one's space suit while on a `space-walk' on the moon; etc., etc. but none of these conditions is a necessary condition for a person's dying. indeed almost everyone dies without having satisfied one of these unusual sufficient conditions. conclusion: sometimes, it is easier to specify sufficient conditions than necessary ones. 6. the concept of converse relations, 7. "is a necessary condition for" and "is a sufficient condition for" are converse relations, 8. four possible combinations, practice exercise #1, practice exercise #2, 9. different kinds (or modes) of necessary condition.

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NECESSARY AND SUFFICIENT CONDITIONS

Necessary and sufficient conditions help us understand and explain the connections between concepts, and how different situations are related to each other.

4.1 NECESSARY CONDITIONS

To say that X is a necessary condition for Y is to say that the occurrence of X is required for the occurrence of Y (sometimes also called an essential condition ). In other words, if there is no X, Y would not exist. Examples:

• Having four sides is necessary for being a square.
• Infection by HIV is necessary for developing AIDS.
• Having the intention to kill someone or to cause grievous bodily harm is necessary for murder.

To show that X is not a necessary condition for Y , we simply find a situation where Y is present but X is not. Examples:

• Eating meat is not necessary for living a healthy life. There are plenty of healthy vegetarians.
• Being a land animal is not necessary for being a mammal. Whales are mammals, but they live in the sea.

In daily life, we often talk about necessary conditions, maybe not explicitly. When we say combustion requires oxygen, this is equivalent to saying that the presence of oxygen is a necessary condition for combustion.

Note that a single situation can have more than one necessary condition. To be a good pianist, it is necessary to have good finger technique. But this is not enough. Another necessary condition is being good at interpreting piano pieces.

4.2 SUFFICIENT CONDITIONS

If X is a sufficient condition for Y , this ...

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Chapter 8 Summary

CHAPTER SUMMARY

Enumerative Induction

• An inductive argument is intended to provide only probable support for its conclusion, being considered strong if it succeeds in providing such support and weak if it does not.
• Inductive arguments come in several forms, including enumerative, analogical, and causal. In enumerative induction, we argue from premises about some members of a group to a generalization about the entire group. The entire group is called the target group; the observed members of the group, the sample; and the group characteristics we’re interested in, the relevant property.
• An enumerative induction can fail to be strong by having a sample that’s too small or not representative. When we draw a conclusion about a target group based on an inadequate sample size, we’re said to commit the error of hasty generalization.
• Opinion polls are enumerative inductive arguments, or the basis of enumerative inductive arguments, and must be judged by the same general criteria used to judge any other enumerative induction.

Analogical Induction

• In analogical induction, or argument by analogy, we reason that since two or more things are similar in several respects, they must be similar in some further respect. We evaluate arguments by analogy according to several criteria: (1) the number of relevant similarities between things being compared, (2) the number of relevant dissimilarities, (3) the number of instances (or cases) of similarities or dissimilarities, and (4) the diversity among the cases.

Causal Arguments

• A causal argument is an inductive argument whose conclusion contains a causal claim. There are several inductive patterns of reasoning used to assess causal connections. These include the Method of Agreement, the Method of Difference, the Method of Agreement and Difference, and the Method of Concomitant Variation.
• Errors in cause-and-effect reasoning are common. They include misidentifying relevant factors in a causal process, overlooking relevant factors, confusing cause with coincidence, confusing cause with temporal order, and mixing up cause and effect.
• Crucial to an understanding of cause-and-effect relationships are the notions of necessary and sufficient conditions. A necessary condition for the occurrence of an event is one without which the event cannot occur. A sufficient condition for the occurrence of an event is one that guarantees that the event occurs.

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Necessary And Sufficient Conditions

Definition of necessary and sufficient conditions.

Necessary and sufficient conditions are concepts in logic and mathematics used to describe the relationship between two statements or conditions. A necessary condition for some state of affairs is a condition that must be present for the state of affairs to occur. Conversely, a sufficient condition is one that, if met, guarantees the state of affairs.

Necessary Condition

To be a bachelor, it is necessary to be unmarried. Being unmarried is a necessary condition of being a bachelor because without being unmarried, one cannot be considered a bachelor. However, being unmarried alone does not make one a bachelor; other conditions, such as being a male, are also required.

Sufficient Condition

Scoring 90% on an exam is a sufficient condition for passing the class. This means if you score 90%, you are guaranteed to pass, but scoring less might still allow you to pass, based on other criteria like participation, other assignments, or a curve.

Why Necessary and Sufficient Conditions Matter

Understanding necessary and sufficient conditions is crucial in various fields, including mathematics, logic, philosophy, and the sciences, for structuring logical arguments, proving theorems, and establishing causality. They are foundational for constructing precise definitions, hypotheses, and theories. For instance, in law, distinguishing between necessary and sufficient conditions can help in the analysis of legal responsibility and the construction of legal statutes.

Frequently Asked Questions (FAQ)

Can a condition be both necessary and sufficient.

Yes, a condition can be both necessary and sufficient for a state of affairs. For example, having a total of 180 degrees is both a necessary and sufficient condition for a figure to be a triangle in Euclidean geometry. This means for any figure to be a triangle, it must have a total of 180 degrees, and any figure with exactly 180 degrees is a triangle.

How can understanding these conditions help in everyday decision-making?

Understanding necessary and sufficient conditions can help in improving critical thinking and decision-making. It enables individuals to identify what must be true for a certain outcome to occur (necessary) and what would produce an outcome if it were true (sufficient). This can aid in evaluating arguments, making strategic choices, and setting clearer objectives.

What is the difference between ‘necessary’ and ‘sufficient’ in practical terms?

In practical terms, a necessary condition is something that needs to be there but might not alone cause the outcome. For example, water is necessary for plants to grow, but by itself is not enough; sunlight and soil nutrients are also needed. A sufficient condition, however, assures the outcome. For example, dumping a large amount of water quickly can be sufficient to extinguish a small fire, assuming it’s done appropriately.

How do these concepts relate to causality?

Understanding necessary and sufficient conditions is fundamental to causality. A necessary condition, while required for an event to occur, might not result in the event every time. In contrast, a sufficient condition, when present, always results in the event, establishing a direct cause-effect relationship. These distinctions help in clarifying causal inferences in scientific research and philosophical inquiry.

Overall, the concepts of necessary and sufficient conditions are vital tools in logical reasoning, aiding in the clear understanding and communication of complex relationships and causal mechanisms across various disciplines.

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Necessary and Sufficient Conditions

A handy tool in the search for precise definitions is the specification of necessary and/or sufficient conditions for the application of a term, the use of a concept, or the occurrence of some phenomenon or event. For example, without water and oxygen, there would be no human life; hence these things are necessary conditions for the existence of human beings. Cockneys, according to the traditional definition, are all and only those born within the sound of the Bow Bells. Hence birth within the specified area is both a necessary and a sufficient condition for being a Cockney.

Like other fundamental concepts, the concepts of necessary and sufficient conditions cannot be readily specified in other terms. This article shows how elusive the quest is for a definition of the terms “necessary” and “sufficient”, indicating the existence of systematic ambiguity in the concepts of necessary and sufficient conditions. It also shows the connection between puzzles over this issue and troublesome issues surrounding the word “if” and its use in conditional sentences.

1. Philosophy and Conditions

2. the standard theory: truth-functions and reciprocity, 3. problems for the standard theory, 4. inferences, reasons for thinking, and reasons why, 5. conclusion, other internet resources, related entries.

An ambition of twentieth-century philosophy was to analyse and refine the definitions of significant terms—and the concepts expressed by them—in the hope of casting light on the tricky problems of, for example, truth, morality, knowledge and existence that lay beyond the reach of scientific resolution. Central to this goal was specifying at least in part the conditions to be met for correct application of terms, or under which certain phenomena could truly be said to be present. Even now, philosophy’s unique contribution to interdisciplinary studies of consciousness, the evolution of intelligence, the meaning of altruism, the nature of moral obligation, the scope of justice, the concept of pain, the theory of perception and so on still relies on its capacity to bring high degrees of conceptual exactness and rigour to arguments in these areas.

If memory is a capacity for tracking our own past experiences and witnessings then a necessary condition for Penelope remembering giving a lecture is that it occurred in the past. Contrariwise, that Penelope now remembers the lecture is sufficient for inferring that it was given in the past. In a well-known attempt to use the terminology of necessary and sufficient conditions to define what it is for one thing to be cause of another thing, J. L. Mackie proposed that causes are at a minimum INUS conditions, that is, “Insufficient but Necessary parts of a condition which is itself Unnecessary but Sufficient” for their effects (Mackie 1965). What, then, is a necessary (or a sufficient) condition? This article shows that complete precision in answering this question is itself elusive. Although the notion of sufficient condition can be used in defining what a necessary condition is (and vice versa), there is no straightforward way to give a precise and comprehensive account of the meaning of the term “necessary (or sufficient) condition” itself. Wittgenstein’s warnings against premature theorising and overgeneralising, and his insight that many everyday terms pick out families, should mandate caution over expecting a complete and unambiguous specification of what constitutes a necessary, or a sufficient, condition.

The front door is locked. In order to open it (in a normal, non-violent way) and get into the house, I must first use my key. A necessary condition of opening the door, without violence, then, is to use the key. So it seems true that

• If I opened the door, I used the key.

Can we use the truth-functional understanding of “if” to propose that the consequent of any conditional (in (i), the consequent is “I used the key”) specifies a necessary condition for the truth of the antecedent (in (i), “I opened the door”)? Many logic and critical thinking texts use just such an approach, and for convenience we may call it “the standard theory” (see Blumberg 1976, pp. 133–4, Hintikka and Bachman 1991, p. 328 for examples of this approach).

The standard theory makes use of the fact that in classical logic, the truth-function “ p ⊃ q ” (“If p , q ”) is false only when p is true and q is false. The relation between “ p ” and “ q ” in this case is often referred to as material implication . On this account of “if p , q ”, if the conditional “ p ⊃ q ” is true, and p holds, then q also holds; likewise if q fails to be true, then p must also fail of truth (if the conditional as a whole is to be true). The standard theory thus claims that when the conditional “ p ⊃ q ” is true the truth of the consequent, “ q ”, is necessary for the truth of the antecedent, “ p ”, and the truth of the antecedent is in turn sufficient for the truth of the consequent. This relation between necessary and sufficient conditions matches the formal equivalence between a conditional formula and its contrapositive (“~ q ⊃ ~ p ” is the contrapositive of “ p ⊃ q ”). Descending from talk of truth of statements to speaking about states of affairs, we can equally correctly say, on the standard theory, that using the key was necessary for opening the door.

Given the standard theory, necessary and sufficient conditions are converses of each other, and so there is a kind of mirroring or reciprocity between the two: B ’s being a necessary condition of A is equivalent to A ’s being a sufficient condition of B (and vice versa). So it seems that any truth-functional conditional sentence states both a sufficient and a necessary condition as well. Suppose that if Nellie is an elephant, then she has a trunk. Being an elephant is a sufficient condition of her having a trunk; having a trunk in turn is a necessary condition of Nellie’s being an elephant. Indeed, the claim about the necessary condition is simply another way of putting the claim about the sufficient condition, just as the contrapositive of a formula is logically equivalent to the original formula.

It is also possible to use “only if” to identify a necessary condition: we can say that Jonah was swallowed by a whale only if he was swallowed by a mammal, for if a creature is not a mammal, it is not a whale. The standard theory usually maintains that “If p , q ” and “ p only if q ” are equivalent ways of expressing the truth-functional “ p ⊃ q ”. Equivalent to (i) above, on this account, is the sentence “I opened the door only if I used the key”—a perfectly natural way of indicating that use of the key was necessary for opening the door.

The account of necessary and sufficient conditions just outlined is particularly apposite in dealing with logical conditions. For example, from the truth of a conjunction, it can be inferred that each component is true (if “ p and q ” is true, then “ p ” is true and “ q ” is true). Suppose, then, that it is true that it is both raining and sunny. This is a sufficient condition for “it is raining” to be true. That it is raining is—contrariwise—a necessary condition for it being true that it is both raining and sunny. A similar account seems to work for conceptual and definitional contexts. So if the concept of memory is analysed as the concept of a faculty for tracking actual past events, the fact that an event is now in the past is a necessary condition of my presently recollecting it. If water is chemically defined as a liquid constituted mainly of H 2 O, then if a glass contains water, it contains mainly H 2 O. That the glass contains mostly H 2 O is a necessary condition of its containing water.

Despite its initial appeal, objections to the standard theory have been made by theorists from a number of backgrounds. In summary, the objections build on the idea that “if” in English does not always express a uniform kind of condition. If different kinds of conditions are expressed by the word “if”, the objectors argue, then it would be wise to uncover these before engaging in attempts to formalize and systematize the concepts of necessary and sufficient . In trying to show that there is an ambiguity infecting “if”-sentences in English, critics have focused on two doctrines they regard as mistaken: first, that there is a reciprocity between necessary and sufficient conditions, and, second, that “if p , q ” and “ p only if q ” are equivalent.

Given any two true sentences A and B , the conditional “If A , then B ” is true. For example, provided it is true that the sun is made of gas and also true that elephants have four legs, then the truth-functional conditional “If elephants have four legs, then the sun is made of gas” is also true. However, the gaseous nature of the sun would not normally be regarded as either a conceptually, or even a contingently, necessary condition of the quadripedality of elephants. Indeed, according to the standard theory, any truth will be a necessary condition for the truth of every statement whatsoever, and any falsehood will be a sufficient condition for the truth of any statement we care to consider.

These odd results would not arise in some non-classical logics where it is required that premisses be relevant to the conclusions drawn from them, and that the antecedents of true conditionals are likewise relevant to the consequents. But even in those versions of relevance logic which avoid some of these odd results, it is difficult to avoid all of the so-called “paradoxes of implication”. For example, a contradiction (a statement of the form “ p and not p ”) will be a sufficient condition for the truth of any statement unless the semantics for the logic in question allow the inclusion of inconsistent worlds (for more details, see logic: relevance , and for an account of relevance in terms of the idea of ‘meaning containment’ see chapter 1 of Brady 2006).

These oddities might be dismissed as mere anomalies were it not for the fact that writers have apparently identified a number of other problems associated with the ideas of reciprocity and equivalence mentioned at the end of the previous section. According to the standard theory, there is a kind of reciprocity between necessary and sufficient conditions, and “if p , q ” sentences can always be paraphrased by “ p only if q ” ones. However, as writers in linguistics have observed, neither of these claims matches either the most natural understanding of necessary (and sufficient) conditions, or the behaviour of “if” (and “only if”) in English. Consider, for example, the following case (drawn from McCawley 1993, p. 317):

• If you touch me, I’ll scream.

While in the case of the door, using the key was necessary for opening it, no parallel claim seems to work for (ii): in the natural reading of this statement, my screaming is not necessary for your touching me. McCawley claims that the “if”-clause in a standard English statement gives the condition—whether epistemic, temporal or causal—for the truth of the “then”-clause. The natural interpretation of (ii) is that my screaming depends on your touching me. To take my screaming as a necessary condition for your touching me seems to get the dependencies back to front. A similar concern arises if it is maintained that (ii) entails that you will touch me only if I scream.

A similar failure of reciprocity or mirroring arises in the case of the door example ((i) above). While opening the door depended, temporally and causally, on using the key first, it would be wrong to think that using the key depended, either temporally or causally, on opening the door. So what kind of condition does the antecedent state? To get clear on this, we can consider a baffling pair of conditional sentences (a modification of Sanford 1989, 175–6):

• If he learns to play, I’ll buy Lambert a cello.
• Lambert learns to play only if I buy him a cello.

Notice that these two statements are not equivalent in meaning, even though textbooks standardly treat “if p , q ” as just another way of saying “ p only if q ”. While (iii) states a condition under which I buy Lambert a cello (presumably he first learns by using a borrowed one, or maybe he hires one), (iv) states a necessary condition of Lambert learning to play the instrument in the first place (there may be others too). Indeed, if we take them together, the statements leave poor old Lambert with no prospect of ever getting the cello from me. If (iv) were just equivalent to (iii), combining the two statements would not lead to an impasse like this.

But how else can we formulate (iii) in terms of “only if”? A natural, English equivalent is surprisingly hard to formulate. Perhaps it would be something like:

• Lambert has learned to play the cello only if I have bought him one.

where the auxiliary (“has”/“have”) has been introduced to try to keep dependencies in order. Yet (v) is not quite right, for it can be read as implying that Lambert’s success is dependent on my having first bought him a cello—something that is certainly not implied in (iii). A still better (but not completely satisfactory) version requires further adjustment of the auxiliary, say:

• Lambert will have learned to play the cello only if I have bought him one.

This time, it is not so easy to read (vi) as implying that I bought Lambert a cello before he learned to play. These changes in the auxiliary (sometimes described as changes in “tense”) have led some writers to argue that conditionals in English involve implicit quantification across times (see, for example, von Fintel 1998). Assessment of this claim lies beyond the scope of the present article (see the entry on conditionals and the detailed discussion in Bennett 2003).

What the case suggests is that different kinds of dependency are expressed by use of the conditional construction: (iv) is not equivalent to (iii) because the consequent of (iii) provides what might be called a reason for thinking that Lambert has learned to play the cello. By contrast, the very same condition—that I buy Lambert a cello—appears to fulfil a different function in (iv) (namely that I first have to buy him a cello before he learns to play). In the following section, the possibility of distinguishing between different kinds of conditions is discussed. The existence of such distinctions is evidence for a systematic ambiguity about the meaning of “if” and in the concepts of necessary (and sufficient ) condition .

The possibility of ambiguity in these concepts raises a further problem for the standard theory. According to it—as von Wright pointed out (von Wright 1974, 7)—the notions of necessary condition and sufficient condition are themselves interdefinable:

A is a sufficient condition of B = df the absence of A is a necessary condition of the absence of B

B is a necessary condition of A = df the absence of B is a sufficient condition of the absence of A

Ambiguity would threaten this neat interdefinability. In the following section, we will explore whether there is an issue of concern here. The possibility of such ambiguity has been explored in work by Downing (1959, 1975), Wilson (1979), and has also been raised more recently in Goldstein et al. (2005), ch. 6. These writers have argued that in some contexts there is a lack of reciprocity between necessary and sufficient conditions understood in a certain way, while in other situations the conditions do relate reciprocally to each other in the way required by the standard theory. If these critics are right, and ambiguity is present, then there is no general conclusion that can safely be drawn about reciprocity, or lack of it, between necessary and sufficient conditions. Instead there will be a need to distinguish the sense of condition that is being invoked in a particular context. Without specification of meaning and context, it would also be wrong to make the general claim that sentences like “if p , q ” are generally paraphrasable as “ p only if q ”. By means of a semi-formal argument, Carsten Held has suggested a way of explaining why necessary and sufficient conditions are not converses, making appeal to a version of truthmaker theory (Held 2016). In what follows, we do not follow this route, but instead explore ways of making sense of the lack of reciprocity between the two kinds of conditions in terms of the difference between inferential, evidential and explanatory uses of conditionals.

Are the following two statements equivalent? (see Wertheimer 1968, 363–4):

• The occurrence of a sea battle tomorrow is a necessary and sufficient condition for the truth, today, of “There will be a sea battle tomorrow.”
• The truth, today, of “There will be a sea battle tomorrow” is a necessary and sufficient condition for the occurrence of a sea battle tomorrow.

Sanford argues that while (vii) is sensible, (viii) “has things backward” (Sanford 1989, 176–7). He writes: “the statement about the battle, if true, is true because of the occurrence of the battle. The battle does not occur because of the truth of the statement” ( ibid .) What he probably means is that the occurrence of the battle explains the truth of the statement, rather than explanation being the other way around. Of course, people sometimes do undertake actions just to ensure that what they had formerly said turns out to be true; so there will be cases where the truth of a statement explains the occurrence of an event. But this seems an unlikely reading of the sea battle case.

Now let S be the sentence “There will be a sea battle tomorrow”. If S is true today, it is correct to infer that a sea battle will occur tomorrow. That is, even though the truth of the sentence does not explain the occurrence of the battle, the fact that it is true licenses the inference to the occurrence of the event. Ascending to the purely formal mode (in Carnap’s sense), we can make the point by explicitly limiting inference relations to ones that hold among sentences or other items that can bear truth values. It is perfectly proper to infer from the truth of S today that some other sentence is true tomorrow, such as “there is a sea battle today”. Since “there is a sea battle today” is true tomorrow if and only if there is a sea battle tomorrow, then we can infer from the fact that S is true today that a sea battle will occur tomorrow.

From this observation, it would appear that there is a gap between what is true of inferences, and what is true of explanations. There is an (inferential) sense in which the truth of S is both a necessary and sufficient condition for the occurrence of the sea battle. However, there is an (explanatory) sense in which the occurrence of the sea battle is necessary and sufficient for the truth of S , but not vice versa . It would appear that in cases like (vii) and (viii) the inferences run in both directions, while explanations run only one way. Whether we read (vii) as equivalent to (viii) will depend on the sense in which the notions of necessary and sufficient conditions are being deployed.

Is it possible to generalize this finding? Our very first example seems to be a case in point. The fact I used the key explains why I was able to open the door without force. That I opened the door without force gives a ground for inferring that I used the key. Here is a further example from McCawley:

• If John wins the race, we will celebrate.

John’s winning the race is a sufficient condition for us having a celebration, and his winning the race is the reason why we will be celebrating. Our celebration, however, is not likely to be the reason why he wins the race. In what sense then is the celebration a necessary condition of John’s winning the race? Again, there is a ground for inferring: that we don’t celebrate is a ground for inferring that John didn’t win the race. English “tense” usage is sensitive to the asymmetry uncovered here, in the way noted in the previous section. The natural way of writing the contrapositive of (ix) is not the literal “If we will not celebrate, then John does not win the race”, but rather something like:

• If we don’t celebrate, John didn’t win the race.
• If we aren’t celebrating, John hasn’t won the race.
• If we don’t celebrate, John can’t have won the race.

Inferential reciprocity and explanatory non-reciprocity seems to be no different in the case of conditionals than in the case of logical and mathematical equations in general. For example, Newton’s classical identity, f = ma , can be rewritten in equivalent forms such as a = f/m or f/a = m . These all state just the same thing, from an algebraic point of view. Now let us suppose that force is a measure of what brings a particle to a certain state. Then we would say that while force causes acceleration, the ratio f/a does not cause, or explain, mass, even though it does determine it (see the Epilogue of Pearl 2000 for a non-technical attempt at tackling the representation of causal intervention by algebraic notations).

There are at least three different relations to be distinguished in connection with conditional statements, each of which bears on questions of necessity and sufficiency. First is the implication relation symbolised by the hook operator, “⊃” or perhaps some relevant implication operator. Such an operator captures some inferential relations as already noted. For example, we saw that from the truth of a conjunction, it can be inferred that each component is true (from “ p and q ”, we can infer that “ p ” is true and that “ q ” is true). Hook, or a relevant implication operator, seems to capture one of the relations encountered in the sea battle case, a relation which can be thought of as holding paradigmatically between bearers of truth values, but can be loosely thought of in terms of states of affairs. For this relation, we are able to maintain the standard theory’s reciprocity thesis with the limitations already noted.

Two further relations, however, are often implicated in reflections on necessary and sufficient conditions. To identify these, consider the different things that can be meant by saying

• If Lambert was present, it was a good seminar.

One scenario in terms of which (xiii) can be understood is where Lambert is invariably a lively contributor to any seminar he attends. Moreover, his contributions are always insightful, hence guaranteeing an interesting time for all who attend. In this case, Lambert’s presence explains or was the reason why the seminar was good. A different scenario depicts Lambert as someone who has an almost unerring knack for spotting which seminars are going to be good, even though he himself is not always active in the discussion. Lambert’s attendance at a seminar, according to this story, provides a reason for thinking that the seminar is going to be good. We might say that according to the first story, the seminar is good because Lambert is at it. In the second case, Lambert is at it because it is good. Examples of this kind were first introduced in Wilson (1979) inspired by the work of Peter Downing (Downing 1959, 1975). Notice that the hook (as understood in classical logic) does not capture the reason for thinking relation, for it permits any truth to be inferred from any other statement whatever.

The reason why and reason for thinking that conditions may help to shed light on the peculiarities encountered earlier. That I opened the door is a reason for thinking that I used the key, not a reason why. In case (iii) above, that he learns to play the instrument is the reason why I will buy Lambert a cello, and that I buy him a cello is (in the same case) a reason for thinking that—but not a reason why—he has learned to play the instrument. Our celebrating is a reason for thinking that John has won the race in case (ix), but not a reason why.

Although there is sometimes a correlation between reasons why, on the one hand, and evidentiary relations, on the other, few generalisations about this can be safely made (although Wilson 1979 puts forward a number of suggestions about the connections between these notions). If A is a reason why B has occurred (and so perhaps also is evidence that B has occurred), then the occurrence of B will sometimes be a reason for thinking—but not a guarantee—that A has occurred. If A is no more than a reason for thinking that B has occurred, then B will sometimes be a reason why—but not a guarantee that— A has occurred. Going back to our initial example, my opening the door without violence was a reason for thinking, that is to say evidence, that I had used the key. That I used the key, however, was not just a reason for thinking that I had opened the door, but one of the reasons why I was able to open the door. What is important is that the “if” clause of a conditional may do any of three things described in the present section. One of these is well captured by classical truth-functional logic, namely (i) introduce a sentence from which the consequent follows in the way modelled by an operator such as hook. But there are two other jobs that “if” may do as well, namely: (ii) state a reason why what is stated in the consequent is the case; (iii) state a reason for thinking that what is stated in the consequent is the case (but not state a reason why it is the case).

In general, if explanation is directional, it may not seem surprising that when A explains B , it is not usually the case that B , or its negation, is in turn an explanation of A (or its negation). John’s winning the race explains our celebration, but our failure to celebrate is not (normally) a plausible explanation of his failure to win. Lambert’s presence may explain why the seminar was such a great success, but a boring seminar is not—in any normal set of circumstances—a reason why Lambert is not at it. This result undermines the usual understanding that if A is a sufficient condition of B , it will typically be the case that B is a necessary condition for A , and the falsity of B a sufficient condition for the falsity of A .

In defence of contraposition, it might be argued that in the case of causal claims there is at least a weak form of contraposition that is valid. Gomes proposes (Gomes 2009) that where ‘ A ’ is claimed to be a causally sufficient condition for ‘ B ’, or ‘ B ’ a causally necessary condition of ‘ A ’, then some form of reciprocity between the two kinds of conditions holds, and so some version of contraposition will be valid. Going back to example (ii), suppose we read this as stating a causal condition—that your touching me would cause me to scream. Gomes suggests that ‘ A ’ denotes a sufficient cause of B , provided that (1) ‘ A ’ specifies the occurrence of an event that would cause another event ‘ B ’, and does this by (2) stating a condition the truth of which is sufficient for inferring the truth of ‘ B ’. In such a case, we could further maintain that ‘ B ’, in turn, denotes a necessary effect of ‘ A ’, meaning that the truth of B provides a necessary condition for the truth of A (Gomes 2009, 377–9). This proposal preserves contraposition by treating causal conditionals as inferential.

While it is possible to distinguish these different roles the “if” clause may play (there may be others too), it is not always easy to isolate them in every case. The appeal to “reasons why” and “reasons for thinking” enables us to identify what seem to be ambiguities both in the word “if” and in the terminology of necessary and sufficient conditions. Unfortunately, the concept of explanation itself is too vague to be very helpful here, for we can explain a phenomenon by citing a reason for thinking it is the case, or by citing a reason why it is the case. A similar vagueness infest the word “because”, as we see in a moment. Consider, for example, cases where mathematical, physical or other laws that are involved (one locus classicus for this issue is Sellars 1948). The truth of “that figure is a polygon” is sufficient for inferring “the sum of that figure’s exterior angles is 360 degrees”. Likewise, from “the sum of the figure’s exterior angles is not 360 degrees” we can infer “the figure is not a polygon”. Such inferences are not trivial. Rather they depend on geometrical definitions and mathematical principles, and so this is a case of mathematically necessary and sufficient conditions. But it appears quite plausible that mathematical results also give us at least a reason for thinking that because a figure is a polygon its exterior angles will sum to 360 degrees. We may even be able to think of contexts in which the fact a figure is a polygon provides a reason why its exterior angles sum to 360 degrees. And it might not be unnatural for someone to remark that a certain figures is a polygon because its exterior angles sum to 360 degrees.

A similar point holds for the theory of knowledge where it is generally held that if I know that p , then p is true. The truth of p is a necessary condition of knowing that p , according to such accounts. In saying this we do not rule out claims stronger than simply saying that the truth of p follows from the fact that we know that p . That a belief is true—for example—may be (part of) a reason for thinking it constitutes knowledge. Other cases involve inferences licensed by physics, biology and the natural sciences—inferences that will involve causal or nomic conditions. Again there is need for care in determining whether reason why or reason for thinking relations are being stated. The increase of mean kinetic energy of its molecules does not just imply that the temperature of a gas is rising but also provides a reason why the temperature is increasing. However, if temperature is just one way of measuring mean molecular kinetic energy, then a change in temperature will be a reason for thinking that mean kinetic energy of molecules has changed, not a reason why it has changed.

As mentioned at the start of the article, the specification of necessary and sufficient conditions has traditionally been part of the philosopher’s business of analysis of terms, concepts and phenomena. Philosophical investigations of knowledge, truth, causality, consciousness, memory, justice, altruism and a host of other matters do not aim at stating explanatory relations, but rather at identifying and developing conceptual ones (see Jackson 1998 for a detailed account of conceptual analysis). But even here, the temptation to look for reasons why or reasons for thinking that is not far away. It might be said that conceptual analysis is like dictionary definition, hence eschewing evidential and explanatory conditions. But at least evidential conditions seem to be natural consequences of definition and analysis. That Nellie is an elephant is not a (or the) reason why she is an animal, any more than that a figure is a square is a reason why it has four sides. But some evidential claims seem to make sense even in such contexts: being an elephant apparently gives a reason for thinking that Nellie is an animal, and a certain figure may be said to have four sides because it is a square, in the evidential sense of “because”.

To specify the necessary conditions for the truth of the sentence “that figure is a square” is to specify a number of conditions including “that figure has four sides”, “that figure is on a plane”, and “that figure is closed”. If any one of these latter conditions is false, then the sentence “that figure is a square” is also false. Conversely, the truth of “that figure is a square” is a sufficient condition for the truth of “that figure is closed”. The inferential relations in this case are modelled to some extent—albeit inadequately, as noted earlier—by an operator such as hook.

Now consider our previous example—that of memory. That Penelope remembers something—according to a standard account of memory—means (among other things) that the thing remembered was in the past, and that some previous episode involving Penelope plays an appropriate causal role her present recall of the thing in question. It would be a mistake to infer from the causal role of some past episode in Penelope’s current remembering, that the definition of memory itself involves conditions that are explanatory in the reason why sense. That Penelope now remembers some event is not a reason why it is in the past. Rather, philosophical treatments of memory seek for conditions that are a priori constitutive of the truth of such sentences as “Penelope remembers doing X ”. The uncovering of such conditions does not explain Penelope’s now remembering things, but simply provides insight into whether, and how, “remember” is to be defined. Reason why and reason for thinking that conditions do not play a role in this part of the philosopher’s enterprise.

Finally, it should be noted that not all conditional sentences primarily aim at giving necessary and/or sufficient conditions. A common case involves what might be called jocular conditionals . A friend of Lys mistakenly refers to “Plato’s Critique of Pure Reason ” and Lys remarks, “If Plato wrote the Critique of Pure Reason , then I’m Aristotle”. Rather than specifying conditions, Lys is engaging in a form of reductio argument. Since it is obvious that she is not Aristotle, her joke invites the listener to infer (by contraposition) that Plato did not write the Critique of Pure Reason .

Given the different roles for “if” just identified, it is hardly surprising that generalisations about necessary and/or sufficient conditions are hard to formulate. Suppose, for example, someone tries to state a sufficient condition for a seminar being good in a context where the speaker and all the listeners share the view that Lambert’s presence is a reason why seminars would be good. In this case, Lambert’s presence might be said to be a sufficient condition of the seminar being good in the sense that his presence is a reason why it is good. Now, is there a similar sense in which the goodness of the seminar is a necessary condition of Lambert’s presence? The negative answer to this question is already evident from the earlier discussion. If we follow von Wright’s proposal, mentioned above, we get the following result: that the seminar is not good is a sufficient condition of Lambert not being present. But this cannot plausibly be read as a sufficient condition in anything like the sense of a reason why. At most, the fact of the seminar not being a good one may be a reason for thinking that Lambert was not at it. So how can we tell, in general, what kind of condition is being expressed in an “if” sentence? As noted in the case of the sea battle, when rewriting in the formal mode captures the sense of what is being said, and when the formulations “if p , q ” and “ p only if q ” seem idiomatically equivalent, then an inferential interpretation will be in order, von Wright’s equivalences will hold, and the material conditional gives a reasonable account of such cases with the limitations pointed out earlier.

As already noted, even the inferential use of “if” is not always associated primarily with the business of stating necessary and sufficient conditions. This observation, together with the cases and distinctions introduced in the present article, show the need for caution when we move from natural language conditionals to analysis of them in terms of necessary and sufficient conditions, and also the need for caution in modelling the latter conditions by means of logical operators. It appears that there are several kinds of conditionals, and several kinds of conditions. So although we can—and do—sometimes use conditional statements to express necessary and sufficient conditions, and can explicate necessary and sufficient conditions by analysis of some of the roles of “if” in natural language conditionals, this does not give us as much as we might hope for. In particular, there seems to be no general formal scheme for translating between conditionals as used in natural language and the statement of any one particular type of condition, or vice versa.

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• Bennett, J., 2003. A Philosophical Guide to Conditionals , Oxford: Oxford University Press.
• Brady, Ross, 2006. Universal Logic , Stanford: CSLI Publications.
• Downing, Peter, 1959. “Subjunctive Conditionals, Time Order and Causation”, Proceedings of the Aristotelian Society , 59: 126–40.
• Downing, Peter, 1975 “Conditionals, Impossibilities and Material Implications”, Analysis , 35: 84–91.
• Gomes, Gilberto, 2009. “Are Necessary and Sufficient Conditions Converse Relations?”, Australasian Journal of Philosophy , 87: 375–87.
• Goldstein, L., Brennan, A., Deutsch, M. and Lau, J., 2005. Logic: Key Concepts in Philosophy , London: Continuum.
• Held, Carsten, 2016. Conditions , download from philsci-archive.pitt.edu.
• Hintikka, J. and Bachman, J., 1991. What If …? Toward Excellence in Reasoning , London: Mayfield.
• Jackson, F., 1998. From Metaphysics to Ethics: A Defence of Conceptual Analysis , Oxford: Oxford University Press.
• Mackie, J. L., 1965. “Causes and Conditions”, American Philosophical Quarterly , 12: 245–65.
• McCawley, James, 1993. Everything that Linguists have Always Wanted to Know about Logic * (Subtitle: * But Were Ashamed to Ask ), Chicago: Chicago University Press.
• McLaughlin, Brian, 1990. On the Logic of Ordinary Conditionals , Buffalo, NY: SUNY Press.
• Pearl, Judea, 2000. Causality: Models, Reasoning, and Inference , Cambridge: Cambridge University Press.
• Sanford, David H., 1989. If P, then Q: Conditionals and the Foundations of Reasoning , London: Routledge.
• Sellars, Wilfrid, 1948. “Concepts as involving laws and inconceivable without them”, Philosophy of Science , 15: 289–315.
• Von Fintel, Kai, 1997. “Bare Plurals, Bare Conditionals and Only”, Journal of Semantics , 14: 1–56.
• Von Wright, G. H., 1974. Causality and Determinism , New York: Columbia University Press.
• Wertheimer, R., 1968. “Conditions”, Journal of Philosophy , 65: 355–64.
• Wilson, Ian R., 1979. “Explanatory and Inferential Conditionals”, Philosophical Studies , 35: 269–78.
• Woods, M., Wiggins, D. and Edgington D. (eds.), 1997. Conditionals , Oxford: Clarendon Press.

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• The Concepts of Necessary and Sufficient Conditions , maintained by Norman Swartz, Philosophy, Simon Fraser University.

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Acknowledgments

I am grateful to Richard Borthwick, Jake Chandler, Laurence Goldstein, Fred Kroon, Y.S. Lo, Jesse Alama, Edward Zalta and Uri Nodelman for their generous help and advice relating to this entry.

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Necessary and Sufficient Conditions

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The discussion of  'necessary and sufficient conditions' is well understood in philosophy, and as a result, I sometimes make the mistake of assuming it is commonly understood in the wider community. This post redresses this by sketching the concept and why it is important.

To say that one thing is a condition for another is to say that the one thing is involved in making the second thing happen.

The most common example of a condition is a cause . For example, striking a billiard ball with a cue causes the ball to move. Thus, the striking of the ball is a 'condition' of the movement of the ball.

But conditions need not be causes. Giving permission is another type of condition. For example, a driver's license gives you permission to drive. This, having a driver's license is a 'condition' for driving.

There are two ways to express conditions:

B if A (alternatively: if A then B) B only if A

The first is called a sufficient condition. The second is a necessary condition.

The idea of a sufficient condition is that it is enough to make something happen. For example, in most cases, pushing on the gas is enough to make the car go forward. It's not the only thing that would do it; you could make the car go forward by pushing it, for example.

The idea of a necessary condition is that something will not happen unless the condition happens. For example, we might say that the car will not go forward unless we have turned off the parking brake. Turning off the brake is thus a necessary condition to the car going forward.

Necessary and sufficient conditions are typically used to explain why something happens. "Why did the car go forward?" we ask. The brake was turned off; that was necessary for the motion to happen. And then somebody pressed on the gas; that was sufficient to make it move forward.

The Logic of Conditions

The logical structures of necessary and sufficient conditions do a dance around each other.

The simplest statement of a sufficient condition is as follows:

If A then B

This is equivalent to:

If not B then not A, and

It is not equivalent to:

If B then A

Meanwhile, the simplest statement of a necessary condition is as follows:

And we often use special words to indicate this special status:

B only if A

Not B unless A

This is also equivalent to:

If not A then not B

And it is not equivalent to:

If not B then not A

The Conditional Fallacy

Why is this important? Because it points to what is probably the most common fallacy involving conditions: not sufficient means not necessary .

For example, we often hear this kind of argument:

Studies show that simply spending money will not improve test scores in schools. So we should be looking at something else, like quality teachers.

What makes this a bit tricky is that the conclusion is often implicit. The conclusion, if spelled out, is that we should be doing something instead of throwing money at the problem.

Here's an example [https://edtechbooks.org/-jcb] of the fallacy being committed. Ewan McIntosh writes, " In 2006 there was $2 trillion spent on education by the world's governments. But money alone is not the reason we see improvement, not always." He then recommends "Getting the right people to become teachers, developing them into effective instructors (and) ensuring that the system is able to offer the best possible instruction for every child ." Presumably, instead of spending money on the problem - after all, Singapore didn't have to.

Here is Tom Hoffman identfying the fallacy [https://edtechbooks.org/-wHC] in McIntosh's reasoning: "I don't have the slightest idea what school budgets look like in Scotland, so maybe over there it is appropriate to put across the message that more funding isn't necessary to improve education, but on this side of the pond, even this study makes it clear that improving American education requires spending more money."

The stiuation is represented thus: spending money is necessary but not sufficient to improve educational outcomes.

What this means is that simply spending money won't solve the problem. There are many ways to spend money that are not effective, as evidenced by many actual spendings of money that are not effective. Purchasing each mathematics class a Lear jet, for example, would certainly spend money. But it would not be very effective.

The response to this fallacy is to say, as Tom Hoffman did, that spending money is necessary in order to solve the problem. What this means is that, while the mere spending of money is no guarantee, nonetheless, the problem will not be solved unless money is spent. The supposition that the problem can be solved without spending money is a fallacy.

As you may imagine, with the logic of conditions being so entwined, it is very easy to get tangled in a mess of necessary and sufficient conditions. This is especially the case when attempting to state whether one thing will cause another to happen.

Many people mistake a cause as the sufficient condition for something to happen (sometimes thought of as the ' efficient cause [https://edtechbooks.org/-CzX] ' or the 'causal agent'). But formally, we should think of a 'cause' as 'a necessary and sufficient condition for an effect'.

That is to say, the description of a cause needs to include, not only the sufficient conditions, but also the necessary conditions, for an effect.

So if we sat that 'A' is a set of necessary and sufficient conditions, then when we say that 'A causes B' we mean that: 'If A then B' and 'If not A then not B' You need both parts to ascribe a cause. You need to show that when A happens, that B also happens, but also, that it is not a coincidence , that is, when A does not happen, B does not happen either.

Some people at this point may argue that only a correlation , and not a cause, has been established. They argue that, in addition to a correlation, a causal argument must also appeal to a general principle or law of nature. This may be the case; if so, then we can simply say here that showing that 'If A then B' and 'If not A then not B' is necessary , but not sufficient, to show that A causes B.

Ceteris Paribus

The phrase ceteris paribus is Latin for 'all other things being equal' and is an important principle for understanding the concept of necessary and sufficient conditions.

Strictly speaking, the description of a cause for any event would be endless. For example, if I wanted to say that 'the car caused the accident' then I would need to say that the car exists and that the accident happened and that the earth exists and that the laws of nature are as we understand and that the accident was not a sub-temporal sentient being and that Merlin did not intervene and... well, you get the idea.

Usually, when we say that one thing is a cause for another, or that one thing is a condition for another, we assume a certain background state of affairs, which continues as it always has. This is especially important when talking about sufficient conditions, but will also come into play when talking about necessary conditions

When I said 'pressing on the gas was sufficient to move the car', I assumed that, as usual, the parking brake was not engaged. Because, after all, were the parking brake engaged, pressing on the gas might not be sufficient to move the car. Really, I should say, 'Releasing the parking brake and pressing on the gas is sufficient to move the car'. But since the parking brake is almost never engaged, it is not usually necessary to say this; I just assume it.

Similarly, when I said that 'releasing the parking brake is necessary to move the car', the presumption was that the parking brake was engaged. But most of the time, releasing the brake is not necessary because the brake was not engaged in the first place. I do not need to state the necessary condition.

This is why the concept of 'control' is so important in scientific experimentation. If you say 'all else being equal', then if you are measuring for results, then you need to know that all else was, in fact, equal.

Expectations

When you say 'all else being equal', you are assuming that a certain state of affairs holds, described in shorthand as 'all else'. But, of course, something changes, for otherwise causation would be impossible.

When you say 'all else ' you mean 'everything not affected by the cause'. But this is essentially a statement of expectations . When you say 'A caused B' what you mean, in full, is that 'A caused B instead of C', where C denotes the alternative that would have been the case, all else being equal, has A not occurred.

Bas van Fraassen [https://edtechbooks.org/-yTY] explains this at length. When you plant sunflower seeds beside the house and they grow to be six feet tall, someone may ask, "Why did the sunflowers grow here?" What they mean is, 'what caused them to grow (instead of to not grow)?' and not 'what caused sunflowers to grow instead of rutabagas?'.

When Tom Hoffman writes, sarcastically, "I don't get Ewan's Scottish spin on [https://edtechbooks.org/-jcb] this McKinsey (i.e., American) study [https://edtechbooks.org/-kyy] of educational systems around the globe," he is speaking of expectations. He is suggesting the production of a given effect involves spending more money in one context, where in the other the production fo the same effect, it is implied, does not mean the spending of more money.

Tricks Involving Ceteris Paribus

This is where ceteris paribus gets tricky. Very often, the presumption 'all other things being equal' does not mean, strictly, ' all other things', but rather, a subset of other things, and specifically (and importantly), the set of necessary conditions for the effect to happen.

Let us suppose that McIntosh said: "We can hire better teachers, but we do not need to spend money in order to do so." This is a bit of a caricature, but it is implied in the suggestion that the problem will not be solved by spending money.

Strictly speaking, this is impossible. It is not possible to hire teachers without spending money. What can only be meant is that it is not necessary to spend more money. He is stating, in other words, that enough money is already being spent to hire quality teachers.

But, of course, this money is currently being spent on something else. So in this case, ' ceteris paribus ' means 'same amount of money spend' but does not mean 'spent on the same things'.

The unstated argument here is that the money being spent elsewhere should be reallocated to spending on quality teachers. But this very necessary condition remains unstated. This is a fallacy; the necessary condition is hidden in the ceribus paribus clause.

A similar fallacy exists elsewhere in the same argument. McIntosh writes,

Less than 1% of African and Middle Eastern children perform at or above the Singaporian average - to be expected, you might believe, because those Singaporeans must hemorrhage cash into their education system. Wrong. Singapore spends less on Primary education than 27 of the 30 OECD countries.

Fair enough. But is McIntosh recommending that finding for education in the UK be adjusted to match the funding provided to education in Singapore? Almost certainly not!

This is a case of shifting ceterus paribus clauses. In Singapore , 'all else being equal' means expenditures at Singapore's levels. But in Britain, this means something very different.

Why is this important?

Because, if the expenses in Britain are not the same as those in Singapore, this means that there is something very different about Singapore which makes it possible to spend much less on education. But if Singapore is very different in precisely this way, then it is a poor analogy and cannot be used to define 'all else being equal', for, in this case, 'all else' is very different .

Arguments involving the use of conditions and causation are often deceptive because of the misuse of necessary and sufficient conditions.

When reading such arguments, you should not be swayed into believing that something is not necessary simply because it is not sufficient.

You should also be wary of hidden, and often shifting, assumptions about necessary conditions implicit in (frequently unstated) ceteris paribus clauses.

When evaluating such arguments, ask yourself simple questions. Like: if they did A, would the result be B? If they did not do A, would B not result?

Trust your intuitions. And keep in mind that if the appeal, by analogy, is to something that is unfamiliar to you - like Singapore, or like Estonia - the reason is most likely to hide some hidden difference that makes them a special case.

Moncton, January 07, 2008

National Research Council of Canada

Stephen Downes is a specialist in online learning technology and new media. Through a 25 year career in the field Downes has developed and deployed a series of progressively more innovative technologies, beginning with multi-user domains (MUDs) in the 1990s, open online communities in the 2000s, and personal learning environments in the 2010s. Downes is perhaps best known for his daily newsletter,  OLDaily , which is distributed by web, email and RSS to thousands of subscribers around the world, and as the originator of the Massive Open Online Course (MOOC), is a leading voice in online and networked learning, and has authored learning management and content syndication software.

Downes is known as a leading proponent of connectivism, a theory describing how people know and learn using network processes. Hence he has also published in the areas of logic and reasoning, 21st century skills, and critical literacies. Downes is also recognized as a leading voice in the open education movement, having developed early work in learning objects to a world-leading advocacy of open educational resources and free learning. Downes is widely recognized for his deep, passionate and articulate exposition of a range of insights melding theories of education and philosophy, new media and computer technology. He has published hundreds of articles online and in print and has presented around the world to academic conferences in dozens of countries on five continents.

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Alessandra Imperio

Il pensiero critico (CT) è considerato un'abilità chiave per il successo nel 21° secolo. Le politiche educative mondiali sostengono la promozione del CT e ricercatori di diverse aree disciplinari sono stati coinvolti in un ampio dibattito sulla sua definizione, senza raggiungere un accordo. Al giorno d'oggi, la ricerca non ha affron-tato compiutamente la valutazione del CT, né il modo in cui dovrebbe essere insegnato. Nel presente lavoro, viene fornita una panoramica sull'argomento, nonché una valutazione delle pratiche, al fine di fornire a ricercatori o professionisti (in particolare quelli della scuola primaria) un riferimento per lo sviluppo di ulteriori teorie e metodi sull'educazione al CT. Il CT è considerato dal punto di vista della filosofia, della psicologia co-gnitiva e delle scienze dell'educazione. Inoltre proponiamo l'inclusione di una quarta prospettiva, che potrebbe essere definita della pedagogia socio-culturale, per le sue importanti implicazioni sull'insegnamento e nelle pratiche valutative. Critical thinking (CT) is considered a key skill for success in the 21st century. Worldwide educational policies advocate the promotion of CT, and scholars across different fields have been involved in a wide debate on its definition, without reaching an agreement. Currently, research has not adequately addressed CT assessment, nor the way in which it should be taught. In the present work, an overview of the topic is provided, as well as an evaluation of the practices, in order to provide researchers or practitioners (particularly those involved in primary school education) a reference for the development of further theories and methods about CT in education. CT is considered from the perspective of philosophy, cognitive psychology, and education sciences. In addition, we propose the inclusion of a fourth perspective, which could be referred as socio-cultural pedagogic perspective, due to its important implications in teaching and assessment practices.

Critical thinking includes the component skills of analyzing arguments, making inferences using inductive or deductive reasoning, judging or evaluating, and making decisions or solving problems. Background knowledge is a necessary but not a sufficient condition for enabling critical thought within a given subject. Critical thinking involves both cognitive skills and dispositions. These dispositions, which can be seen as attitudes or habits of mind, include openand fair-mindedness, inquisitiveness, flexibility, a propensity to seek reason, a desire to be wellinformed, and a respect for and willingness to entertain diverse viewpoints. There are both general-and domain-specific aspects of critical thinking. Empirical research suggests that people begin developing critical thinking competencies at a very young age. Although adults often exhibit deficient reasoning, in theory all people can be taught to think critically. Instructors are urged to provide explicit instruction in critical t...

Jonathan Heard

Donald Jenner

A very short comment on the mistaken use of "critical thinking" deriving from merely psychological foundations.

Daniel Strauss

Before criticism is justified, an account of the applicable criteria should be given. This task concerns first of all the well-known logical principles of identity, contradiction and the excluded middle. They connect critical thinking to the conceptual element of rationality and to the normed nature of logical thinking, manifest in logically sound (norm-conformative) thinking and antinormative thinking—briefly also accounting for the dialectical tradition. An analysis of these principles requires an understanding of the uniqueness of, and coherence between, the logical and non-logical aspects in the light of contraries like logical-illogical, polite-impolite and frugal-wasteful. It also questions the idea of autonomy and examines the switch from principles to values. When a school of thought does not accept all the logical principles, the criteria for scientific thinking are challenged, for example in intuitionistic logic which rejects the universal validity of the principle of the excluded middle. It is then argued that the principle of sufficient reason and that of the excluded antinomy points at a more than logical foundation for critical thinking and ultimately calls for a non-reductionist ontology.

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3.3: Conditional Arguments

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• Page ID 95017

• Jason Southworth & Chris Swoyer
• Fort Hays State & University University of Oklahoma

Conditional Arguments that Affirm

Arguments that have a conditional as one premise and either the antecedent or the consequent of that very conditional as the second premise are called conditional arguments. The first type of conditional argument we will study has the antecedent of the conditional as the second premise.

• If you own a Switch , then you must buy Animal Crossing .
• You own a Switch .
• Therefore, you must buy Animal Crossing.

The first premise of this argument is a conditional and the second premise says that the antecedent of that conditional is true. The second premise just repeats and affirms the antecedent in the first premise. We say that such arguments affirm the antecedent. All arguments that affirm the antecedent are deductively valid . It is impossible for an argument with this format to have all true premises and a false conclusion. This format is sometimes known by its Latin name modus ponens .

In case you are skeptical that this argument structure is valid, remember we can always use the method of counterexample from chapter 2 to check them. Assume that if you own a Switch you must buy Animal Crossing (maybe some strange law is in place) and assume that you own a Switch . Can you think of any way that you don’t have to buy Animal Crossing? No. If you didn’t buy it while owning a Switch , then the first premise would be false, but we were assuming it was true. So, if the premises are true then the conclusion must be true, showing that affirming the antecedent is valid.

We can also have arguments where the second half of the conditional – the consequent – is repeated as a premise:

• If Norman is in Oklahoma, then Norman is south of Kansas.
• Norman is south of Kansas.
• Therefore, Norman is in Oklahoma.

The first premise is a conditional and the second premise says that the consequent of the conditional is true. Such arguments affirm the consequent . Each and every argument that has this format is deductively invalid. It is possible for such arguments to have all true premises and a false conclusion.

Just to be safe, let’s check again, using the method of counterexample. Assume that if Norman is in Oklahoma, then Norman is south of Kansas, and that Norman is south of Kansas. Can you think of any way these premises can be true and it not be the case that Norman is in Oklahoma? Yes. Plenty of places are south of Kansas without being Oklahoma. Norman could be in Texas or Mexico or Brazil (or any number of other places better than Oklahoma). It doesn’t matter that we know Norman is in Oklahoma, because validity is just asking us to take for granted that the premises are true and check to see if the conclusion follows from them.

Conditional Arguments that Deny

We have studied one kind of sentence, the conditional. Now we need to introduce another kind, the negation . The negation of a sentence is another sentence which says that the first sentence is false. It says the opposite of what the first sentence says; it denies it. We could express the negation of the sentence:

It is raining.

by any of the following sentences:

• It is not the case that it is raining.
• It is not true that it is raining.
• It is not raining.
• It isn’t raining.
• It ain’t raining.
• Ain’t rainin’.

Arguments that have a conditional as one premise and either the negation of that conditional’s antecedent or the negation of the conditional’s consequent are also conditional arguments . So, there are two conditional arguments that affirm and two more that deny, for a total of four.

Here is a conditional argument in which the second premise is the negation of the antecedent of the first premise:

• Norman is not south of Kansas.
• Therefore, Norman is not in Oklahoma.

The first premise is a conditional and the second premise says that the consequent of the conditional is false. Such arguments deny the consequent . Each and every argument that has this format is deductively valid. This format is sometimes known by its Latin name, modus tollens .

Let’s use the method of counterexample again to double check that this structure is valid. Assuming that if Norman is in Oklahoma, then Norman is south of Kansas, and that Norman is not south of Kansas, could it be possible for Norman to be in Oklahoma? No. If being south of Kansas is necessary for Norman to be in Oklahoma, and we know that Norman isn’t south of Kansas, then there is no way Norman is in Oklahoma.

By contrast, consider this argument:

• Norman is not in Oklahoma.
• Therefore, Norman is not south of Kansas.

The first premise is a conditional and the second premise says that the antecedent of the conditional is false. Such arguments deny the antecedent . All arguments having this format are deductively invalid . Denying the antecedent is always a fallacy.

For completion, let’s go back to the method of counterexample one more time. Assume that if Norman is in Oklahoma then Norman is south of Kansas, and that Norman is not in Oklahoma. Does this mean that Norman can’t be south of Kansas? No. Just like before it could be in Texas, etc.

Here are two more examples:

• If he builds it, they will come. But they didn’t come. So, he didn’t build it.

We can repackage the argument into standard form like this:

• If he builds it, they will come.
• They didn’t come.
• Therefore, he didn’t build it.

It is impossible for both premises of this argument to be true while its conclusion is false, and so is deductively valid. The argument denies the consequent .

• If the sawdust is the work of carpenter ants, then we’ll need something stronger than Raid to fix the problem. But fortunately, it’s not the work of carpenter ants, so we won’t need anything stronger than Raid.

In standard form:

• If the sawdust is the work of carpenter ants, then we’ll need something stronger than Raid.
• The sawdust is not the work of carpenter ants.
• Therefore, we won’t need anything stronger than Raid.

This argument denies the antecedent . Hence, it is invalid. But we should be able to see this without knowing the label: if you knew that the two premises were true, you still could not be sure whether the conclusion was true or not. The sawdust might be the work of termites (in which case we’ll definitely need something stronger than Raid.)

It is important to remember that you will always be able to work through the argument using the method of counterexample to determine validity and invalidity. But, thinking in terms of shortcuts, you are going to save yourself a lot of time of you memorize these rules.

Affirming the antecedent and denying the consequent are always valid. Denying the antecedent and affirming the consequent are always invalid.