Pursuing Truth: A Guide to Critical Thinking

Chapter 4 propositional logic.

Categorical logic is a great way to analyze arguments, but only certain kinds of arguments. It is limited to arguments that have only two premises and the four kinds of categorical sentences. This means that certain common arguments that are obviously valid will not even be well-formed arguments in categorical logic. Here is an example:

  • I will either go out for dinner tonight or go out for breakfast tomorrow.
  • I won’t go out for dinner tonight.
  • I will go out for breakfast tomorrow.

None of these sentences fit any of the four categorical schemes. So, we need a new logic, called propositional logic. The good news is that it is fairly simple.

4.1 Simple and Complex Sentences

The fundamental logical unit in categorical logic was a category, or class of things. The fundamental logical unit in propositional logic is a statement, or proposition 5 Simple statements are statements that contain no other statement as a part. Here are some examples:

  • Oklahoma Baptist University is in Shawnee, Oklahoma.
  • Barack Obama was succeeded as President of the US by Donald Trump.
  • It is 33 degrees outside.

Simple sentences are symbolized by uppercase letters. Just pick a letter that makes sense, given the sentence to be symbolized, that way you can more easily remember which letter means which sentence.

Complex sentences have at least one sentence as a component. There are five types in propositional logic:

  • Conjunctions
  • Disjunctions
  • Conditionals
  • Biconditionals

4.1.1 Negations

Negations are “not” sentences. They assert that something is not the case. For example, the negation of the simple sentence “Oklahoma Baptist University is in Shawnee, Oklahoma” is “Oklahoma Baptist University is not in Shawnee, Oklahoma.” In general, a simple way to form a negation is to just place the phrase “It is not the case that” before the sentence to be negated.

A negation is symbolized by placing this symbol ‘ \(\neg\) ’ before the sentence-letter. The symbol looks like a dash with a little tail on its right side. If \(\textrm{D}\) = ‘It is 33 degrees outside,’ then \(\neg \textrm{D}\) = ‘It is not 33 degrees outside.’ The negation symbol is used to translate these English phrases:

  • it is not the case that
  • it is not true that
  • it is false that

A negation is true whenever the negated sentence is false. If it is true that it is not 33 degrees outside, then it must be false that it is 33 degrees outside. if it is false that Tulsa is the capital of Oklahoma, then it is true that Tulsa is not the capital of Oklahoma.

When translating, try to keep the simple sentences positive in meaning. Note the warning on page 24, about the example of affirming and denying. Denying is not simply the negation of affirming.

4.2 Conjunction

Negations are “and” sentences. They put two sentences, called conjuncts, together and claim that they are both true. We’ll use the ampersand (&) to signify a negation. Other common symbols are a dot and an upside down wedge. The English words that are translated with the ampersand include:

  • nevertheless

For example, we would translate the sentence ‘It is raining today and my sunroof is open’ as ‘ \(\textrm{R} \& \textrm{O}\) .’

4.3 Disjunction

A disjunction is an “or” sentence. It claims that at least one of two sentences, called disjuncts, is true. For example, if I say that either I will go to the movies this weekend or I will stay home and grade critical thinking homework, then I have told the truth provided that I do one or both of those things. If I do neither, though, then my claim was false.

We use this symbol, called a “vel,” for disjunctions: \(\vee\) . The vel is used to translate - or - eitheror - unless

4.4 Conditional

The conditional is a common type of sentence. It claims that something is true, if something else is also. Examples of conditionals are

  • “If Sarah makes an A on the final, then she will get an A for the course.”
  • “Your car will last many years, provided you perform the required maintenance.”
  • “You can light that match only if it is not wet.”

We can translate those sentences with an arrow like this:

  • \(F \rightarrow C\)
  • \(M \rightarrow L\)
  • \(L \rightarrow \neg W\)

The arrow translates many English words and phrases, including

  • provided that
  • is a sufficient condition for
  • is a necessary condition for
  • on the condition that

One big difference between conditionals and other sentences, like conjunctions and disjunctions, is that order matters. Notice that there is no logical difference between the following two sentences:

  • Albany is the capital of New York and Austin is the capital of Texas.
  • Austin is the capital of Texas and Albany is the capital of New York.

They essentially assert exactly the same thing, that both of those conjuncts are true. So, changing order of the conjuncts or disjuncts does not change the meaning of the sentence, and if meaning doesn’t change, then true value doesn’t change.

That’s not true of conditionals. Note the difference between these two sentences:

  • If you drew a diamond, then you drew a red card.
  • If you drew a red card, then you drew a diamond.

The first sentence must be true. if you drew a diamond, then that guarantees that it’s a red card. The second sentence, though, could be false. Your drawing a red card doesn’t guarantee that you drew a diamond, you could have drawn a heart instead. So, we need to be able to specify which sentence goes before the arrow and which sentence goes after. The sentence before the arrow is called the antecedent, and the sentence after the arrow is called the consequent.

Look at those three examples again:

The antecedent for the first sentence is “Sarah makes an A on the final.” The consequent is “She will get an A for the course.” Note that the if and the then are not parts of the antecedent and consequent.

In the second sentence, the antecdent is “You perform the required maintenance.” The consequent is “Your car will last many years.” This tells us that the antecedent won’t always come first in the English sentence.

The third sentence is tricky. The antecedent is “You can light that match.” Why? The explanation involves something called necessary and sufficient conditions.

4.4.1 Necessary and Sufficient Conditions

A sufficient condition is something that is enough to guarantee the truth of something else. For example, getting a 95 on an exam is sufficient for making an A, assuming that exam is worth 100 points. A necessary condition is something that must be true in order for something else to be true. Making a 95 on an exam is not necessary for making an A—a 94 would have still been an A. Taking the exam is necessary for making an A, though. You can’t make an A if you don’t take the exam, or, in other words, you can make an a only if you enroll in the course.

Here are some important rules to keep in mind:

  • ‘If’ introduces antecedents, but Only if introduces consequents.
  • If A is a sufficient condition for B, then \(A \rightarrow B\) .
  • If A is a necessary condition for B, then \(B \rightarrow A\) .

4.5 Biconditional

We won’t spend much time on biconditionals. There are times when something is both a necessary and a sufficient condition for something else. For example, making at least a 90 and getting an A (assuming a standard scale, no curve, and no rounding up). If you make at least a 90, then you will get an A. If you got an A, then you made at least a 90. We can use a double arrow to translate a biconditional, like this:

  • \(N \rightarrow A\)

For biconditionals, as for conjunctions and disjunctions, order doesn’t matter.

Here are some English phrases that signify biconditionals:

  • it and only if
  • when and only when
  • just in case
  • is a necessary and sufficient condition for

4.6 Translations

Propositional logic is language. Like other languages, it has a syntax and a semantics. The syntax of a language includes the basic symbols of the language plus rules for putting together proper statements in the language. To use propositional logic, we need to know how to translate English sentences into the language of propositional logic. We start with our sentence letters, which represent simple English sentences. Let’s use three borrowed from an elementary school reader:

We then build complex sentences using the sentence letters and our five logical operators, like this:

We can make even more complex sentences, but we will soon run into a problem. Consider this example:

\[ T \mathbin{\&} J \rightarrow S\]

We don’t know this means. It could be either one of the following:

  • Tom hit the ball, and if Jane caught the ball, then Spot chased it.
  • If Tom hit the ball and Jane caught it, then Spot chased it.

The first sentence is a conjunction, \(T\) is the first conjunct and \(M \rightarrow S\) is the second conjunct. The second sentence, though, is a conditional, \(T \mathbin{\&}M\) is the antecdent, and \(S\) is the consequent. Our two interpretations are not equivalent, so we need a way to clear up the ambiguity. We can do this with parentheses. Our first sentence becomes:

\[ T \mathbin{\&} (J \rightarrow S) \]

The second sentence is:

\[ (T \mathbin{\&} J) \rightarrow S\]

If we need higher level parentheses, we can use brackets and braces. For instance, this is a perfectly good formula in propositional logic:

\[ [(P \mathbin{\&} Q) \vee R] \rightarrow \{[(\neg P \leftrightarrow Q) \mathbin{\&} S] \vee \neg P\} \] 6

Every sentence in propositional logic is one of six types:

  • Conjunction
  • Disjunction
  • Conditional
  • Biconditional

What type of sentence it is will be determined by its main logical operator. Sentences can have several logical operators, but they will always have one, and only one, main operator. Here are some general rules for finding the main operator in a symbolized formula of propositional logic:

  • If a sentence has only one logical operator, then that is the main operator.
  • If a sentence has more than one logical operator, then the main operator is the one outside the parentheses.
  • If a sentence has two logical operators outside the parentheses, then the main operator is not the negation.

Here are some examples:

Informally, we use ‘proposition’ and ‘statement’ interchangeably. Strictly speaking, the proposition is the content, or meaning, that the statement expresses. When different sentences in different languages mean the same thing, it is because they express the same proposition. ↩︎

It may be a good formula in propositional logic, but that doesn’t mean it would be a good English sentence. ↩︎

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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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CRITICAL THINKING – Fundamentals: Necessary and Sufficient Conditions

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April 28, 2016

In critical thinking / free will , philosophy / skepticism.

In this video, Kelley Schiffman (Yale University) discusses one of the most basic tools in the philosophers’s tool kit: the distinction between necessary and sufficient conditions. Through the use of ordinary language glosses and plenty of examples this mighty distinction is brought down to earth and presented in a ready-to-use fashion.

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5.2 Logical Statements

Learning objectives.

By the end of this section, you will be able to:

Specific types of statements have a particular meaning in logic, and such statements are frequently used by philosophers in their arguments. Of particular importance is the conditional , which expresses the logical relations between two propositions. Conditional statements are used to accurately describe the world or construct a theory. Counterexamples are statements used to disprove a conditional. Universal statements are statements that assert something about every member of a set of things and are an alternative way to describe a conditional.

Conditionals

A conditional is most commonly expressed as an if–then statement, similar to the examples we discussed earlier when considering hypotheses. Additional examples of if–then statements are “If you eat your meat, then you can have some pudding” and “If that animal is a dog, then it is a mammal.” But there are other ways to express conditionals, such as “You can have pudding only if you eat your meat” or “ All dogs are mammals.” While these sentences are different, their logical meaning is the same as their correlative if–then sentences above.

All conditionals include two components—that which follows the “if” and that which follows the “then.” Any conditional can be rephrased in this format. Here is an example:

Statement 1: You must complete 120 credit hours to earn a bachelor’s degree. Statement 2: If you expect to graduate, then you must complete 120 credit hours.

Whatever follows “if” is called the antecedent ; whatever follows “then” is called the consequent . Ante means “before,” as in the word “antebellum,” which in the United States refers to anything that occurred or was produced before the American Civil War. The ante cedent is the first part of the conditional, occurring before the consequent. A consequent is a result, and in a conditional statement, it is the result of the antecedent (if the antecedent is true).

Necessary and Sufficient Conditions

All conditionals express two relations, or conditions : those that are necessary and those that are sufficient. A relation is a relationship/property that exists between at least two things. If something is sufficient, it is always sufficient for something else . And if something is necessary, it is always necessary for something else. In the conditional examples offered above, one part of the relation is required for the other. For example, 120 credit hours are required for graduation, so 120 credit hours is necessary if you expect to graduate. Whatever is the consequent—that is, whatever is in the second place of a conditional—is necessary for that particular antecedent. This is the relation/condition of necessity. Put formally, Y is a necessary condition for X if and only if X cannot be true without Y being true . In other words, X cannot happen or exist without Y. Here are a few more examples:

But notice that the necessary relation of a conditional does not automatically occur in the other direction. Just because something is a mammal does not mean that it must be a dog. Being a bachelor is not a necessary feature of being unmarried because you can be unmarried and be an unmarried woman. Thus, the relationship between X and Y in the statement “if X, then Y” is not always symmetrical (it does not automatically hold in both directions). Y is always necessary for X, but X is not necessary for Y. On the other hand, X is always sufficient for Y.

Take the example of “If you are a bachelor, then you are unmarried.” If you know that Eric is a bachelor, then you automatically know that Eric is unmarried. As you can see, the antecedent/first part is the sufficient condition, while the consequent/second part of the conditional is the necessary condition. X is a sufficient condition for Y if and only if the truth of X guarantees the truth of Y. Thus, if X is a sufficient condition for Y, then X automatically implies Y. But the reverse is not true. Oftentimes X is not the only way for something to be Y. Returning to our example, being a bachelor is not the only way to be unmarried. Being a dog is a sufficient condition for being a mammal, but it is not necessary to be a dog to be a mammal since there are many other types of mammals.

The ability to understand and use conditionals increases the clarity of philosophical thinking and the ability to craft effective arguments. For example, some concepts, such as “innocent” or “good,” must be rigorously defined when discussing ethics or political philosophy. The standard practice in philosophy is to state the meaning of words and concepts before using them in arguments. And oftentimes, the best way to create clarity is by articulating the necessary or sufficient conditions for a term. For example, philosophers may use a conditional to clarify for their audience what they mean by “innocent”: “If a person has not committed the crime for which they have been accused, then that person is innocent.”

Counterexamples

Sometimes people disagree with conditionals. Imagine a mother saying, “If you spend all day in the sun, you’ll get sunburnt.” Mom is claiming that getting sunburnt is a necessary condition for spending all day in the sun. To argue against Mom, a teenager who wants to go to the beach might offer a counterexample , or an opposing statement that proves the first statement wrong. The teenager must point out a case in which the claimed necessary condition does not occur alongside the sufficient one. Regular application of an effective sunblock with an SPF 30 or above will allow the teenager to avoid sunburn. Thus, getting sunburned is not a necessary condition for being in the sun all day.

Counterexamples are important for testing the truth of propositions. Often people want to test the truth of statements to effectively argue against someone else, but it is also important to get into the critical thinking habit of attempting to come up with counterexamples for our own statements and propositions. Philosophy teaches us to constantly question the world around us and invites us to test and revise our beliefs. And generating creative counterexamples is a good method for testing our beliefs.

Universal Statements

Another important type of statement is the universal affirmative statement . Aristotle included universal affirmative statements in his system of logic, believing they were one of only a few types of meaningful logical statements ( On Interpretation ). Universal affirmative statements take two groups of things and claim all members of the first group are also members of the second group: “All A are B.” These statements are called universal and affirmative because they assert something about all members of group A. This type of statement is used when classifying objects and/or the relationships. Universal affirmative statements are, in fact, an alternative expression of a conditional.

Universal Statements as Conditionals

Universal statements are logically equivalent to conditionals, which means that any conditional can be translated into a universal statement and vice versa. Notice that universal statements also express the logical relations of necessity and sufficiency. Because universal affirmative statements can always be rephrased as conditionals (and vice versa), the ability to translate ordinary language statements into conditionals or universal statements is helpful for understanding logical meaning. Doing so can also help you identify necessary and sufficient conditions. Not all statements can be translated into these forms, but many can.

Counterexamples to Universal Statements

Universal affirmative statements also can be disproven using counterexamples. Take the belief that “All living things deserve moral consideration.” If you wanted to prove this statement false, you would need to find just one example of a living thing that you believe does not deserve moral consideration. Just one will suffice because the categorical claim is quite strong—that all living things deserve moral consideration. And someone might argue that some parasites, like the protozoa that causes malaria, do not deserve moral consideration.

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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

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):

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):

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:

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:

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):

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:

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:

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

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.

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

conditionals | conditionals: counterfactual | definitions | logic: classical | logic: conditionals | logic: modal | logic: relevance

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.

Copyright © 2017 by Andrew Brennan < A . Brennan @ latrobe . edu . au >

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Wireless Philosophy

Course: wireless philosophy   >   unit 1.

  • Fundamentals: Introduction to Critical Thinking
  • Introduction to Critical Thinking, Part 1
  • Introduction to Critical Thinking, Part 2
  • Fundamentals: Deductive Arguments
  • Deductive Arguments
  • Fundamentals: Abductive Arguments

Necessary and Sufficient Conditions

  • Instrumental vs. Intrinsic Value
  • Implicit Premise
  • Justification and Explanation
  • Normative and Descriptive Claims
  • Fundamentals: Validity
  • Fundamentals: Truth and Validity
  • Fundamentals: Soundness
  • Fundamentals: Bayes' Theorem
  • Fundamentals: Correlation and Causation
  • (Choice A)   True A True
  • (Choice B)   False B False

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  • What Is Critical Thinking? | Definition & Examples

What Is Critical Thinking? | Definition & Examples

Published on May 30, 2022 by Eoghan Ryan . Revised on May 31, 2023.

Critical thinking is the ability to effectively analyze information and form a judgment .

To think critically, you must be aware of your own biases and assumptions when encountering information, and apply consistent standards when evaluating sources .

Critical thinking skills help you to:

  • Identify credible sources
  • Evaluate and respond to arguments
  • Assess alternative viewpoints
  • Test hypotheses against relevant criteria

Table of contents

Why is critical thinking important, critical thinking examples, how to think critically, other interesting articles, frequently asked questions about critical thinking.

Critical thinking is important for making judgments about sources of information and forming your own arguments. It emphasizes a rational, objective, and self-aware approach that can help you to identify credible sources and strengthen your conclusions.

Critical thinking is important in all disciplines and throughout all stages of the research process . The types of evidence used in the sciences and in the humanities may differ, but critical thinking skills are relevant to both.

In academic writing , critical thinking can help you to determine whether a source:

  • Is free from research bias
  • Provides evidence to support its research findings
  • Considers alternative viewpoints

Outside of academia, critical thinking goes hand in hand with information literacy to help you form opinions rationally and engage independently and critically with popular media.

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definition of sufficient condition in critical thinking

Critical thinking can help you to identify reliable sources of information that you can cite in your research paper . It can also guide your own research methods and inform your own arguments.

Outside of academia, critical thinking can help you to be aware of both your own and others’ biases and assumptions.

Academic examples

However, when you compare the findings of the study with other current research, you determine that the results seem improbable. You analyze the paper again, consulting the sources it cites.

You notice that the research was funded by the pharmaceutical company that created the treatment. Because of this, you view its results skeptically and determine that more independent research is necessary to confirm or refute them. Example: Poor critical thinking in an academic context You’re researching a paper on the impact wireless technology has had on developing countries that previously did not have large-scale communications infrastructure. You read an article that seems to confirm your hypothesis: the impact is mainly positive. Rather than evaluating the research methodology, you accept the findings uncritically.

Nonacademic examples

However, you decide to compare this review article with consumer reviews on a different site. You find that these reviews are not as positive. Some customers have had problems installing the alarm, and some have noted that it activates for no apparent reason.

You revisit the original review article. You notice that the words “sponsored content” appear in small print under the article title. Based on this, you conclude that the review is advertising and is therefore not an unbiased source. Example: Poor critical thinking in a nonacademic context You support a candidate in an upcoming election. You visit an online news site affiliated with their political party and read an article that criticizes their opponent. The article claims that the opponent is inexperienced in politics. You accept this without evidence, because it fits your preconceptions about the opponent.

There is no single way to think critically. How you engage with information will depend on the type of source you’re using and the information you need.

However, you can engage with sources in a systematic and critical way by asking certain questions when you encounter information. Like the CRAAP test , these questions focus on the currency , relevance , authority , accuracy , and purpose of a source of information.

When encountering information, ask:

  • Who is the author? Are they an expert in their field?
  • What do they say? Is their argument clear? Can you summarize it?
  • When did they say this? Is the source current?
  • Where is the information published? Is it an academic article? Is it peer-reviewed ?
  • Why did the author publish it? What is their motivation?
  • How do they make their argument? Is it backed up by evidence? Does it rely on opinion, speculation, or appeals to emotion ? Do they address alternative arguments?

Critical thinking also involves being aware of your own biases, not only those of others. When you make an argument or draw your own conclusions, you can ask similar questions about your own writing:

  • Am I only considering evidence that supports my preconceptions?
  • Is my argument expressed clearly and backed up with credible sources?
  • Would I be convinced by this argument coming from someone else?

If you want to know more about ChatGPT, AI tools , citation , and plagiarism , make sure to check out some of our other articles with explanations and examples.

  • ChatGPT vs human editor
  • ChatGPT citations
  • Is ChatGPT trustworthy?
  • Using ChatGPT for your studies
  • What is ChatGPT?
  • Chicago style
  • Paraphrasing

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  • Types of plagiarism
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Critical thinking refers to the ability to evaluate information and to be aware of biases or assumptions, including your own.

Like information literacy , it involves evaluating arguments, identifying and solving problems in an objective and systematic way, and clearly communicating your ideas.

Critical thinking skills include the ability to:

You can assess information and arguments critically by asking certain questions about the source. You can use the CRAAP test , focusing on the currency , relevance , authority , accuracy , and purpose of a source of information.

Ask questions such as:

  • Who is the author? Are they an expert?
  • How do they make their argument? Is it backed up by evidence?

A credible source should pass the CRAAP test  and follow these guidelines:

  • The information should be up to date and current.
  • The author and publication should be a trusted authority on the subject you are researching.
  • The sources the author cited should be easy to find, clear, and unbiased.
  • For a web source, the URL and layout should signify that it is trustworthy.

Information literacy refers to a broad range of skills, including the ability to find, evaluate, and use sources of information effectively.

Being information literate means that you:

  • Know how to find credible sources
  • Use relevant sources to inform your research
  • Understand what constitutes plagiarism
  • Know how to cite your sources correctly

Confirmation bias is the tendency to search, interpret, and recall information in a way that aligns with our pre-existing values, opinions, or beliefs. It refers to the ability to recollect information best when it amplifies what we already believe. Relatedly, we tend to forget information that contradicts our opinions.

Although selective recall is a component of confirmation bias, it should not be confused with recall bias.

On the other hand, recall bias refers to the differences in the ability between study participants to recall past events when self-reporting is used. This difference in accuracy or completeness of recollection is not related to beliefs or opinions. Rather, recall bias relates to other factors, such as the length of the recall period, age, and the characteristics of the disease under investigation.

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5: Necessary and Sufficient Conditions

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  • Michael Shaffer
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p \rightarrow q

\[\mathrm{S}(p, q) \equiv (p \rightarrow q)\]

\[\mathrm{N}(q, p) \equiv (p \rightarrow q)\]

In effect, D1 and D2 are then intended to be the standard logical interpretations of our ordinary language concepts of necessary and sufficient conditions framed in terms of classical propositional logic. [2] They are based on the idea that necessary and sufficient conditions can be exhaustively defined in terms of the conditional understood as material implication and represented by the “→” of classical propositional logic with the following familiar truth conditions: [3]

Of course, material implication plays an important role in reasoning in general, particularly with respect to the following valid inferential forms in classical propositional logic, as we saw in Chapter 3.

Affirming the Antecedent ( Modus Ponens )

1. \(A \rightarrow B\) 2. \(A\) 3. \(/ \therefore B\)

Denying the Consequent ( Modus Tollens )

1. \(A \rightarrow B\) 2. \(\neg B\) 3. \(/ \therefore \neg A\)

These inference forms have important connections to the concepts of necessary and sufficient conditions, and to how we reason using them. In the case of affirming the antecedent, the first premise can be understood to be the claim that A is sufficient for B, and the second premise the claim that the condition A obtains. So, from these claims it validly follows that B obtains. In the case of denying the consequent, the first premise can be read as the claim that B is a necessary condition for A and the second premise as the claim that B does not obtain. From these premises it validly follows that A does not obtain.

However, the following inferential forms involving material implication are invalid in classical propositional logic:

Affirming the Consequent

1. \(A \rightarrow B\) 2. \(B\) 3. \(/ \therefore A\)

Denying the Antecedent

1. \(A \rightarrow B\) 2. \(\neg A\) 3. / \(\therefore \neg B\)

These invalid inference forms also are importantly related to the concepts of necessary and sufficient conditions. In the case of affirming the consequent, the first premise can be read as the claim that A is a necessary condition for B and the second premise as the claim that B is true. But, from these premises it does not validly follow that A is also true. The fact that B is necessary for A does not ensure it is also sufficient for A. In the case of denying the antecedent, the first premise can be read as the claim that A is a sufficient condition for B and the second premise as the claim that A is not true. From these premises it does not validly follow that B is not true, as some other condition that suffices for B might, in fact, obtain.

Moreover, where NS( p , q ) means “ p is necessary and sufficient for q , and q is necessary and sufficient for p ,” such jointly necessary and sufficient conditions take the following form: [4]

\[\mathrm{NS}(p, q) \equiv [(p \rightarrow q) \,\& \,(q \rightarrow p)]\]

(p \rightarrow q) \,\&\, (q \rightarrow p)

\[\mathrm{NS}(p, q) \equiv (p \equiv q)\]

This concept is just the idea that the truth values of p and q are always the same, and the notion of logical equivalence has the following truth conditions:

Sets of jointly necessary and sufficient conditions are, then, just definitions regimented as sentences of this sort. For example, it turns out that being a bachelor and being an unmarried male are jointly necessary and sufficient conditions for one another. Now why, specifically, are the concepts of necessary and sufficient conditions, so understood, of such central significance in contemporary analytic philosophy?

Conceptual Analysis and Necessary and Sufficient Conditions

The central account of the methods of contemporary analytic philosophy is predicated on the claim that philosophical methodology is intuition-driven conceptual analysis that aims to determine true sets of necessary and sufficient conditions. In fact, according to a significant number of philosophers, such conceptual analysis is the only method of philosophy. For the purposes at hand, this account of the methods of philosophy will be referred to as the standard philosophical method (SPM). Conceptual analyses take the form of specifications of the content of a pre-theoretical concept (the analysans) through the articulation of a set of necessary and sufficient conditions (the analysandum or analysanda), and here we find the locus of the connection between the concepts of necessary and sufficient conditions and philosophical methodology. This methodological account of philosophy can be more completely characterized as follows:

  • Conceptual analyses take the form of proposed definitions (i.e. sets of necessary and sufficient conditions) of analysanda.
  • The adequacy of any analysandum can be tested against concrete and/or imagined cases.
  • Whether or not a proposed analysandum is adequate with respect to a given case can be determined by the use of a priori intuition, with a priori intuition being a distinct, reliable and fallible non-sensory mental faculty. [5]
  • Intuition allows us to reliably access knowledge about concepts.
  • The method of reflective equilibrium is the particular method by which intuitions can be used to confirm/disconfirm analysanda. [6]

According to the defenders of SPM, this is essentially the orthodox methodology of analytic philosophy, and it has been assumed to be adequate for the solution of philosophical problems by a significant number of both practicing and prominent philosophers throughout the recent history of philosophy. For example, this is the contention made by Colin McGinn in a recent book. McGinn is not in the least bit tentative in his blanket defense of SPM as the one and only method of philosophy. With this aim in mind, early in his 2012 book he makes the following extended declaration about philosophy:

… it is not a species of empirical enquiry, and it is not methodologically comparable to the natural sciences (though it is comparable to the formal sciences). It seeks the discovery of essences. It operates “from the armchair”: that is, by unaided (usually solitary) contemplation. Its only experiments are thought-experiments, and its data are possibilities (or “intuitions” about possibilities). Thus philosophy seeks a priori knowledge of objective being—of non-linguistic and non-conceptual reality. We are investigating being as such, but we do so using only a priori methods. (McGinn 2012, 4)

As should be immediately apparent, this is a clear, straightforward, and ringing endorsement of SPM as it has been understood here. To buttress this contention we need only take note of his other claims that “…the proper method for uncovering the essence of things is precisely conceptual analysis,” (McGinn 2012, 4) and that “philosophy, correctly conceived, simply is conceptual analysis” (McGinn 2012, 11). In effect, he believes then that we arrive at such analyses by considering possible cases and asking ourselves whether the concept applies or not in those cases—that is by consulting our “intuitions” about such cases (McGinn 2012, 5). What is also important for the purposes at hand is his acknowledgment that this account of philosophical methodology “was really the standard conception for most of the history of the subject, in one form or another” (McGinn 2012, 7). So, not only does McGinn endorse SPM as the sole methodology of contemporary philosophy, but he also claims that it is the enduring methodology of philosophical inquiry throughout its history. [7]

One important clarification regarding McGinn’s version of SPM concerns the nature of the object of analysis (the analysans) and, more importantly, the nature of the analysandum itself as they are typically understood (i.e. as definitions of a particular sort framed as sets of necessary and sufficient conditions). Carl Hempel usefully makes a crucial distinction in this regard, which we can use to illuminate the standard view of such definitions:

The word “definition” has come to be used in several different senses….A real definition is conceived of as a statement of the “essential characteristics” of some entity, as when man is defined as a rational animal or a chair as a separate moveable seat for one person. A nominal definition, on the other hand, is a convention which merely introduces an alternative—and usually abbreviated—notation for a given linguistic expression, in the manner of a stipulation. (Hempel 1952, 2)

Moreover, he tells us further that some real definitions are to be understood as meaning analyses , or as analytic definitions , of the term in question. The validation of such claims requires only that we know the meanings of the constituent expressions, and no empirical investigation is necessary to determine the correctness of the analysandum (Hempel 1952, 8).

This is, of course, precisely what McGinn has in mind with respect to conceptual analysis. It is, then, worth making the obvious point that conceptual analysis is the operation of analyzing concepts via proposing definitions, but to point that out is not enough to fully grasp the view. It is true that SPM is a method that takes as inputs our concepts, but it involves the clear recognition that the definitions involved are to be understood as meaning analyses rather than as nominal or stipulative (i.e. “dictionary”) definitions. So, for example, the question of whether knowledge is justified true belief is just the question of the analysis of the concept of knowledge in terms of definitions constituted by sets of necessary and sufficient conditions understood as a meaning analysis . Conceptual analysis is then a method of doing something with concepts that we already possess— wherever they have ultimately come from. [8] It is defining a pre-theoretical concept by offering a synonymous expression. It then appears to be the case that the defenders of SPM must believe that concepts have the form of sets of necessary and sufficient conditions, that such analyses are meaning analyses, and that analyses of our pre-analytic concepts are informative. Typical analysanda are thus kinds of decompositions of pre-analytic concepts. They are conceptual truths with the form of analytic definitions.

So, for McGinn and other like-minded thinkers, analysanda have a very simple logical form, and we can see this via the example of the analysis of the concept of knowledge. Where K x is “ x is knowledge”, J x is “ x is justified”, T x is “ x is true” and B x is “ x is believed”, the standard analysis of knowledge looks like this:

This analysis is supposed to tell us the true nature, or essence, of the concept of knowledge in terms of a finite set of defining essential features, with the logical form of a set of jointly necessary and sufficient conditions. So, providing such an analysis involves decomposing the analysans into a list of features, thus exposing in some important sense the content of the concept.

A Problem with the Orthodox View and SPM

Many recent critics have attacked SPM in terms of (2)-(5) by challenging the reliability of the faculty of intuition. This is the main line of criticism against SPM offered by many defenders of what is called experimental philosophy , and it is an interesting criticism of orthodox philosophy indeed. However, some critics have alternatively attacked SPM by challenging (1) on the basis of the theory of concepts it assumes; specifically, the idea that concepts can be adequately captured by sets of necessary and sufficient conditions. [9] One version of this latter form of criticism is particularly relevant to this chapter. This criticism is based on the contention that SPM wrongly assumes that concepts take the form of necessary and sufficient conditions at all. Call this the potential vacuity problem .

The Potential Vacuity Problem

The problem of potential vacuity arises as follows, and is based on Ludwig Wittgenstein’s infamous remarks about the theory of concepts assumed in SPM. He addressed the matter of the reliability of SPM in his Philosophical Investigations and The Blue and Brown Books , and therein Wittgenstein attacks the foundation of the project of conceptual analysis by attempting to undermine (1) via examination of the claim that concepts have the form of sets of necessary and sufficient conditions. [10] First, Wittgenstein rejected the notion that most, or even perhaps any, concepts can be defined precisely via the specification of sets of necessary and sufficient conditions, and that this is a problem central to orthodox philosophy. This important revelation was made by noting that philosophical attempts at conceptual analysis have systematically failed to produce the goods. He tells us explicitly that,

We are unable to clearly circumscribe the concepts we use; not because we don’t know their real definition, but because there is no real “definition” to them. (Wittgenstein 1958, 25)

Second, he sought to replace the notion of concepts understood as sets of necessary and sufficient conditions with an alternative theory of concepts. This alternative account of concepts is based on the notion of a “family resemblance relation.”

To see the first point more clearly, let us look at Wittgenstein’s favorite example from his Philosophical Investigations . Wittgenstein specifically argued that the concept of a game cannot be correctly analyzed in terms of a set of necessary and sufficient conditions. This is because games do not share some set of defining features in common. Rather, the members of the set of games are only similar to one another in some respects, and it is these relations of similarity that constitute the family of games. As we have seen, SPM assumes the following principle:

(CON) For any concept C, there exists a set of necessary and sufficient conditions that constitutes the content of C.

Wittgenstein’s attack on SPM is mounted via an attack on CON, and this is the fundamental ground of the potential vacuity problem. Essentially, the gist of the problem is that if there are no (or even just very few) concepts that can be correctly regimented as sets of necessary and sufficient conditions, there can be no (or very few) correct conceptual analyses in the sense of SPM. The basis of Wittgenstein’s criticism then can be understood as follows: it is clear from the consideration of examples across the history of philosophy that most or all philosophical attempts to analyze concepts by providing sets of necessary and sufficient conditions have failed. This is because, for any proposed set of necessary or sufficient conditions intended to be the correct analysis of a concept, there are instances of that concept that do not meet the set of proposed defining conditions.

Think back to Wittgenstein’s favorite example of the concept of a game. Poker and soccer are both plausibly taken to be games and so we might, for example, posit that something is a game, if and only if, that activity involves a winner and a loser. But, the game patty cake is another plausible case of a game and does not have a winner and a loser. So, this definition of a game in terms of a set of necessary and sufficient conditions fails. Wittgenstein claims that this example generalizes, and the presumptive best explanation for the failed philosophical attempts to articulate the contents of concepts in terms of sets of necessary and sufficient conditions is that the contents of concepts are not captured by sets of necessary and sufficient conditions (i.e. the denial of CON). In other words, Wittgenstein holds that for any (or, at least most) attempt(s) to specify the contents of concepts in terms of necessary and sufficient conditions, we will find counter-examples.

As a replacement for CON, Wittgenstein introduces the notion of a family resemblance class. The central idea is that the cases that fall under a concept are related to one another not by a defining set of necessary and sufficient conditions, but rather by complex overlapping similarity conditions that relate groups of members of the total set of cases that fall under the concept. However, no one set of conditions holds for all and only the members that exhibit that concept. Thus, if Wittgenstein is correct, the reason that there are no correct conceptual analyses is due to the fact that concepts cannot be analysed in terms of necessary and sufficient conditions. SPM is, thus, potentially (if not actually) vacuous.

Prospective Solutions to the Potential Vacuity Problem

Does Wittgenstein’s criticism signal defeat of the SPM, then? Not necessarily. Colin McGinn (2012) proposes a solution to the problem. First, notice that Wittgenstein’s criticism is a direct denial of (1). [11] McGinn responds by biting the bullet against Wittgenstein and arguing that, although they are very often difficult to articulate, concepts are properly characterized by sets of necessary and sufficient conditions. Pace Wittgenstein, our failure to articulate definitive examples of such analyses is no reason to suppose that there are no such things. More cleverly, he shows how Wittgenstein’s criticism can be effectively rebutted in the following way. As we have seen, Wittgenstein’s claim that concepts cannot be captured by sets of necessary and sufficient conditions is supposed to follow from his investigation of the concept of a game. But, as McGinn points out, from the fact that it is difficult to produce the goods in this (or any other) case, it does not necessarily follow that there are no such analyses (McGinn 2012, 21-28).

Second, Wittgenstein uses this point in support of the claim that concepts actually have the structure of a set of family resemblance relations between paradigm and non-paradigm elements in the extension of a concept. What McGinn then shows is that Wittgenstein’s own theory of concepts in terms of family resemblances presupposes that concepts can be captured by a special type of necessary and sufficient conditions: for any concept C, the non-paradigmatic members of C bear a family resemblance relation to the paradigmatic case(s) of C. [12] So, it would appear to be the case that according to Wittgenstein, something is necessarily a concept, if and only if, it is a set of entities related by family resemblance relations to one or more paradigm cases. As such, McGinn rightly claims that Wittgenstein does not reject SPM. Rather, in his treatment of the concept of game he is “favoring a particular form of it—one in which the analysis takes the form ‘family-resembles paradigm games’ (such as chess, tennis, etc.)” (McGinn 2012, 18-19). However, this response does nothing to defuse the problem that such specifications of conceptual contents cannot plausibly be necessary truths, as McGinn and other defenders of SPM typically believe. This is because family resemblance relations cannot plausibly be understood to be necessary truths. In other words, it is clearly not the case that resemblance relations between objects are such that they are true in all possible worlds. [13] This is the case because resemblances are not purely objective relations between objects. They are perceiver relative, and so vary depending on what features one focuses on. For example, a pen resembles a pencil when one focuses on the function of writing. But, a pen and a pencil do not resemble one another when one focuses instead on the feature of containing ink.

Exercise One

For each pair, decide whether the first member of the pair is either a necessary condition for the second, a sufficient condition, or neither .

Example: Bob’s car is blue/Bob’s car is coloured

Answer: Bob’s car being blue is sufficient for it being coloured, as its being blue ensures that it is coloured. However, it isn’t a necessary condition , for Bob’s car could be coloured without being blue—it could be red, for example.

  • Bob drew the eight of Spades from an ordinary deck of playing cards. Bob drew a black card from a deck of ordinary playing cards.
  • Alice has a brother-in-law. Alice is not an only child.
  • Alice’s daughter is married. Alice is a parent.
  • Alice’s daughter is married. Alice is a grandmother.
  • Some women pay taxes. Some taxpayers are women.
  • All women pay taxes. All taxpayers are women.
  • Being a mammal. Being warm blooded.
  • Being warm blooded.
  • Being a mammal.

Exercise Two

For each claim, rewrite it in terms of necessary and/or sufficient conditions.

Example: You can’t play football without a ball

Answer: Having a ball is necessary for playing football.

  • You must pay if you want to enter.
  • A cloud chamber is needed to observe subatomic particles.
  • If something is an electron it is a charged particle.
  • Your car is only cool if it’s a Honda.
  • Being a triangle just is being a three-sided, two-dimensional shape.

Exercise Three

Test for yourself the traditional philosophical assumption that concepts are defined by necessary and sufficient conditions . Try to provide necessary and sufficient conditions for the following concepts, and then test these set of conditions with potential counterexamples:

  • Health (mental and physical)

Potential counterexamples to your analysis of these concepts in terms of necessary and sufficient conditions can either take the form of:

  • Cases that the concept should apply to, but which don’t fulfill your necessary and sufficient conditions.
  • Cases that the concept should not apply to, but which do fulfill your necessary and sufficient conditions.
  • As given previously in Chapter 3. ↵
  • See, for example, Copi, Cohen and Flage (2007, 196, 446, 449) and Fisher (2001, 241). ↵
  • The concept of the material conditional introduced here is just a formalization of what we were previously and informally calling “conditionals”. ↵
  • NS( p , q ) is then equivalent to S( p , q ) & S( q , p ) & N( p , q ) & N( q , p ). ↵
  • A priori knowledge is knowledge totally independent of any experience. ↵
  • Recent defenses of SPM include: Bealer (1996), Jackson (1998), and McGinn (2012). For closely related views, see Braddon-Mitchell and Nola (2009). See Shaffer (forthcoming) for extensive discussion of this view. Reflective equilibrium is the method of bringing intuitively true cases into conformity with a rule or principle. ↵
  • See McGinn (2012, 4-11) for a summary of significant historical examples of the use of SPM, including some of those discussed here in more detail. ↵
  • Strictly speaking, conceptual analyses can also involve some degree of alteration in the content of the pre-theoretical concepts, as often happens when such analysis involves making a concept more precise. ↵
  • See Moore (1968) and Wittgenstein (1953), for example. Moore’s paradox of analysis appears to show that such analyses are uninformative, and Wittgenstein claims that concepts have the form of family resemblances, rather than sets of necessary and sufficient conditions. See also Brennan (2017) and Shaffer (2015) for additional worries about the nature of necessary and sufficient conditions. ↵
  • See Wittgenstein (1953), Lakoff (1987), Ramsey (1998), Rosch and Mervis (1998), and McGinn (2012, Ch. 3) for more on this matter. ↵
  • Wittgenstein’s criticism also has important additional application to views, like that of McGinn, where conceptual truths are understood to be necessary truths. This is because if concepts are not captured by sets of necessary and sufficient conditions, and only have the form of sets of cases related by family resemblances, then it is not easily understood how they could possibly be necessarily true definitions. This is simply because relations of resemblance between things appear to be contingent relations. ↵
  • Paradigm members of a family resemblance class are the obvious central cases, whereas non-paradigmatic cases are less central and obvious cases of that class. So, for example, a robin is a paradigmatic case of the class of birds, whereas a penguin is (plausibly) a non-paradigmatic case of a bird. ↵
  • This understanding of necessary truth as claims that are true in all possible worlds is the standard concept of a necessary truth. Such truths cannot be false in any consistent arrangement of what could possibly exist. ↵

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Skills and dispositions of critical thinking: are they sufficient?

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sufficient condition

Definition of sufficient condition

Examples of sufficient condition in a sentence.

These examples are programmatically compiled from various online sources to illustrate current usage of the word 'sufficient condition.' Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Send us feedback about these examples.

Word History

1885, in the meaning defined at sense 1

Dictionary Entries Near sufficient condition

sufficientness

Cite this Entry

“Sufficient condition.” Merriam-Webster.com Dictionary , Merriam-Webster, https://www.merriam-webster.com/dictionary/sufficient%20condition. Accessed 22 Mar. 2024.

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COMMENTS

  1. Necessary and Sufficient Conditions

    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 ...

  2. 2.3: Necessary and Sufficient Conditions

    3. Necessary and Sufficient. 4. Neither Necessary nor Sufficient. Necessary and Sufficient conditions are things that are both enough for and required for something else. If X is a necessary and sufficient condition for Y, then: If X obtains, then Y must obtain (so any time X obtains, Y also obtains) And.

  3. 3.2: Necessary and Sufficient Conditions

    3.2: Necessary and Sufficient Conditions. One sentence is a sufficient condition for another sentence if the truth of the first would guarantee the truth of the second. The truth of the first is enough - all you need, sufficient - to ensure the truth of the second. Having your head cut off is a sufficient condition for being dead.

  4. 1.4: Necessary and Sufficient Conditions

    A Miniguide to Critical Thinking (Lau) 1: Chapters 1.4: Necessary and Sufficient Conditions ... Expressions such as If X then Y, or X is enough for Y, can also be understood as saying that X is a sufficient condition for Y. Note that some state of affairs can have more than one sufficient condition. Being blue is sufficient for being colored ...

  5. Chapter 4 Propositional Logic

    4.4.1 Necessary and Sufficient Conditions. A sufficient condition is something that is enough to guarantee the truth of something else. For example, getting a 95 on an exam is sufficient for making an A, assuming that exam is worth 100 points. A necessary condition is something that must be true in order for something else to be true.

  6. Chapter 4: Necessary and Sufficient Conditions

    CHAPTER 4 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 … - Selection from An Introduction to Critical Thinking and Creativity: Think More, Think Better [Book]

  7. Necessary and Sufficient Conditions

    The hook operator ("⊃") captures much of what is meant by reference to necessary and sufficient conditions in such contexts. 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.

  8. Chapter 8 Summary

    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.

  9. Reasoning About Alternatives and Necessary and Sufficient Conditions

    Since critical thinking is aimed at knowledge, the kinds of reasons it requires are epistemic reasons, which are more commonly called "evidence." In this chapter, the authors study three very familiar and very useful forms of reasoning, methods for drawing conclusions from the evidence that we can rely on when thinking critically about what ...

  10. CRITICAL THINKING

    [3:14] In this video, Kelley Schiffman (Yale University) discusses one of the most basic tools in the philosophers's tool kit: the distinction between necessary and sufficient conditions. Through the use of ordinary language glosses and plenty of examples this mighty distinction is brought down to earth and presented in a ready-to-use fashion. [Video and text source: Wireless Philosophy ...

  11. 5.2 Logical Statements

    As you can see, the antecedent/first part is the sufficient condition, while the consequent/second part of the conditional is the necessary condition. X is a sufficient condition for Y if and only if the truth of X guarantees the truth of Y. Thus, if X is a sufficient condition for Y, then X automatically implies Y. But the reverse is not true.

  12. Necessary and sufficient conditions

    Necessary Conditions. Sufficient Conditions. Describing how two Things are Connected. The Write-off Fallacy. Different Kinds of Possibility. Exclusive and Exhaustive Possibilities. An Introduction to Critical Thinking and Creativity: Think More, Think Better. Related; Information; Close Figure Viewer. Return to Figure. Previous Figure Next ...

  13. Critical Thinking

    Critical Thinking. Critical thinking is a widely accepted educational goal. Its definition is contested, but the competing definitions can be understood as differing conceptions of the same basic concept: careful thinking directed to a goal. Conceptions differ with respect to the scope of such thinking, the type of goal, the criteria and norms ...

  14. [M06] Necessity and sufficiency

    The concepts of necessary and sufficient conditions help us understand and explain the different kinds of connections between concepts, and how different states of affairs are related to each other. §1. Necessary conditions. To say that X is a necessary condition for Y is to say that it is impossible to have Y without X.

  15. Critical thinking problem with necessary and sufficient conditions

    The definitions of the following ideas are standardized in any course on scientific reasoning or critical thinking: Giving a definition means finding a necessary and sufficient condition. For example, the definition for "Brother" is "A male sibling." ... A sufficient condition for some state of affairs S is a condition that, if satisfied ...

  16. Florida International University

    Natural Kinds and Definitions . If the thing under consideration has an essential nature, a "sine qua non" then an adequate definition for the concept would identify all the necessary and sufficient qualities of the thing defined such that the definition picks out all and only members of the set named by the concept. But more than that, the definition would not simply assemble a set ...

  17. 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. ... "I opened the door")? Many logic and critical thinking texts use just such an approach, and for convenience we may call it ...

  18. 2.4: Necessary and Sufficient Conditions

    The concepts of necessary and sufficient conditions help us understand and explain the different kinds of connections between concepts, and how different states of affairs are related to each other. To say that X is a necessary condition for Y is to say that it is impossible to have Y without X. In other words, the absence of X guarantees the ...

  19. Necessary and Sufficient Conditions (practice)

    Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

  20. What Is Critical Thinking?

    Critical thinking is the ability to effectively analyze information and form a judgment. To think critically, you must be aware of your own biases and assumptions when encountering information, and apply consistent standards when evaluating sources. Critical thinking skills help you to: Identify credible sources. Evaluate and respond to arguments.

  21. 5: Necessary and Sufficient Conditions

    5: Necessary and Sufficient Conditions. The concepts of necessary and sufficient conditions play central and vital roles in analytic philosophy. For example, being an unmarried male is a necessary condition for being a bachelor and being a bachelor is a sufficient condition for being an unmarried male.

  22. Skills and dispositions of critical thinking: are they sufficient?

    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.

  23. Sufficient condition Definition & Meaning

    The meaning of SUFFICIENT CONDITION is a proposition whose truth assures the truth of another proposition.