necessary and sufficient in logic and critical thinking

5.2 Logical Statements

Learning objectives.

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

• Identify the necessary and sufficient conditions in conditionals and universal affirmative statements.
• Describe counterexamples for statements.
• Assess the truth of conditionals and universal statements using counterexamples.

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:

• Being unmarried is a necessary condition for being a bachelor . If you are a bachelor, then you are unmarried.
• Being a mammal is a necessary condition for being a dog. If a creature is a dog, then it is a mammal.

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

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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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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19 Logic and Critical Thinking

Introduction [1].

This chapter is a primer on basic logical concepts that often appear in various critical thinking textbooks—concepts such as entailment, contraries, contradictories, necessary and sufficient conditions, etc. The chapter will not provide a historical genealogy of these concepts—in some sense critical thinking, argumentation theory, and formal logic all trace their roots back to at least Aristotle over two thousand years ago. As a result, for many of these concepts, determining whether the concept was a logic concept co-opted by critical thinking, or a critical thinking concept co-opted and changed by logic and then co-opted back again, is extremely difficult. Regardless, a brief orientation of the relationship of critical thinking and logic is in order.

Critical thinking, at least as it is most often justified, is a practical, skill-building exercise with the goal of improving our reasoning. This motivation, of understanding and improving our reasoning, has also been the motivation behind the development of logic over the past several thousand years. While we could study and understand each piece of reasoning individually, it is much more efficient to look for reasoning patterns that recur over and over again, to distinguish those patterns that are good from those that are bad, and so to find principles underpinning our reasoning that help us distinguish good reasoning from bad reasoning across the board. This push to generalize and theorize with the patterns of reasoning generated numerous formal logical systems, including the syllogistic and modal logics of Aristotle.

However, logic, especially formal logic, has not been constrained solely by the goal of understanding and improving our reasoning. Like abstract mathematics, the formal structures underpinning logical systems, are rich and complex enough to generate study all their own, with no concern for the original motivation that may have pushed us to study patterns of human reasoning. Regardless, many of logic’s concepts are still useful in organizing any study of reasoning.

In what follows I begin with a fairly substantial discussion of the core concept needed to understand the traditional logical concepts such as entailment or contradictory or necessary condition—the concept of a possibility. Once we have this notion in play, the definitions of the standard logical concepts, which I provide in Section 2, are quite straightforward. In the final section, I discuss the potential for misapplication of various concepts or distinctions.

1. Possibilities

1.1 possibility and reasoning.

The core concept of logic is the concept of a possibility (a case, a scenario, an option, a way things could be). While logicians and philosophers continue to work on illuminating the nature of possibilities, we can, even without a precise definition, still intuitively grasp the notion. You could stop reading right now or you could keep going. England didn’t win, but England could have won, if they had scored their penalty kick. That die, when rolled, will land on one of six possible sides. There are many things that might happen if the bill is passed into law. According to 18th century philosopher Gottfried Leibniz, God surveyed all the possible ways the universe might be and, being omnibenevolent, chose to create the best one (this one!?).

We appeal to possibilities all the time in our reasoning. Indeed, if there were but one way things could be and we knew completely what that way was like, then we would not need to reason at all—we would just know how things were going to unfold. But given that (i) we do not know completely how things are or how the future is going to unfold and (ii) we assume there are multiple possibilities for how the future might unfold, we need to reason about the ways things could be in order to learn how things are and how to best manage whatever the future brings. For example, the detective investigating a suspicious death gets a new piece of evidence—the deceased was killed by a rare poison. As a result, some scenarios are closed off as viable explanations of the death—e.g., the deceased was deliberately killed by someone who did not have access to the poison. Other scenarios, ones that may not have been in the detective’s awareness before the new piece of evidence was acquired, become relevant—e.g., that someone who knew or at least had access to the poison was responsible for the death. As a result, a new line of inquiry opens for the detective: find out who had access to the poison. Similarly, a doctor runs a series of tests to try to eliminate certain possible explanations for a given patient’s symptoms. Given certain results the possible explanations get narrowed down to one (and hopefully a treatment is available); given other results multiple possibilities remain and the doctor has to decide which tests may be required for progress to be made; unexpected results, while eliminating some possibilities may open up new possibilities that the doctor had not originally been considering. Finally, you are trying to decide when and in which order to run a list of errands. You take into account the likely lines at each location at different times of day, and the likely traffic at different times of day. After evaluating the possibilities, you choose the best option for you.

1.2 Types of possibil i ties

1.2 .1 physical & epistemic possibilities.

Given the ubiquity of possibilities in our reasoning, theorists often classify the possibilities. For example, physicists are interested in distinguishing the physical possibilities (the possibilities consistent with the laws of physics) from the physical impossibilities (the possibilities inconsistent with the laws of physics). Other general types of possibilities include epistemic possibilities—scenarios consistent with what we know; moral possibilities—those consistent with a given moral code; legal possibilities—situations consistent with what is permitted by a given legal code. We can even combine these types—epistemic physical possibilities are those that are consistent with the laws of physics as we currently know them. If what we know about the world changes at a fundamental level, what once was epistemically physically possible (measuring time independently of motion or gravity) may become epistemically physically impossible. Like for the detective and the doctor above, new, unexpected evidence may require an adjustment by the scientist in what possibilities are under consideration as viable explanations.

1.2 .2 Equally probable possibilities

Two other sorts of possibilities deserve mention. Probabilistic reasoning depends on possibilities of a very special sort—equally probable possibilities. To determine the probability that a fair coin will come up heads we assume that there are two equally likely possibilities, “heads’’ and “tails” (we usually ignore the extremely unlikely, though still physically possible situation in which the coin lands and stays on its edge). Failing to consider the relevant equally likely possibilities can make our probabilistic reasoning go awry. You will either win the lottery or you will not. There are two possibilities here, but treating them as equally likely is certainly an obvious mistake. Assuming the lottery is fair, the relevant equally likely possibilities are that each individual ticket (or set of numbers) will be the winner. If your ticket is one of many, then the probability you will win the lottery is much lower than the probability of your losing. Less obvious, but equally problematic is the following sort of case:

Three drawers contain the following mixture of coins—one contains two gold coins, one contains two silver coins, and one contains one gold coin and one silver coin. Without looking you pick a drawer, open it, and pick out a coin. When you open your eyes, you see the coin is gold. What is the probability that the other coin in that drawer is gold?

Many will reason as follows. The coin came from either the gold/gold drawer or the gold/silver drawer. Each drawer is equally likely and if it came from the gold/gold drawer the other coin is gold. But if it came from the gold/silver drawer the other coin is silver. Hence, the probability the other coin is gold is ½ or 50%. Unfortunately, the two possible drawers are not the relevant equally likely possibilities (no more than your winning or losing were the relevant equally likely possibilities in the lottery case). The relevant possibilities are opening a drawer and picking out a coin without looking. There are six different equally likely ways that could happen, one for each coin. Once you gain the new evidence that the coin you picked is gold when you open your eyes, you can eliminate three of the six possibilities, i.e. the ones in which you pick a silver coin. Of the three possibilities left two are such that the other coin is gold, i.e., the two possibilities in which you pick one of the two coins from the gold/gold drawer. Only in the gold/silver case is the other coin silver. Hence, the probability of the other coin being gold is 2/3. The moral here is that accurate probabilistic reasoning requires identifying and using the relevant equally likely possibilities from amongst all the sorts of possibilities that may present themselves—not always an easy task.

1.2 .3 Practical possibilities

Another significant type of possibility, especially in our everyday reasoning, is practical possibility—possibilities that are consistent with our means, desires, and will (or perhaps our epistemic practical possibilities—the possibilities that, given what we know or believe, are consistent with our means, desires, and will). When deciding how to get to an important meeting across town you are likely to not even consider the possibility that you flap your arms and fly, or the possibility that you use your personal matter/energy transport device, or even the possibility that you sprint all the way there. The first is physically impossible; the second, while perhaps physically possible, is beyond our current technological means; and the third, while certainly physically possible, is quite likely beyond your will and most certainly contrary to your strong desire to not arrive at the important meeting sweating profusely and gasping for breath. Instead you consider what your actual transportation options are (your own car, Uber, taxi, walk, subway, or some combination), how much time you have, how much money you are willing to spend, and then you try to find the optimal possibility (usually constrained by the desire to not spend too much time actually calculating the optimal possibility). Mundane decisions about which possibility to actualize like this happen all the time: what to eat this week, which movie to go see, what to do after dinner, when to get your hair cut, etc. Though mundane, they are still of interest to critical thinking or argumentation theorists since businesses and advertisers spend billions of dollars and devote millions of work-hours to trying to influence your desires and will in order to persuade you to choose their product.

Of more social significance are your individual choices that impact larger groups—in particular (if you live in a democracy) your voting choices, your decisions about how much effort you put into monitoring the outcome of your voting choices, and what the individuals or policies you voted for end up doing. In an optimal world, your political representatives would enact policies that benefit the most people in the most cost efficient, affordable, and just way. Of course, there may be little agreement about what is the most affordable, or just, or beneficial option, especially if what elected representatives take to be the best option is what will get them re-elected rather than what is actually good for their constituents. Regardless of the complexities and intricacies of public policy debate and decision-making, at the core is an attempt to find and agree upon a practical possibility, from amongst the myriad available, to actualize for our mutual benefit.

Given so many types of overlapping sets of possibilities, many of which differ for different individuals or groups of individuals—your set of practical possibilities does not likely match that of your neighbor even if the two sets overlap significantly; compare your set with someone of quite different socio-economic means and the sets overlap even less—and it is no surprise that numerous problems can arise when reasoning with and about possibilities. Individuals can consider too many possibilities, or more commonly, fail to consider all the relevant possibilities. For example, human beings are quite prone to confirmation bias—taking confirming instances as justifying an already-accepted theory or explanation rather than actively seeking out or testing for disconfirming instances. Detectives, or doctors, or researchers can become so fixated on the explanation they already believe to be correct that they are blind to the alternate explanations that are still consistent with the evidence available. In the case of probabilistic reasoning, we already saw cases of considering the wrong set of possibilities. Reasoners can also illegitimately shift the set of possibilities under consideration or shift the value assigned to various possibilities mid-reasoning. An egregious example can occur in public policy debates over the negative consequences of potential policies. When negative consequence X is a potential consequence of the opposition’s preferred policy it is judged to be likely enough to count as a reason against the policy, but when negative consequence X is a potential consequence of one’s own preferred policy, it is judged not to be likely enough to count as a reason against the policy. Identifying the correct set of possibilities and correct relative values of those possibilities is essential to reasoning correctly in numerous situations and yet identifying and ranking possibilities is often an extremely difficult task.

1.2 .4 Logical possibilities

One way to try to sidestep some of these problems is to determine what reasoning holds no matter what the possibilities in question are—to determine the patterns of reasoning that work in all the possibilities. After all, if a piece of reasoning works no matter what possibility you are considering, then you do not need to worry whether you are considering the right set of possibilities or not. Hence, one goal of formal logic is to be able to identify the structure that defines all the ways things could be, i.e., the logical possibilities.

The rough and ready notion of a “logical possibility” is a possibility that has no contradiction in it. Whilst it is not logically possible for an individual to both exist at a particular time and place and not exist at that time and place, which is contradictory, it is logically possible that the person exist in Montana in one instant, and then exist on one of the moons of Jupiter, say Io, in the next. There is no contradiction in the possibility that you exist in Montana in one instant and on Io in the next. But this possibility, while logically possible, is not physically possible. Given the distance from Montana to Io, we would need to violate the physical restriction on moving matter or energy (currently travelling below the speed of light) faster than the speed of light to get from Montana to Io from one instant to the next, so such travel is physically impossible.

Earlier I said that philosophers are still investigating and debating the nature of possibilities. But, whatever they are, there is one actualized one and lots of unactualized ones. In Leibniz’s argument that this world is the best of all possibilities, God examines all the possibilities and then actualizes the best one. Even if you doubt Leibniz’s argument, of all the myriad ways this universe could be, it is in fact one way, namely, the possibility that is actualized. The detective has numerous possibilities in mind about who is responsible for the deceased’s death; the detective hopes that by finding more evidence the possibilities can be reduced to one, the actual one. When you are deciding what to do tomorrow, you consider numerous possibilities and then engage in actions that make one (hopefully the one you wanted) actual.

But since there are lots of unactualized possibilities and only one actual possibility, how do we distinguish the unactualized possibilities from each other? Quite simply by what is true and false at each possibility. I flip a coin twice. There are four possible outcomes. Heads for the first flip and heads for the second; heads for the first and tails for the second; tails for the first, and heads for the second; and tails for both. Suppose the coin comes up tails on the first and heads on the second—that is the possibility that got actualized. How do we distinguish the three non-actualized possibilities? Well, in the first and second it is true that the coin first came up heads, but in fourth it is false that the coin first came up heads. But possibilities one and two differ in what is true and false of the second coin flip.

1.3 Declarative sentences and propositions

Given that we distinguish possibilities by what is true and false if they are actualized, one proposal for understanding possibilities is just as sets of declarative sentences. For example, the first coin flip possibility would be the set {“the first flip of the coin came up heads”, “the second flip of the coin came up heads”}. While initially appealing, the problem with this proposal is that sentences are not as well behaved as is needed to demarcate possibilities. Why?

Sometimes different sentences describe the same possibility or state of affairs. For example, “George is a bachelor” and “George is an unmarried male of marriageable age” describe the same state of affairs, but are different sentences since they are composed of different words. But since they are different sentences, sets that differ only in regards to which of these two sentences they contain are still different sets, and so different possibilities. Yet, we agreed the sentences were just two different ways of talking about the same possibility.

Alternatively, sometimes the same sentence can be used in different ways to describe different possibilities. For example, the sentence “The movie was a bomb” used in the United States likely describes a state of affairs in which the movie was bad, but the same sentence used in the United Kingdom likely describes a state of affairs in which the movie was good. But if one sentence can be used in different ways to describe different possibilities, then, once again, we cannot identify possibilities merely with sets of sentences.

To avoid the vagaries of sentences, logicians usually resort to propositions—what it is that declarative sentences express. “George is a bachelor” and “George is an unmarried male of marriageable age” express the same proposition about George’s marital status. “England won the World Cup in 1966” expresses a true proposition about the English national soccer team; “2+2 = 5” expresses a false proposition about the sum of 2 and 2. We use declarative sentences to express propositions directly, but other language use often involves them. For instance, when we ask, “did Hungary win the World Cup in 1938?” we wonder whether the proposition that Hungary won the 1938 World Cup is true or false. If we get the correct answer (they did not win—they lost to Italy 4-2 in the finals), then we stop wondering whether it is true or false and start believing it is false (and if the belief if strong enough and acquired in the correct way, we might even know that the proposition is false).

Instead of treating possibilities as sets of sentences, many logicians treat possibilities (or at least model possibilities) as sets of propositions. There are technical details that might require modifying even this proposal, but since the resolution of these details is unlikely to be relevant to the critical thinking project, we can take possibilities to be sets of propositions. The propositions that are members of a particular possibility are said to be true or obtain at that possibility. Propositions that are not members of a particular possibility are false at that possibility or do not obtain at that possibility. Armed with the concepts of (i) a possibility and (ii) propositions being true at or obtaining at possibilities, we can define many of the logical concepts that pervade logic and critical thinking textbooks. So even though some of the logical concepts that are forthcoming are, in some textbooks, defined in terms of sentences, the more common way is to define them in terms of propositions.

2. Logical concepts

2.1 types of propositions.

I begin by discussing some common types of propositions that arise in our reasoning. The most basic is a simple proposition, propositions expressed by such declarative sentences as “George is a bachelor” or “the sky is blue” or “Romeo loves Juliet.” Simple propositions attribute something to some object(s) or thing(s). In the first case, of George, that he is a bachelor, and in the third case, of Romeo, that he loves another object, namely Juliet. N egations of simple propositions, propositions expressed by such declarative sentences as “George is not a bachelor” or “Hungary did not win the 1938 World Cup” say that the simple proposition does not obtain. Of course, we do not speak declaratively solely by affirming either simple propositions or the denial of simple propositions; we combine or modify our simple propositions such as in:

• “George is a bachelor, and so is Todd”;
• “Mary loves Antonio, but he does not love her back”;
• “George went to Sophie’s house or he went to the movies”;
• “If the butler did not do it, then the cook did”;
• “If I take the subway, I will be on time for my meeting”;
• “Every student in this class is eligible”;
• “Someone deliberately killed the deceased”;
• “In order to be on time for your meeting, you must take the subway”;
• “England did not win the game, but they might have if they had scored their penalty kick in the last minute.”

The first two sentences express conjunctions . For a conjunction to be true, both sub-parts of the conjunction have to be true. So for “George is a bachelor and so is Todd” to be true, both “George is a bachelor” and “Todd is a bachelor” must be true. [For ease of exposition I will often omit the phrase “the proposition expressed by” before mentioning sentences as I just did above.]

The sentence “George went to Sophie’s house or he went to the movies” expresses a disjunction. There are two sorts of disjunctions—inclusive and exclusive. For an inclusive disjunction to be true, at least one of the sub-parts must be true. For an exclusive di s junction to be true, exactly one of the sub-parts must be true. If our sentence about George expresses an exclusive disjunction, then for it to be true George needs to be in exactly one of two places—at Sophie’s house or at the movies. This is likely to be the usage of someone trying to tell us where George is at a particular moment. If, on the other hand, the sentence expresses an inclusive disjunction, then it will be true if George went to one of those locations and is still true if George went to both. This is likely to be the usage of someone just trying to lay out where George might have gone over a period of time. While some languages have different words for expressing inclusive and exclusive disjunctions. English relies on context or background knowledge, sometimes with limited success, to try to distinguish which type of disjunction is being expressed. Legal documents, in order to avoid the ambiguity of ‘or’ in English, often spell out exclusive disjunctions as “A or B and not both A and B” while representing inclusive disjunctions as “A and/or B”.

Sentences such as: “ If the butler did not do it, then the cook did” and “ If I take the subway, [ then ] I will be on time for my meeting,” express onditional propositions. Conditionals are frequently used in natural languages such as English, yet there is little agreement on how they are to be analyzed logically. (Some theorists even go so far as to deny that conditional sentences express propositions at all.) Usually the disagreement concerns determining exactly what it takes for conditionals to be true, but there is widespread agreement that declarative conditionals are false if the ‘if’­–part, the antecedent , is true, and the ‘then’–part, the consequent , is false. If it is true that I take the subway, and false that I will be on time for my meeting, then the conditional “If I take the subway, I will be on time for my meeting,” is false. As a consequence, logic has defined a minimal version of the conditional, called the material conditional . Material conditionals are false if the antecedent is true and the consequent is false, but true otherwise—in other words, material conditionals are the most permissive when considering what it takes for a conditional to be true. There has been much debate about whether indicative conditionals such as “If the butler did not do it, then the cook did” just express material conditionals or rather express something stronger. Despite the disagreement, the most common articulation of conditionals in introductory logic texts is in terms of material conditionals, and it is most often this sort of conditional that is coopted into critical thinking texts. One merely needs to keep in mind that the work on understanding conditionals is far from finished.

“ Every student in this class is eligible” expresses a universal proposition—a proposition that attributes something to every member of a specified group. For a universal proposition to be true there can be no instance of a member of the group not having the specified attribute. If “Every student in this class is eligible” is true, then there is no student in the class who is not eligible. Oftentimes the group is not fully identified in the sentence used to express the proposition. For example, saying “All the beer is in the fridge” or “All horses have heads” are unlikely to be taken as expressing that every single beer in the universe is in a particular fridge or that there is no single instance of a headless horse anywhere. Depending on the context of use, likely plausible interpretations of those sentences would be: “All the beer we brought home from the store (and which has not already been drunk) is in the fridge” and “Typical, normal live horses have heads.” But once the group is fully specified, for a universal proposition to be true, every member of the group must have the attributed property or properties.

Instead of saying that everything in a given group has a stated attribute, we often merely want to convey that at least one thing or some things in the group have a particular property as in “Someone deliberately killed the deceased.” Such propositions are existential propositions. They are true when at least one object in a specified group has a specified attribute. For example, “Some student is eligible” is true just so long as at least one student is indeed eligible.

So far, most of our examples of propositions can be true or false given a single possibility. Suppose we restrict ourselves to just the actual possibility—then it is either true that George is a bachelor at the actual possibility or it is not; if, at the actual possibility, there is no student in the class who is not eligible, then the universal “Every student in class is eligible” is true at the actual possibility and otherwise false. But some of our declarative sentences are not just about one possibility; rather, they depend on multiple possibilities. Sentences such as the last two on our list, which express modal propositions are examples. (They are called “modal” because expressions such as “must”, “can”, “might”, “would”, etc., were said to indicate the “mode” of the component proposition.)

Different modal expressions have different truth conditions. Consider, for example, the sentence—“In order to be on time for your meeting you must take the subway.” For it to be true, all the possible ways (probably some set of practical possibilities constrained by the background in which the sentence is uttered) in which you make the meeting on time include your taking the subway. In the case of England losing, but winning if they’d scored their penalty, the first part is a negation that is true just so long as England won is false. So the first part tells us what the actual possibility is like. But the second part tells us what the relevantly similar possibilities except for England scoring their penalty, are like—namely, that England won in at least one of those possibilities. Compare that with the stronger claim that England would have won if they had scored their penalty—that claim will be true just so long as England wins in all the relevantly similar possibilities. (Part of the debate about conditionals is whether even conditionals without explicit modal terms, such as ‘might’ or ‘would’ or ‘must’, etc. are really expressing propositions concerning multiple possibilities, and not just the actual one—again, a debate I will not be able to resolve here.)

This list is not at all meant to be exhaustive of the type of propositions we express via our declarative sentences. Rather, it is meant to give a flavor for the sorts of propositions dealt with in first and second logic courses, the sorts of propositions that logicians attempt to model and define clearly and precisely in their basic systems. Why are logicians interested in these sorts of propositions? Because they show up in many of the reasoning patterns that we use over and over. For example, if I tell you George is either at Sophie’s or at the movies, and you tell me he is not at movies, we both hopefully reason that we should check for George at Sophie’s house. Another example: If the IRS says that all taxpayers satisfying their three specified conditions can claim a particular deduction, and you determine that you satisfy those three conditions, you should reason that you can take that particular deduction. It is by recognizing these types of propositions, and the patterns that result in combining them, that formal logic, which focuses on the patterns, gets its impetus. But regardless of whether one is focusing on the goodness of patterns or more generally on the patterns and content of reasoning, both critical thinking theorists and logicians need to take special care in determining what proposition a given sentence in a particular context expresses, for without understanding the correct proposition we will not be considering and evaluating the correct possibilities.

Even though understanding and classifying what propositions various declarative sentences express is an ongoing project, there is another classification scheme that logicians often appeal to—necessary truths (also called tautologies), necessary falsehoods (also called contradictions), and contingent propositions. The definitions are as follows:

Necessary Truth : A proposition that is true in all possibilities.

Necessary Falsehood : A proposition that is false in all possibilities.

Contingent Proposition : A proposition that is true in some, but not all, possibilities.

Sentences such as: “Either Socrates corrupts the youth of Athens or he does not”, or “If it is raining, then it is raining” express necessary truths. For every possibility there is, either Socrates corrupts the youth of Athens in that possibility or he does not. Some have wondered if there any non-trivial tautologies, since the standard examples, such as the ones I just gave, seem to be pretty trivial, uninformative sentences. Many theoreticians hold that the truths of mathematics are all necessary truths and many of those truths are certainly non-trivial—they often take a lot of work for us to know that they are true. Others point out that even if many necessary truths seem trivial or uninformative, they are still very useful. Plato, for example, uses the Socrates sentence in part of his dialogue concerning whether Socrates should have been found guilty of a particular offense. Plato starts with the obvious truth that either Socrates corrupts the youth or he does not, but proceeds to argue that in either case, Socrates should not be found guilty.

Sentences such as “At a particular moment in time, Socrates is over six-feet tall and Socrates is not over six-feet tall” express necessary falsehoods. For any possibility, and any moment of time in that possibility, Socrates cannot be both over six-feet tall and not over six-feet tall. Necessary falsehoods, or contradictions as they are more commonly called, are useful as sign-posts of something having gone drastically wrong in our reasoning. If we can show that someone’s position contains or leads to a contradiction, then we show that they aren’t even talking about a genuine possibility at all, but rather an impossibility. Good reasoners generally want to avoid being committed to impossibilities, so they try to avoid being committed to contradictions in their reasoning.

Most of the propositions we deal with in our everyday reasoning are contingent ones. “The coin landed heads on the first flip” is true in some possibilities, but false in others. “George will arrive on time” is true in some, but false in others. Even complex propositions, such as “If I take the subway, I will make it to the meeting on time” are likely to be true in some possibilities (smooth running reliable subway system) and false in others (an unreliable or scanty subway system). The challenge for good reasoners, of course, is to try to figure out, on the basis of what we already know, and the acquisition of new evidence, which propositions are in fact true at the actual possibility and which are not true. The detective, the doctor, the scientist, the everyday reasoner, are all reasoning using various possibilities in order to try to determine which propositions are true or false at the actual possibility.

2.2 Relations amongst propositions

Given that reasoning is the moving from given propositions to other propositions, and logicians are trying to understand correct reasoning, many of the important concepts of logic concern not just types of propositions, but the relations amongst propositions, I finish this section with definitions, examples, and discussion of eight such relations.

Necessary condition

One proposition, A , is a necessary condition for another propos i tion , B, if there is no possibility in which B obtains and A does not.

If A is a necessary condition for B, then you cannot have B without A. For example, if it is true that meeting the eligibility requirements is a necessary condition for legitimately holding office, then there is no possibility in which one legitimately holds office and does not meet the eligibility requirements. But if it is false that meeting the eligibility requirements is a necessary condition for legitimately holding office, then there is at least one possibility in which one legitimately holds office and does not meet the eligibility requirements.

Sufficient condition

One proposition, A , is a sufficient condition for another propos i tion B, if there is no possibility in which A obtains and B does not.

If A is a sufficient condition for B, then A guarantees B. For example, if it is true that getting a perfect score on every assessment is sufficient for passing the course, then there is no possibility in which one gets a perfect score on every assessment and one does not pass the course. If it is false, then there is at least one possibility in which one gets a perfect score on every assessment and still does not pass the course.

In many elementary logic or critical thinking textbooks, necessary and sufficient conditions are treated as material conditionals. For example, “George attending class is sufficient for George passing the course” is treated as “If George attends class, then George passes the course.” But necessary and sufficient conditions cannot be material conditionals, since denying a sufficient or necessary condition is not the same as denying a material conditional. For example, saying “George attending class is not sufficient for George passing” is not the same as denying the material conditional “If George attends class, then George passes the course” is true. Denying the material conditional is just saying that it is actually the case that George attends class, but does not pass the course, i.e., that the antecedent is true and the consequent is false. But denying that George’s attending is sufficient for George’s passing is not saying that George attends and does not pass, but rather says that there is a possibility, not necessarily the actual one, in which George attends, but does not pass. In other words, necessary and sufficient conditions are describing what is true of a range of possibilities.

Equivalence

Two propositions are equivalent just so long as there is no poss i bility in which one is true and the other is false .

In other words, for each possibility, the two propositions are either both true or both false. For example, “All Euclidean triangles have three sides” and “All Euclidean triangles have three interior angles” are both true in all possibilities and false in none, so they are logically equivalent to each other. Similarly, “Either Peter failed to make the team or Abigail failed to make the team” is logically equivalent to “Abigail and Peter did not both make the team.” If the first proposition is true, then, on an inclusive disjunction reading, at least one of the two did not make the team, so it is true that they did not both make the team. If on the other hand the first proposition is false, then it is false Pater failed to make the team (and so made it) and it is false Abigail failed to make the team (and so also made it), in which case both made the team and the second proposition is also false. Since the propositions are true in the same possibilities and false in the same possibilities they are logically equivalent.

Equivalence of proposition is not to be confused with the equivalence of sentences. Two sentences are equivalent, such as “George is a bachelor” and “George is a unmarried male of marriageable age” just in case they express the same proposition. Two distinct propositions, on the other hand, are equivalent just in case they are true or false in exactly the same possibilities. Of course, without a clear notion of the identity conditions of propositions, it is often hard to determine whether we have two sentences expressing one proposition, or two sentences expressing two distinct propositions that are equivalent to each other. [Like possibilities, theorists are still debating how to understand propositions. For example, here I have defined possibilities as sets of propositions, but some theorists reverse the order of dependence and define propositions as sets of possibilities, i.e., the possibilities at which they are true. Either way, having defined one concept in the terms of the other, the theorist still owes us an account of the undefined concept—a task theorists continue to pursue.]

Consistency

Two propositions are consistent with each other just in case there is at least one possibility in which both are true.

For example, “Sphere A is completely red” is consistent with “Cube B is completely blue” just so long as there is a possibility in which both are true. But “Sphere A is completely red” is inconsistent with “Sphere A is partly blue” since there is no possibility in which both are true.

Two propositions are contrary to each other if there is no possibi l ity in which both are true.

Contrariness is a kind of inconsistency. As we just saw, “Sphere A is completely red” is inconsistent with “Sphere A is partly blue” because there is no possibility in which both are true, i.e., because they are contrary to each other. But even though both propositions cannot be true together, they both could be false together, such as in possibilities in which “Sphere A is completely green” is true. But there is an even stronger kind of inconsistency, than mere contrariness.

Contradictor y

Two propositions are contradictory to each other if there is no possibility in which both are true or both are false .

“Sphere A is completely red” is contradictory to “Sphere A is not completely red” since if one is true, the other is false and if one is false, the other is true. Similarly, if it is true that “Snow guarantees skiing” then it is false that “There is a possibility in which there is snow and no skiing” and vice versa.

One important reason to keep these two kinds of inconsistency separate is that reasoners sometimes treat inconsistency as if it were just the same as being contradictory—they reason that if two states of affairs are inconsistent, then if one is false, the other one must be true. But such reasoners miss or ignore the possibility that two inconsistent propositions might still both be false, and as we saw in the previous section, ignoring or missing relevant possibilities is prone to generate reasoning errors. Hence, knowing whether two propositions are consistent, or contrary, or contradictory gives us important information about which possibilities are still relevant to whatever inquiry or reasoning we are pursuing using those propositions.

Since logicians are motivated by the goal of distinguishing good reasoning from bad reasoning and at least one part of good reasoning is that what we reason from adequately supports what we reason to, logicians are very interested in relations of adequate support. One very special kind of adequate support is entailment.

Entailmen t

P roposition A entails proposition B just so long as there is no po s sibility in which A is true and B is false.

For example, “Sam’s car weighs over 1000kg” entails “Sam’s car weighs at least 500kg”—any possibility in which Sam’s car is over 1000kg it is clearly at least 500kg. “Sam’s car is a red hatchback” entails “Sam’s car is red” and “Sam possesses a car” and “Sam’s car is a hatchback”. Instead of talking about what a single proposition entails, logicians are often interested in what a group or set of propositions entails. [A set of propositions entails another proposition just so long as there is no possibility in which all members of the set are true and the other proposition is false.] For example, “George went to Sophie’s house or to the movies” and “George did not go to the movies” entail “George went to Sophie’s house.” On the other hand, “If Sally attends class, then she passes the course” and “Sally passes the course” does not entail “Sally attends class,” since there are possibilities in which Sally can study well enough on her own and there is no attendance requirement, such that while it is true that “If Sally attends, then she passes the course” and true that “she passes the course”, it is false that “she attends class”.

Logic, especially formal logic, is primarily interested in entailment and other consequence relations. But at the elementary levels of logic at least the concept of entailment is applied to a concept that is also of interest to critical thinking and argumentation theorists—the concept of an argument. In logic, arguments are often modeled as a set of a set of propositions (the premises) and another proposition (the conclusion). [But see Chapters 8 and 9 of this volume for a more detailed discussion of the concept of an argument.] Logicians define validity , a property of arguments, in terms of whether or not the entailment relation holds between the premises and the other proposition, the conclusion. If the premises entail the conclusion, then the argument is valid, i.e., there is no possibility in which the premises are true and the conclusion false, and otherwise the argument is invalid. [Validity here is not to be confused with the notion of ‘valid’ that is used in everyday speech to signify that something is “good” or “worthy of further consideration”, as in: “She made a valid point, when she said ….”. Nor is it to be confused with the notion of ‘valid’ that is used in survey research to signify the goodness or utility of a measuring instrument or the results of such an instrument—for that concept see Chapter 19 of this volume.]

In the previous section, I said that one of the motivations for studying logic was to try to find properties of good reasoning that would hold in all the possibilities. Entailment (and so validity) is one such property. If the arguments you make are valid, i.e. if your reasons entail your conclusion, then your reasoning, at least in terms of support, is good reasoning. Of course, other aspects of that reasoning might be problematic, but at least you know that your reasons, if true, guarantee your conclusion, no matter what set of possibilities is the relevant set.

But consider: Most of the coins on the table are heads-up and that quarter is a coin on the table, so it is heads up. “Most of the coins on the table are heads-up” and “That quarter is a coin on the table” do not entail that “That quarter is heads up” and yet in many situations we would likely say that the first two propositions give very strong reasons to believe the third. In other words while entailment is a sure sign of inferential goodness in reasoning, the lack of entailment does not necessarily mean there is a lack of inferential goodness. Sometimes we say our reasoning is good enough, even if our reasons do not entail what we infer from them. If, in the possibilities in which our reasons are true, enough of them also have what we infer to be true, then we can say that the inferential link is good because the reasons sufficiently support our conclusion. The general definition of sufficient support is as follows:

Sufficient Support

Propostion A (or a set of propositions) sufficiently supports a proposition B just so long as, in enough of the possibilities in which A (or the set of propositions) is true, B is also true.

What counts as “enough” often varies from context to context. For example, in civil litigation, the conclusion of wrongdoing has to be supported by a preponderance of the evidence, i.e., the possibilities in which the defendant did what they are accused of, should be the case in more than 50% of the possibilities in which the provided evidence is true. But in criminal cases, the conclusion of wrongdoing should be supported beyond a reasonable doubt (which, at least if we take the vast majority of judges’ views on what that means, is above 80%). Statistical significance for supporting various hypotheses in the sciences is often set at 95% or higher. Determining what should count as “enough” in various contexts is often extremely challenging. At the very least, some of what counts as “enough” depends on the importance of the outcome. For example, since criminal sanctions are so much higher than civil sanctions, we demand more assurance that the evidence supports the conclusion of wrongdoing in the criminal case than in the civil case.

Logicians, I said, are primarily interested in consequence relations such as entailment. Different types of logic study these relations in different domains. For example, temporal or tense logics are interested in determining the consequence relations amongst uses of temporal phrases, such as, “in the future”, “in the past” and “now”. Modal logics study the consequence relations amongst propositions containing modal terms such as “must”, “can”, etc. But in addition to distinguishing types of logics by the types of propositions being modeled, logics are also categorized in terms of the type of consequence relation being studied. At the most general level, there are two types of logic—deductive and inductive. Deductive logic is concerned with entailment. Inductive logic is concerned with consequence relations weaker than entailment. Unsurprisingly, since there are many consequence relations weaker than entailment, inductive logic is a much less unified field of study than deductive logic. As we shall see in the next section, there are other uses of the terms ‘deductive’ and ‘inductive’, but these are generally misuses—the key difference between inductive and deductive logic is the type of consequence relation being studied.

I conclude this section with a final point about these eight definitions. They have all been given in terms of possibilities in general, i.e., logical possibilities. But for each definition, we could restrict the possibilities we are talking about and get restricted versions of these definitions. For example, physically necessary truths are those that hold in all the possibilities in which the physical laws hold. Morally necessary truths are those that are true in all the possibilities with the same moral code, etc. A set of propositions would physically entail another proposition if there is no physical possibility in which the propositions in the set are true and the other proposition is false. Two propositions are morally contradictory if there is no moral possibility in which both are true or both are false.

Even though explicit talk of these restricted kinds of logical concepts is rare, the theoretical apparatus is available and useful for trying to get clear on what various reasoners or arguers are in fact claiming. For example, in common discourse, when someone says that A entails B, I suspect they rarely mean that there is no possibility whatsoever in which A obtains and B does not; rather, for some contextually determined (though usually unspecified) group of possibilities there is no possibility in which A obtains and B does not. Similarly, for necessary and sufficient conditions; when someone says that snow is necessary for skiing, they probably do not mean that there is no possibility whatsoever in which there is skiing but no snow (there are in fact numerous possibilities—water skiing, roller skiing, sand skiing, skiing on artificial pellets, etc.), but rather that our typical conception of skiing requires snow. In the sciences, they are rarely concerned with logically necessary and sufficient conditions, but rather with causally necessary and sufficient conditions—conditions that require or guarantee something else in all the possibilities consistent with the causal laws. The moral for critical thinking is that even when one encounters terms such as ‘entails’ or ‘contradictory’ or ‘necessary condition’ they may not be being used in their strictly logical sense, but rather being used over a subset of relevant possibilities.

3. Logic and the activity of reasoning

3.1 logic and reasoning.

I conclude with some final comments about the application, and misapplication, of logical concepts in the study of reasoning. Logical systems are models. In particular, they are models of consequence relations between propositions. Some of the models are quite limited. For example, standard sentential or propositional logic systems ignore the internal structure of simple propositions and focus solely on connectives such as ‘and’, ‘or’ or ‘if,…then’. Others add elements to model ‘must’ and ‘can’ while still ignoring everything else, and so on. The hope is to ultimately get a model, or group of models, that illuminates the standards of good reasoning, at least with regards to inferential support. Like most models, logical models can be very helpful when properly applied within the domain they model. Trying to use the model outside the proper domain, however, can have drastic consequences. For example, claiming that the standard sentential logic system is a good model for explaining instances of good reasoning utilizing modal claims is clearly a mistake. (This is true not just for logical models. For example, using the “model” of the north star as a fixed point is extremely useful for general terrestrial navigation, but using the same model for routing certain sorts of messages, which requires quite precise location determination, gets poor results.) Similarly, since logic focuses on support relations and good reasoning usually involves not just adequate support, but good reasons as well, it is a mistake to think logic is the whole story of good reasoning. Indeed, logic has little to nothing to say about what makes reasons good reasons, but rather focuses on what can legitimately be inferred from whatever good reasons we find.

3.2 Arguments and explanations

Clearly the target domain we are trying to understand and improve—the activities of reasoning, arguing, justifying, persuading, etc., are much more complicated than any of the various logical systems that logicians produce to model certain aspects of those activities. And yet many theorists still try to find distinctions in the models that are really only distinctions in the activities and not really the concern of logic at all. For example, logicians and argumentation theorists have spent a lot of time trying to distinguish arguments from explanations. But suppose I lay out several reasons (including some reasons about what I think will happen in the next six months) why you should believe a particular company will fail in the next six months. Six months go by and the company fails and someone else asks “Why?” and I trot out my reasons again. Nothing has changed about the propositions involved, so, from the perspective of logic, there is one object, one set of propositions, here. Yet, how that object has been used has changed. Initially the reasons are used to argue that the company will fail. After the fact, the reasons are used to explain the company’s failure. We argue for propositions we are not sure of (or to convince others of propositions they are not sure of), but we explain propositions we are sure of, some of which may have been proved to us by argument, in order to understand why they are true. [Note that unlike my example, there are plenty of cases where the reasons one might give to argue for a proposition, which turns out to be true, need not be the reasons given when explaining why the proposition is true. For example, if something unexpected happens in the six months that contributes to the company’s failure that is likely to be a part of the act of explaining even though it was not part of the act of arguing.] The fact that there is a difference between acts of arguing and acts of explaining does not mean that, in the domain of logic, we should find separate kinds of things—arguments on the one hand and explanations on the other.

3.3 Inferring and implying

Going in the other direction, no one doubts that, considered in terms of propositions and support relations the inference from A to B and the implication of B by A are the same thing. But it is a mistake to think that the act of inferring is the same as the act of implying. You assert a group of facts (with the intention that I draw conclusions from those facts). I, being a good reasoner, draw those conclusions. You imply those conclusions and I infer those conclusions. Put another way, if I ask someone what they are inferring, I am asking about reasoning going on in their head, but if I ask someone what they are implying, I am asking about reasoning they hope to be going on in other people’s heads. Put yet another way, reasons do not infer conclusions, but rather imply them. People, when considering those reasons on their own, infer those conclusions, but do not imply them.

3.4 Deductive and inductive

Sometimes concepts are misapplied in both the model and the target domain. For example, some logic textbooks and critical thinking textbooks try to distinguish deductive arguments from inductive arguments, but from the perspective of logic there is nothing about the sets of propositions that compose arguments that make one kind of set deductive and another set inductive. For every group of reasons and a given conclusion we can ask whether the reasons entail the conclusion or not (the domain of deductive logic) or whether those very same reasons offer some support weaker than entailment or not (the domain of inductive logic.) Nor is it clear that we reason deductively or inductively—when we reason, we infer one or more propositions from others. Of that reasoning we did, we might wonder whether it is good or bad. The answer to that question will, in part, depend on what counts as good enough support in the situation in which I am using the reasoning. If the context requires entailment and the reasons do entail the conclusion, then the reasoning is adequate with regards to its support relation. If the reasons do not entail the conclusion, then it will fail to be adequate in such a situation. Similarly for a required support relation weaker than entailment—if reasons support the conclusion at or above the required level, then the support relation is adequate, whereas if it is below the required level the support relation is not adequate. The reasoning is one act of reasoning—whether the actual support relation of that reasoning is adequate or not depends on the situation. But none of this suddenly makes it the case that there are two distinct kinds of reasoning going on (even if there is a felt difference between realizing some reasons entail a conclusion versus realizing some premises only strongly support a conclusion.)

3.5 Linked vs. convergent arguments

One final example. The push for general principles often takes something that may track a real distinction or property in a certain specific set of cases and try to generalize it to all cases. For example, there is a strong intuition that reasons such as: “If you pass the test, then you will pass the course” and “You pass the test” work together to support the conclusion “you will pass the course” whereas reasons such as “You read all the supplemental material” and “You took good notes” and “You went to the tutor consistently” independently support the conclusion that “You are prepared for the test.” This intuition is strong enough, that numerous textbooks, especially those that use argument diagramming as a tool, try to distinguish arguments with linked premise structures from arguments with convergent premise structures. The problem here is two-fold. On the one hand, attempts to actually provide a rule for determining when a set (or subset) of reasons are linked or not have, to date, all failed, at least if we trust the intuitions that generated the drive to generalize the phenomena in the first place. On the other hand, the underlying judgments of whether premises are working together or are independent seem to vary from person to person and context to context enough to suspect that the distinction may not be tracking a real phenomenon that deserves to be represented or captured in our logical models.

4. Last word

Despite these injunctions to take care with the proper application of various concepts that have made their way into various textbooks, the core logical concepts of Section 2, such as sufficient support or co n sistency or necessary condition are useful in any study of reasoning. Even if good reasoners need to be careful and work diligently to determine which propositions are being expressed, and which possibilities are relevant, and what the needed standard of sufficient support is in a given situation, once these tasks are accomplished, we can evaluate our reasoning for inconsistencies and determine whether our reasons entail or at least adequately support our conclusions.

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Studies in Critical Thinking Copyright © by G.C. Goddu is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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CRITICAL THINKING – Fundamentals: Necessary and Sufficient Conditions

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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Necessary and Sufficient Conditions

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

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

1. Philosophy and Conditions

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

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

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

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

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

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

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

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

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

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

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

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

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

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

• If you touch me, I’ll scream.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

• If John wins the race, we will celebrate.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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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Unit 1: Critical thinking

About this unit, fundamentals.

• Fundamentals: Introduction to Critical Thinking (Opens a modal)
• Fundamentals: Deductive Arguments (Opens a modal)
• Fundamentals: Abductive Arguments (Opens a modal)
• Fundamentals: Validity (Opens a modal)
• Fundamentals: Truth and Validity (Opens a modal)
• Fundamentals: Soundness (Opens a modal)
• Fundamentals: Bayes' Theorem (Opens a modal)
• Fundamentals: Correlation and Causation (Opens a modal)
• Introduction to Critical Thinking, Part 1 7 questions Practice
• Introduction to Critical Thinking, Part 2 6 questions Practice
• Deductive Arguments 7 questions Practice
• Necessary and Sufficient Conditions 7 questions Practice
• Instrumental vs. Intrinsic Value 7 questions Practice
• Implicit Premise 7 questions Practice
• Justification and Explanation 7 questions Practice
• Normative and Descriptive Claims 7 questions Practice
• Validity 7 questions Practice
• Soundness 7 questions Practice
• Fallacies: Formal and Informal Fallacies (Opens a modal)
• Fallacies: Fallacy of Composition (Opens a modal)
• Fallacies: Fallacy of Division (Opens a modal)
• Fallacies: Introduction to Ad Hominem (Opens a modal)
• Fallacies: Ad Hominem (Opens a modal)
• Fallacies: Affirming the Consequent (Opens a modal)
• Fallacies: Denying the Antecedent (Opens a modal)
• Fallacies: Post Hoc Ergo Propter Hoc (Opens a modal)
• Fallacies: Appeal to the People (Opens a modal)
• Fallacies: Begging the Question (Opens a modal)
• Fallacies: Equivocation (Opens a modal)
• Fallacies: Straw Man Fallacy (Opens a modal)
• Fallacies: Slippery Slope (Opens a modal)
• Fallacies: Red Herring (Opens a modal)
• Formal and Informal Fallacies 7 questions Practice
• Division and Composition 7 questions Practice
• Ad Hominem, Part 1 7 questions Practice
• Ad Hominem, Part 2 7 questions Practice
• Denying the Antecedent and Affirming the Consequent 7 questions Practice
• Post Hoc Ergo Propter Hoc 7 questions Practice
• Begging the Question 7 questions Practice

Cognitive biases

• Cognitive Biases: Alief (Opens a modal)
• Cognitive Biases: Anchoring (Opens a modal)
• Cognitive Biases: Pricing Biases (Opens a modal)
• Cognitive Biases: Reference Dependence and Loss Aversion (Opens a modal)
• Cognitive Biases: Mental Accounting (Opens a modal)
• Cognitive Biases: Peak-End Effect (Opens a modal)
• Cognitive Biases: The GI Joe Fallacy (Opens a modal)

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2: Language, Meaning, and Definition

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Critical thinking, on one understanding of the idea, is the ability to ask the right questions. Some of the right questions are questions about the words used in an argument or used to express a position. What do they mean? No, what, specifically , do they mean? When someone says something like “immigration is a problem.” What do they mean by “immigration”? Are they referring to illegal immigration? Legal Immigration? All immigration? A specific nationality? A specific subset of illegal immigrants? What do they mean by “a problem”? Do they mean “we need to find out how to support these people as they struggle to survive?” or do they mean “we need to protect ourselves from these people”? We don’t really know exactly what they mean until we’ve clarified it with them (or sometimes looked at the other things they’re saying and inferred what they mean).

• 2.1: Breakdown of meaning
• 2.2: How does meaning work? Definition and Concepts
• 2.3: Necessary and Sufficient Conditions
• 2.4: Chapter 2 - Key Terms
• 2.E: Chapter Two (Exercises)