# 5 Necessary and Sufficient Conditions

Michael Shaffer

The concepts of necessary and sufficient conditions play central and vital roles in analytic philosophy. For example, being an unmarried male is a necessary condition for being a bachelor and being a bachelor is a sufficient condition for being an unmarried male. That these concepts are vital to philosophy is beyond question, and it is primarily because the orthodox account of the methodology of analytic philosophy involves the contention that philosophy aims to yield accurate specifications of sets of necessary and sufficient conditions, such as the claim that all bachelors are unmarried men. It is, then, obviously and deeply important to philosophy that we have an adequate logical grasp of these concepts. In terms of both propositional and first-order logic the concepts of necessary and sufficient conditions are intimately related to the concept of the conditional (i.e. a statement of the form “if p, then q”) as the following canonical account makes clear.[1] Where S(p, q) means “p is a sufficient condition for q” and N(q, p) means “q is a necessary condition for p”, $p \rightarrow q$ means “if p, then q,” and $p \equiv q$ means “p and q are logically equivalent,” the following two definitions are supposed to represent these two important ideas:

(D1) $\mathrm{S}(p, q) \equiv (p \rightarrow q)$

(D2) $\mathrm{N}(q, p) \equiv (p \rightarrow q)$

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

Truth table for material conditional
$A$ $B$ $A \rightarrow B$
T T T
T F F
F T T
F F T

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

Affirming the Antecedent (Modus Ponens)

1. $A \rightarrow B$
2. $A$
3. $/\therefore B$

Denying the Consequent (Modus Tollens)

1. $A \rightarrow B$
2. $\neg B$
3. $/\therefore \neg A$

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

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

Affirming the Consequent

1. $A \rightarrow B$
2. $B$
3. $/\therefore A$

Denying the Antecedent

1. $A \rightarrow B$
2. $\neg A$
3. $/ \therefore \neg B$

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

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

(D3) $\mathrm{NS}(p, q) \equiv [(p \rightarrow q) \,\& \,(q \rightarrow p)]$

However, since the formula $(p \rightarrow q) \,\&\, (q \rightarrow p)$ is equivalent to the formula $(p \equiv q)$ in classical propositional logic, sets of such necessary and sufficient conditions can be more compactly defined in terms of logical equivalence as follows:

(D4) $\mathrm{NS}(p, q) \equiv (p \equiv q)$

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

Truth table for logical equivalence
$A$ $B$ $A \equiv B$
T T T
T F F
F T F
F F T

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

## Conceptual Analysis and Necessary and Sufficient Conditions

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

(1) Conceptual analyses take the form of proposed definitions (i.e. sets of necessary and sufficient conditions) of analysanda.

(2) The adequacy of any analysandum can be tested against concrete and/or imagined cases.

(3) Whether or not a proposed analysandum is adequate with respect to a given case can be determined by the use of a priori intuition, with a priori intuition being a distinct, reliable and fallible non-sensory mental faculty.[5]

(4) Intuition allows us to reliably access knowledge about concepts.

(5) The method of reflective equilibrium is the particular method by which intuitions can be used to confirm/disconfirm analysanda.[6]

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

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

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

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

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

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

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

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

x is Kx $\equiv$ x is Jx & x is Tx & x is Bx

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

## A Problem with the Orthodox View and SPM

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

### The Potential Vacuity Problem

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

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

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

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

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

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

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

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

### Prospective Solutions to the Potential Vacuity Problem

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

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

## EXERCISES

### Exercise One

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

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

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

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

### Exercise Two

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

Example: You can’t play football without a ball

Answer: Having a ball is necessary for playing football.

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

### Exercise Three

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

• Spoon
• Garden
• Success
• Health (mental and physical)

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

1. Cases that the concept should apply to, but which don’t fulfill your necessary and sufficient conditions.
2. Cases that the concept should not apply to, but which do fulfill your necessary and sufficient conditions.

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