5 Necessary and Sufficient Conditions
Michael Shaffer
The concepts of necessary and sufficient conditions play central and vital roles in analytic philosophy. 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”, [latex]p \rightarrow q[/latex] means “if p, then q,” and [latex]p \equiv q[/latex] means “p and q are logically equivalent,” the following two definitions are supposed to represent these two important ideas:
(D1) [latex]\mathrm{S}(p, q) \equiv (p \rightarrow q)[/latex]
(D2) [latex]\mathrm{N}(q, p) \equiv (p \rightarrow q)[/latex]
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]
| [latex]A[/latex] | [latex]B[/latex] | [latex]A \rightarrow B[/latex] |
|---|---|---|
| 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)
- [latex]A \rightarrow B[/latex]
- [latex]A[/latex]
- [latex]/\therefore B[/latex]
Denying the Consequent (Modus Tollens)
- [latex]A \rightarrow B[/latex]
- [latex]\neg B[/latex]
- [latex]/\therefore \neg A[/latex]
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 d