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.