5.1.2. The conditional as a truth-functional connective
We have looked at ψ if φ as a way of hedging the claim ψ by limiting our liability, leaving ourselves in danger of error only in cases where φ is true. If this perspective on the conditional is correct, we cannot go wrong in asserting ψ if φ except in cases where ψ is false while φ is true. Thus, the forecaster who predicts that it will rain tomorrow if the front goes through is wrong only if it does not rain even though the front goes through. That suggests that the truth conditions of the conditional are captured by the table below. The only cases where φ → ψ has a chance of being false are those where φ is true; and, in these cases, φ → ψ has the same truth value as ψ.
| φ | ψ | φ | → | ψ |
|---|---|---|---|---|
| T | T | T | ||
| T | F | F | ||
| F | T | T | ||
| F | F | T |
This can be seen in another way by diagramming the propositions expressed by conditionals, as in Figure 5.1.2-1. Adapting the example used with this sort of illustration before, 5.1.2-1B represents the proposition expressed by The number shown by the die is less than 4 if it is odd.
| A | B |
Fig. 5.1.2-1. Propositions expressed by two sentences (A) and a conditional (B) whose consequent rules out the possibilities at the right of A.
The possibilities ruled out by the main clause or consequent of the conditional form the hatched region at the right of 5.1.2-1A and those ruled out by the antecedent or condition form the lower half. In 5.1.2-1B, the region at the right is whittled down to the portion containing possibilities left open by the antecedent, showing how the conditional weakens the claim made by the consequent alone (in the example, The number shown by the die is less than 4). Since the consequent is the second component of the conditional φ → ψ, the rows of the truth table correspond to the top left and right and bottom left and right regions of 5.1.2-1A, respectively.
Apart from compositionality, the principles of implication and equivalence for the conditional are quite different from those we saw for conjunctions and disjunctions.
Covariance with the consequent. A conditional implies the result of replacing its consequent with anything that component implies. That is, if ψ ⊨ χ, then φ → ψ ⊨ φ → χ.
Contravariance with the antecedent. A conjunction implies the result of replacing its antecedent with anything that implies that component. That is, if χ ⊨ ψ, then ψ → φ ⊨ χ → φ.
Curry’s law. A conjunct of a conditional’s antecedent may be made instead a condition on its consequent. That is, (φ ∧ ψ) → χ ≃ φ → (ψ → χ).
Compositionality. Conditionals are equivalent if their corresponding components are equivalent. That is, if φ ≃ φ′ and ψ ≃ ψ′, then φ → ψ ≃ φ′ → ψ′.
The asymmetry of the conditional (e.g., the fact that it is false in the second row of its table but true when the values of its components are reversed in the third) means that we would not expect it to obey a principle of commutativity. That asymmetry is also responsible for the fact that it obeys a principle of covariance for one component but contravariance for the other. It makes sense that a conditional varies in the same direction as its consequent since it’s hedged assertion of that consequent. And it varies in the opposite direction from its antecedent because a condition that rules out more and will be harder to fulfill, so a commitment to the truth of the consequent will happen in fewer possibilties.
The asymmetry of the conditional also makes it no surprise that a principle of associativity does not hold because such a principle would involve several shifts between the roles of antecedent and consequent. A principle for regrouping that can be stated is here named after the logician Haskell Curry who made extensive use of an analgous operation on functions (and also directed people’s attention to the analogy between certain operations on functions and principles governing conditionals). The operation on functions is sometimes called “currying,” and you might think of the transition from the left to right of Curry’s law as a matter of taking a pair of conditions clumped together as a conjunction in the antecedent of a conditional and combing them out into separate antecedents. The principle is also sometimes referred as an “import-export” principle because it tells us how to export a component of the antecedent to the consequent or import a component of the consequent into the antecedent. Curry’s law holds because each side can be false only when both φ and ψ are true and χ is false. And this shows that both sides have the effect of hedging χ by the two conditions φ and ψ. Such a statement might then be called a double conditional.