How can we reason about "if P then Q" or "P only if Q" statements in propositional logic?

Question

When you have a propositional sentence of the form ***P* ⊃ *Q*** — which we might read as "if *P*, then *Q*" — how can you tell when it is true, or false, based on the truth-values of *P* and *Q* in classical logic? When is this different from ***Q* ⊃ *P***? And what is the connection to 'necessary' conditions, or sufficient conditions?

Answer 1

The general approach here is to refer to a truth table. The center column refers to the truth value of the sentence depending on the values of P and Q.

P ⊃ Q

F T F

T F F

F T T

T T T

Note that the only case where the sentence evaluates to false is if P is true and Q is false.

Suppose the claim is "if it rains, then the earth will explode." It rains, and we are all still around, so we say that the compound claim is false.

In all other cases, the sentence is presumed to be true: if it rains and the earth explodes, if it doesn't rain and the earth explodes, or if it doesn't rain and the earth doesn't explode.

The antecedent (left term) is considered a sufficient cause of the consequent (right term). The consequent can happen even if the antecedent doesn't, but if the antecedent does happen, then the consequent will.

On the other hand, the consequent is considered a necessary cause relative to the antecedent. The antecedent won't happen unless the consequent does. This seems counter-intuitive, but the thing to keep in mind is that the idea of "causation" here isn't really very strict -- one doesn't necessarily or directly have to cause the other -- but it is more about how and whether the different states can exist simultaneously.