## Conditional Thoughts

I was just reading A counterexample to Modus Ponens, a paper written by Vann McGee, and I think the paper fails to provide what the title promises. However, in thinking about it for myself, I came up with an interesting thought (note: it doesn’t really go anywhere, I just thought it was interesting).

According to modern sentential logic and predicate logic (etc.) a sentence whose main connective is a material conditional (eg. if P then Q) is false if and only if the antecedent is true and the consequent false. Thus, the conditional is true if the antecedent is false. The conditional is also always true if the consequent is true. Therefore, we can take it to be an analytic truth that:

If Q then if P then Q: [Q ⊃(P ⊃Q)]

This is supposed to work for any P, including where P might be “Q is false” so that, paradoxically, where we assume Q, we can derive logically that if Q is false then Q is true (without incurring contradiction). However, we only do this in this case because we say that ‘Q is false’ is false, and thus the antecedent in ‘if P then Q’ is false, and therefore the whole conditional is true. Let’s consider the following though:

Q: every proposition other than ‘Malcolm is in the middle’ is false.

P: Some propositions are true (where ‘some’ means ‘at-least-two’).

If Q then if P then Q. Even if the antecedent P is false, the consequent Q is true, if the antecedent Q is true. However, if the consequent Q is true, then there are at least two true propositions other than ‘Malcolm is in the middle’. First, there is the proposition Q, and second there is the proposition if Q then if P then Q (and we might say that there is also the proposition “if P then Q,” and we might as well say that if P1 then Q (where P1 is P⊃Q), and if P2 then Q (where P2 is P1⊃Q) and … if Pn then Q (where Pn is Pn-1⊃Q). However, the possibility of there being values for P such that, given the assumption of Q, if P then Q is logically true (since Q entails ~P, and the antecedent’s being false makes the whole conditional true). The trouble is that the whole operation of inferring from a proposition along with a conditional (Modus Ponens) involves assumptions with which Q is logically incompatible.

Somebody will object by saying that Q is not only false, it is an analytic truth that Q (every proposition other than ‘Malcolm is in the middle‘ is false) is false. This is because it is not logically possible that only one proposition be true.

However, saying [Q ⊃(P ⊃Q)] is an analytic truth is to say that the inference from Q to (P⊃Q) is truth preserving. From a contradiction anything follows, and if from Q everything follows then Q is logically false. Thus we can simply say that it is an analytic truth that [Q ⊃(P ⊃Q)], given what we have come to mean by conditional statements, is truth preserving, and we can say that [Q ⊃(P ⊃Q)] is itself an analytic truth for any possible Q.