Using arrow in looking at arguments involving ‘if … then … ’

There is a lot more that could be said about whether arrow and ‘If … then … ’ have the same truth-table. But we don’t have space to take that further here. What we can note is that arrow can be a useful tool to use in looking at arguments involving ‘If … then … ’, if we are careful.

Where an ‘If … then … ’-claim is true, the corresponding arrow-claim will have to be true. (It seems right to say that where ‘If P, then Q’ is true, it can’t be the case that ‘P’ is true and ‘Q’ is false. And that rules out the one kind of situation in which ‘(P (rightarrow) Q)’ is false). This means that if we’re looking at an argument with (positive, i.e. not negated) ‘If … then … ’ premises and the version with arrow-sentences substituted tests as valid, then the original argument will be valid.

Remember what we said earlier about the parallels between key inferences involving arrow and inferences involving ‘If … then … ’. The form ‘(P (rightarrow) Q), P; therefore, Q’ is valid. And so is the form ‘If P then Q, P; therefore Q’. The form ‘(P (rightarrow) Q), ~Q; therefore, ~P’ is valid. And so is ‘If P then Q, It’s not the case that Q; therefore It’s not the case that P’. So we don’t need to worry about using arrow to investigate an argument just because we see structures like these involving ‘If … then … ’.

We do have to be more wary if the premises of an argument expressed in natural language include negated conditionals (e.g. ‘It is not the case that if P then Q’). Here’s the reason. The sentence ‘~(P (rightarrow) Q)’ is only true where ‘P’ is true and ‘Q’ is false, but it seems like we think that some sentences of the form ‘It is not the case that if P then Q’ can be true without it having to be that ‘P’ is true and ‘Q’ is false. This means we should hold back from evaluating arguments with this kind of sentence as a premise.