Introducing deductive validity

Here, we’re going to introduce the key idea for the course. It’s worth pausing to emphasize just how important this idea is. It’s at the heart of everything else we’ll do from now on. Our key concept is that of deductive validity. Here’s a first go at outlining the notion. A deductively argument is one in which, if the premises are all true, then the conclusion must be true. The intuitive idea here is of a kind of argument where it’s not just that the truth of the premises would somehow make it more likely or plausible that the conclusion is true. Instead, it’s that the truth of the premises would rule out the possibility of the conclusion being false.

If the premises are true, there’s no wriggle room, no way for the conclusion to be false. If the premises are true then the conclusion has to be true. Talk of possibilities here is helpful. We can use it to give a sharper, clearer statement of the kind of feature were interested in. Here’s the revised version, putting deductive validity in terms of possibility. An argument is deductively valid if, and only if, it’s not possible for it to be the case that both, 1) all of its premises are true and 2) it’s conclusion is false, as it were, at the same time. This will be our official definition of deductive validity. Note that in what follows, we’ll often shorten ‘deductively valid’ to ‘valid’.

A quick note on ‘if and only if’. It has, pretty obviously, two components. We can understand these by working through the definition of deductive validity. First, ‘if’ bit. If an argument is such that it’s not possible for all of its premises to be true and its conclusion false, then it’s valid; that is in every case in which an argument is like that, it’s valid. And now the ‘only if’ bit.

An argument is valid only if it’s such that it’s not possible for all of its premises to be true and its conclusion false; that is, in every case in which an argument is valid, it has that feature Putting the two bits together, that’s one way for an argument to be valid and indeed it’s the only way.