Logic

Can we prove anything, or is the best that we can hope to achieve an invitation to compare the plausibility of the premises of an argument with the plausibility of the negation of the conclusion of that argument?

Comments

To prove Y from X is to show that if X is true then Y must be. We could say that to prove Y, period, is to prove Y from assumptions that are true. (Should those assumptions turn out to be false, we might say "We thought we had proved Y, but it turns out we were wrong: we had proved Y from assumptions we now know to be false.")

So, can we prove anything? Sure. Why not? We may well have already. If in fact we've correctly derived propositions from true assumptions, then we've proved them.

Perhaps you're worried by the fact that the bare derivation of Y from X doesn't tell us whether we should accept X and, therefore, Y -- or whether we should reject Y and, therefore, reject X. That's true: the bare derivation doesn't tell us whether we've proved Y (in the above sense). The derivation only gives us grounds for accepting Y if its premises are true. In judging that Y is true, we express our confidence that the truth of X is far more likely than the truth of not-Y. Perhaps your worry is that if this is so, then we can't know whether we've proved Y, for to know that requires knowing more than the derivation of Y from X: we also need to know that X is true. But why shouldn't we be able to know that? I can see why you might really worry about this if you also thought that the only way one could know anything to be true is to prove it, for then the same problem will keep arising. But why accept that? Don't I know there's a computer in front of me, absent any proof?

(In this connection, see also Question 29 and Question 121.)

There's also this point: If one has shown that, if X is true, then Y is true, then one has proven, without making any assumptions, that, if X, then Y. One might say that, if one has given an argument, then one has assumed that it is legitimate to use whatever principles of argument one applied in the argument. But, as Lewis Carroll once observed in a famous paper titled "What the Tortoise Said to Achilles", that claim leads to the conclusion that argument itself is impossible. We have to distinguish between an argument's justifiably employing certain principles and its assuming that those principles may legitmately be employed.

I take it, however, that the question was not concerned with whether we can prove this kind of claim but, perhaps, with whether we can prove anything that would actually be regarded as contentious. Can we prove, for example, that abortion is immoral? or, conversely, that it is morally permissible? In such cases, the answer may well be "No".

Read another response by Alexander George, Richard Heck

Read another response about Logic