So when I think about logically valid inferences like the fact that if all men1 are mortal and Socrates is a man, then Socrates is mortal, I conjure up the incorrigible belief that they are absolutely certain (when their premises are true). But why, exactly? For any story you could tell me I could ask “why does that make these things certain?”. For example, if you try to explain the rules of deduction, I could simply ask why such rules should have purchase with me2. It would be difficult for me to do this with a straight face however, and without losing you as a friend, but still. I could.
But why do we take the reliability of logic and math for granted?
I think it’s because the veracity of things like simple arithmetic and obviously valid inferences are strongly intuitive. In fact, our intuitions concerning simple math and logic are so powerful that just to understand what it means to add two and two or what it means for something to be true of a class of objects, causes the one who understands to believe that the sum of two and two is four or that what is true of every object in a class is necessarily true of any given object in it. The veracity of the enterprise is simply indubitable for the sane man or woman.
In this way I actually think that intuition is the foundation of reason. I intuit that it is the case that if it is true that P implies Q and you grant P, then Q follows inescapably. Assuming that intuition to be correct, logic can proceed. But there is no independent evidence for the truth of what is being assumed. Any “evidence” you could give me for the reliability of logic and math would assume the veracity of the things you’re trying to prove.
Not all intuitions are this powerful, however. And many of our intuitions turn out to be wrong. Therefore the goal of philosophy is to, with sincerely attempted intellectual honesty and other virtues, iron out our tree of intuitions and what we’ve inferred from them in conjunction with observation, so that there aren’t any kinks in our weltanschauung.
1 That is, those men who are not Elijah, Enoch, Jesus, those to be raptured, or the sheep to be welcomed into the Kingdom.
2 In fact, in every logical inference lies a hidden presupposition that cannot be argued for without infinitely begging the question. Namely, the veracity of logical validity. So when presented with the facts that all men are mortal and Socrates is a man, to believe on the basis of those two facts that therefore Socrates is mortal, assumes that if all men are mortal and Socrates is a men, then Socrates is mortal.
But it gets worse. Let us temporarily grant the assumption that “if all men are mortal and Socrates is a man, then Socrates is mortal”, which permits us to infer that Socrates is mortal, granted that all men are mortal and Socrates is a man. The new meta-syllogism formed by the original in conjunction with the assumption duplicates the same assumption on a higher order: that validity is veridical. It assumes that if it is the case that if all men are mortal and Socrates is a man, then Socrates is mortal, and if all men are mortal and Socrates is a man, then Socrates is mortal!
Similarly to believe on the basis of it being the case that if it is true that if all men are mortal and Socrates is a man then Socrates is mortal, and so if all men are indeed mortal and Socrates is indeed a man then Socrates is indeed mortal, therefore given that all men are mortal and Socrates is a man, then Socrates is mortal after all.
And so on.