Logic is a faith.

Prove to me that anything can prove anything else. You can’t.

Is Logic logical?

“There are some who… assert that it is possible for the same thing to be and not to be… But we have now posited that it is impossible for anything at the same time to be and not to be, and by this means have shown that this is the most indisputable of all principles. Some indeed demand that even this shall be demonstrated, but this they do through want of education, for not to know of what things one should demand demonstration, and of what one should not, argues want of education. For it is impossible that there should be demonstration of absolutely everything (there would be an infinite regress, so that there would still be no demonstration)…”
— Aristotle, Metaphysics, IV.4

When we say that something “proves” something else, or that something called “Logic” compels us to accept something, what do we mean?

A full (even a partial) treatment of the nature of Logic is far too much to chew on in one post, so let’s just zero in on one load-bearing part of it: the Law of Contradiction, which holds that (in Aristotle’s words) “it is impossible for anything at the same time to be and not to be”. Much depends on this law. It’s not the whole of Logic, but one certainly needs it to function.

The concept of proof certainly depends on the Law of Contradiction. Consider the proof forms modus ponens and modus tollens:

• Modus ponens —

  • A entails B.

  • A is true.

  • Therefore, B is true.

• Modus tollens —

  • A entails B.

  • B is false.

  • Therefore, A is false.

Both of these modes of proof rely on the Law of Contradiction in order to exclude the possibility that the conclusion is both truly entailed and also not entailed at the same time, or that B is nonetheless false even as it is also true, or that A is true even though also false. In a way, you might even say that these two modi are nothing but extensions or implications of the Law of Contradiction (combined with a definition of the word “entails”). Without these methods of reasoning — and, therefore, without the Law of Contradiction — there really isn’t much you can reason about at all.

But who taught you the Law of Contradiction? Do you remember the first time somebody told you about it?

Probably not, because you were probably a baby, and your mother probably explained it in the context of trying to make you understand that once you have eaten the ice cream cone, the ice cream cone no longer exists to be eaten again.

But did you learn the Law of Contradiction from this talking-to? If suckled by a she-wolf and raised with no human parent, would you have come to it anyway?

The Law of Contradiction feels built into us. Is it literally so? Is it hard-wired into our brains, present either at babyhood or in a form that ineluctably blooms during childhood development?

Or would we learn the Law of Contradiction by induction/experience, regardless of both our hardware and our upbringing? Perhaps in the natural laboratory of the world, we conduct an enormous number of implicit experiments in which things have the opportunity to be true and not true at the same time, and in all of the experiments we deem most trustworthy (for other interesting reasons we won’t go into for now), we get the same result: Things are not true and false at the same moment.

This is a very strong heuristic reason to trust Logic. But do note, it is not a proof of Logic. Principles derived from nothing but induction are always vulnerable to being overturned by a single counterexample. All swans are white… until you discover Australia and the black swan.

“Hardware” or “software,” the Law of Contradiction feels like an inescapable part of our mental infrastructure. But that doesn’t make it objectively always true, any more than the fact that we can’t see ultraviolet light makes ultraviolet nonexistent. Our minds are prone to all kinds of errors; optical illusions can fool us even when we’re aware they’re illusions. So the depth of our belief in the Law of Contradiction is not (in itself) evidence of its truth.

Instead of merely deferring to the Law of Contradiction as a birthright, or coming to it by induction, it would be far stronger to prove it. But that cannot be done.

In a philosophical journal, Lewis Carroll once had great fun demonstrating the infinite regress of modus ponens. His dialogue demonstrates, basically, that if I say, “That proves it” about something, you can come back to me with “Prove that it proves it.” Expanding upon that Lewis Carroll sketch in his magnificent and indescribable book, Gödel, Escher, Bach, Douglas Hofstadter comments on the dialogue’s meaning (emphasis added):

“This little debate shows the difficulty of trying to use logic and reasoning to defend themselves. At some point, you reach rock bottom, and there is no defense except loudly shouting, ‘I know I'm right!’ Once again, we are up against the issue which Lewis Carroll so sharply set forth in his Dialogue: you can't go on Defending Your patterns of reasoning forever. There comes a point where faith takes over… One can never give an ultimate, absolute proof that a proof in some system is correct. Of course, one can give a proof of a proof, or a proof of a proof of a proof, but the validity of the outermost system always remains an unproven assumption, accepted on faith.”

Logic is a faith. It is a faith that we accept pragmatically, because it works, but not because it has some magical grounding or Certificate of Reality® that other faiths lack.

This fact should have implications for our dispositions, attitudes, methods of discourse. It should make us humble. It should give us pause. Nothing about the workings of reason, literally nothing, is on a level we can take absolutely for granted. At the very least, this should wipe the smirks off the faces of rationalists who believe their precious Logic is perfect and that they soar far above the heads of mere plebians who would dare take anything on faith. Because they, too, take something on faith — Logic itself. Logic runs around making demands of everyone else which it cannot itself meet. Logic shouts: “Justify yourself without circularity — not on your own terms, but in terms of something external to yourself.” Yet Logic refuses to do likewise.

We sincerely should forgive and tolerate this bullying behavior from Logic, since, on a day-to-day basis, it makes itself so bloody useful. But let’s understand the leeway we’re giving it.

Logic is a good faith.

“‘By what right,’ Akiba protested, ‘do you presume to call my attitude blind? …faith is indispensable both as a base on which thought may stand, and as a check-rein when logic goes astray. He who wishes to trace a circle must first select out of all space one point about which to draw it. The choice of the point makes possible the line which circumscribes it. The utility of the circle in practice will determine ultimately whether the point has been well placed. So with faith. It is the axis about which we move — an axis that must be posited as an act of will. The fate of man determines whether he has located it properly.”
— Milton Steinberg, As a Driven Leaf

Let me be clear about what I’m not saying. I’m not saying that since Logic depends, at bottom, on an act of faith, therefore we can abandon Logic. Among the many faiths on offer, Logic is an extremely compelling faith! (Nor is it mutually exclusive with many other faiths. We all hold many faiths at a time.)

As a practical matter, I’m not even sure it would be possible to abandon Logic even if one tried; it might be built too deeply into the human mind.

Nor am I saying: “Feel free to have faith in whatever you want; it’s all the same, since even Logic is grounded in faith.” That doesn’t work for me. Some leaps of faith seem a lot smaller than others! I prefer my leaps to be as small as possible. There’s a ladder here, based on how much I directly perceive, how much is built on logical structures that rest on those direct perceptions, and how many layers of possibility there are for pieces I might be missing altogether, flaws in my perceptions, theoretical flaws in the logical principles, or flaws in my understandings of the relationships between all of it. On the absolute ground, the thing I feel the most basic level of certainty about is that I exist and am someone with real first-person consciousness/perspective. Following Descartes, it’s the only thing I feel sure I can’t be fooled about. Then there are other beliefs that I still feel quite sure of, without being as fully certain of them as I am of my own existence as a first-person conscious mind, and that would include the Law of Contradiction and, by extension, the rest of Logic. It also includes the idea that the world I perceive has a strong relationship to an objective world around me (i.e., I am not in The Matrix). Moving up the ladder, still further away from truly basic beliefs, are things like “It is Monday,” or, “The United States exists.” Other beliefs feel much further from truly basic beliefs for me, and things like “God chose the Jewish People and gave us the Torah” feels very far away indeed from being ground-level “basic beliefs.”

So I don’t hold that anything goes. Logic is a small leap of faith; to believe in the historical truth of the Torah’s Revelation on Mount Sinai is a huge leap of faith — at least, to my intuition. Because I don’t have any direct perception of Revelation, and the apologetics for why I should trust in its historicity feel extremely dicey to me. I am aware that not everyone feels this way. Philosopher Alvin Plantinga is very keen to assert that if you do have a basic, ground-level perception of God, you can hold your head up high and reasonably assert that this is a properly basic belief. I have no beef with the validity of Plantinga’s syllogism; it’s the premise that doesn’t apply to me. If Alvin Plantinga directly perceives God, that’s very nice for him and I have no desire to talk him out of it. But it doesn’t help me, because I simply don’t share that perception!

But let’s notice something here — these assertions I am making about more and less basic beliefs, more and less direct forms of perception, and more and less reasonable leaps of faith are all aesthetic or intuitive evaluations. They are not, and cannot be, grounded in Logic, since Logic itself is among the things being evaluated. We’re talking about gut-level inclinations. We’re talking about degrees of strain to our credulity. We’re talking about taste here, about personality. And we’re talking about them as load-bearing parts of the foundation of how we reason.

So much of life turns out to be intuition and aesthetics. We refuse to believe in the Flying Spaghetti monster not based on sober analysis but based on an internal scoff. We make the jump from “things have never before been true and untrue at once” to “things can never be true and untrue at once” simply because our intuitions support this ungrounded leap. It feels neater, cleaner, and truer to us. It works for us both practically and aesthetically.

We’re all feeling our way forward. All “proof” really means is: “My intuitions on this one are now super mega-locked in.” In betting on Logic, we’re betting on a horse who looks massively stronger to us than its competitors, and who has won every race so far. I remain convinced it’s a good bet!

But let’s tilt our fedoras a bit and remember that we’re all gamblers here.