Q&A: Intuitionistic Logic
Intuitionistic Logic
Question
I didn’t fully understand exactly how logics that try to replace “regular” logic actually rely on it and therefore aren’t relevant. If the explanation is too long for a responsum, I’d be happy for a reference to a source that explains this.
Answer
It’s hard to explain this here, and I don’t remember a place where I gathered the issue together systematically. I’ve commented on it in several places, for example in my article “What Is ‘Halut’?” There I explained why Łukasiewicz’s three-valued logic, which supposedly replaces ordinary (two-valued) logic, cannot really do that. When I construct its own truth tables, I use two-valued logic to do so (proofs by contradiction, which rely on the law of the excluded middle—the very law this “logic” rejects). I explained that this does not replace our logic, but rather builds a logical structure that has certain uses in very specific contexts. It is called “logic” because of the mathematical structure of that theory. But from our point of view it does not play the role of logic. See there also regarding quantum logic.
Discussion on Answer
I hope you understood what you wrote. I didn’t. In any case, the two meanings you suggested are not what I mean here (and in my view they also aren’t correct).
My claim is that three-valued logic is not part of our logic, but another structure created within it for some specific concrete purpose. Just as quantum theory is a physical theory that uses logical structures that appear to depart from binary logic. But they do not. The claims of quantum theory are proved using mathematics built on classical logic (including theorems proved by contradiction). As a rule, one does not discover logic in the laboratory. Logic is imposed on us and cannot be experimentally refuted.
So too, when you construct a truth table for three-valued logic, you can prove by contradiction that the value in a certain slot is “true” or “false” or “third.” But a proof by contradiction presupposes the law of the excluded middle, which this logic denies. This is not a matter of language but of logic. If the law of the excluded middle is not correct, one cannot use it.
Think about a case in which there is only one possible value for a necessary proposition (a tautology). The logic governing that proposition is one-valued: “true,” and that’s it. Its negation leads to contradictions and is impossible. Does that undermine binary logic? Not at all. Binary logic is the framework of thought within which everything takes place. Within it there is a domain of tautologies, propositions for which only one truth value is relevant. That is not a change of logic. The same applies to propositions for which one can speak of three truth values. If you treat statistics as a kind of multiple-valued logic, then there can be many “truth values” for each proposition (the entire continuum between 0 and 1). Does that undermine our logic? Is the statement “this is a heap” a statement that can be discussed in binary logic? No, because there are different degrees of heap-ness. Does that undermine logic? Not at all.
Could you please refer me to a source where they constructed a truth table in three-valued logic and filled in one of its slots using a proof by contradiction? I’d like to read that being done and look carefully at the details.
I don’t have time right now to get into it. I explained it well enough, and that suffices.
No problem. I’m not asking you to get into it. I’m asking for a source that does get into it—a book, article, essay. Something that does what you said is done in three-valued logic: constructs a truth table and fills in one of its slots using a proof by contradiction.
Michi,
I don’t think it’s clear what you’re claiming. I’ll put two possible meanings of your central claim on the table and contrast them, and leave it to you to choose between them or offer a meaning of your own.
Are you claiming that the axioms of this three-valued logic include all the axioms of standard logic? If so, that is simply factually wrong. That is not a correct sense in which this logic presupposes “regular logic.”
There is another sense in which one might claim this. When we talk to each other, we assume that both sides assume that the other side is using something like “regular logic.” That lets us understand that “no” is not “yes,” and vice versa, that someone who says “there is a cat here” has not just told us that there is a chair there. In short, it makes communication possible, and in practice that assumption works—it allows us to understand each other.
I assume that when constructing the truth tables you mentioned, one uses language, whether in talking to oneself or in communicating those truth tables to others. Is that the sense in which, when constructing the truth tables you mentioned, one “presupposes” that logic? That sounds like a correct sense in which one does presuppose it, but why should we care? That doesn’t give it superior status over other logics any more than any other linguistic convention gives it superiority over others. Do the sounds of the word “cat” enjoy some special status over other sounds apart from the convention that in Hebrew those are the sounds used to communicate the concept of a cat? Of course not. In other languages other sounds are used: cat (English), gato (Spanish), katze (German), and so on.
I’ll end by saying that I don’t think we really presuppose “regular logic” in our language, but rather some distorted version of it. If that’s the direction you were aiming at, it is even more problematic. As I said, I don’t think it’s even clear what you are claiming.