Today I listened to Hu Ge’s “intuitionistic logic.” The first class naturally began with a bit of introduction and overview before moving on to the logical system itself. What I was interested in was the introductory part. Hu Ge works in mathematics, so the philosophical disputes were brushed past in a sentence and did not take up much time.
Hu Ge’s sketch of intuitionistic logic was still quite concise and on point: intuitionism is aimed at mathematics, not at everyday communication; in everyday communication, the law of excluded middle remains reliable, because the world with which daily life comes into contact is finite. And intuitionistic logic is not some paranoid eccentricity; in fact, it is very simple. It merely replaces the “existence” problem that Platonists talk about in relation to mathematical propositions with a “provability” problem. Platonic mathematics requires positing a world of forms, so that to say a proposition is true means that it exists in that “world”; intuitionism rejects that “world,” and for it to say a proposition is true means only that it can be proved. After such a change, the resulting mathematics moves toward intuitionistic logic.
Hu Ge also understood quite clearly the misunderstandings to which intuitionism is often subject.
Still, there are many philosophical issues worth discussing.
For example, Hu Ge emphasized that the logic of intuitionists is confined to mathematics (because mathematics involves infinity), while in ordinary discussion they still use classical logic. Because if each participant in a discussion used a different logic, the discussion would not be able to proceed. Hence classical logic is said to be the most universal.
Of course, Hu Ge’s understanding of “universal” is not bad. He says that what is universal is what comes about through layer upon layer of abstraction; the most universal things are also the most impoverished things, and formal logic, in the end, can only say tautologies—that is, nonsense. Only those utterances that are not universal enough, that have limitations, are meaningful. Hu Ge thinks that the pair of concepts “true/false” is the highest level of abstraction, and that everything has truth or falsity.
This sounds rather good: “black/white” is an abstraction of the color of objects but does not apply to electrons; “provable” applies to mathematics but not to other contexts; yet “true/false” applies to talk across all domains. Is that really so? From the standpoint of intuitionism or later Wittgenstein, of course not. Words like “true/false” and “existence” are also words in human language, like “table” and “chair”; their meaning lies in use, not in things-in-themselves.
From the intuitionistic point of view, logical rules come after language; they are a summary induction from the actual ways language is used, and this “induction,” like the induction carried out in natural science, is empirical—that is, simplified and fallible. I introduced all this at length in my earlier long essay “In Defense of Intuitionism.”
In fact, in everyday communication people often do not obey logical rules, and they frequently say things that are ambiguous or even self-contradictory, yet communication can still proceed. But wasn’t the original goal of logicism precisely to regulate ordinary discussion with precise logical rules? If the purpose of logic is to regulate everyday language, then one cannot say that in everyday discussion people must necessarily use classical logic; we could equally well take some variant logic as the standard for regulating everyday discussion. After all, everyday discussion is often illogical—regardless of which logic one uses—and classical logic has no privileged status here. And if logic is not meant to regulate everyday language, but instead to induce certain regularities from everyday language, then given that everyday language so often fails to conform to classical logic, why say that classical logic is the most basic?
As Hu Ge also explicitly pointed out, intuitionists are fundamentally opposed to “logic”; even Heyting, after founding the intuitionistic logical system, was “expelled” by the intuitionist camp (this was the first I had heard of it). But even Heyting regarded logic only as “hygiene.” The implication is that it is not an indispensable driving force in the growth of science, but merely an auxiliary means of correction; and medicine, on the one hand, inevitably has toxic side effects, while on the other hand, excessive use may foster dependence—both of which are harmful to health.
Therefore, one cannot say that everyday language always operates with logic (whatever logic it may be); rather, logic comes into play only in certain situations in everyday disputes where special means are needed to clarify matters further (though people often also use means such as lyricism, demonstration, association, and analogy to resolve difficulties in everyday discussion). In fact, logic is not the foundation of everyday language; on the contrary, everyday language is the foundation of logic. Any logic whatsoever, or any comparison between different logics, must ultimately be understood through everyday language (Hu Ge mentioned the difficulties involved in comparing the semantics of different logics, and mentioned the need to place them within some shared context for comparison, but that shared context is undoubtedly everyday language).
2008-02-18
Translated from the Chinese original with AI assistance. The original text is authoritative.
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