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I have some questions related to multivalued logic. I am new to this forum, (I study mathematics) so I would be grateful to any useful advice. I am doubtful on even posting this question on Philosophy SE.

Can 3 valued logic (True, False, Indeterminate) be the basis of mathematics? If accepted so, can normal mathematics like proofs, relations, sets, function, calculus, analysis and algebra be developed? Much of modern math is based on the bivalence principle and the law of the excluded middle??

Also, what if fuzzy logic (developed by Zadeh) and set theory and fuzzy arithmetic are used as the foundations of mathematics? What would be the demerits to this approach? And can it be applied fields of study like physics? And probably new fields like QM and GR?

Apart from other questions, my questions is can we develop normal mathematics from this framework??

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    I would suggest researching constructivist mathematics. It's similar to what you're proposing. It takes all "normal" logical foundations, but does not accept the law of the excluded middle. So it doesn't explicitly have n truth values, but simply says that there's not necessarily only true and false, so it's a little more general. You get a lot of interesting results, but a lot of the motivation lies in the proof theory. The proofs often require much more "construction" and "agency", making them feel more intuitive.
    – Joseph_Kopp
    Commented Sep 29 at 18:04
  • Are you asking out of idle curiosity or do you have in mind some specific characteristics of mathematics that traditional logic can't handle?
    – David Gudeman
    Commented Sep 30 at 9:26
  • The Q is about maths, but in digital electronic logic 3 states are routinely used. See tri-state or Three-state logic. These are real, manufactured circuits. And in computer programming a system can be modelled to have any number of discrete states.
    – Weather Vane
    Commented Sep 30 at 18:39
  • One can, in principle, base a version of mathematics on any non-classical logic, but some are more popular than others. Constructive (intuitionistic logic) and inconsistent mathematics (paraconsistent logic) are the most popular. Mathematics based on fuzzy logic has been developed, see Bělohlávek et al., Fuzzy Logic and Mathematics, ch. 5, but is not very popular. I have not seen versions with finitely many truth values explored in the literature.
    – Conifold
    Commented Sep 30 at 23:46
  • en.wikipedia.org/wiki/Three-valued_logic
    – Idiosyncratic Soul
    Commented Oct 1 at 0:46

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I could find only one reference to "fuzzy logicism" online here (see also about fuzzy mathematics, although you sound like you're familiar with this already?), as part of a course on vagueness (if I understand the webpage). But so it would have to be (neo-)logicism that would "do the deed" (which you are looking for). The linked-to SEP entry on neo-logicism does appear to address something along your lines:

The anti-realist, by contrast, insists that all truths are knowable; and is quick to point out that we do not have any effective method for deciding the truth or falsity of statements in mathematics. Anti-realists, accordingly, reject the Law of Excluded Middle (and all other strictly classical rules that are intuitionistically equivalent to it), and advocate the use of intuitionistic or constructive logic, rather than classical logic. ... The pursuit of analyticity in the foundations of arithmetic is one that could be very well served by the proof-theoretic methods favored by the Dummettian anti-realist’s theory of meaning.

But now to be somewhat flippant, or seemingly dismissive, it would be very hard (or impossible, really) to start with, "There are more than two truth values," and derive all of mathematics therefrom. So 3+-valued logics (or intuitionistic logics) could not really serve, "just like that," as the basis of mathematics, but a basis at most. That's not really much of a serious obstacle to the idea behind your question, of course, per the quotes regarding Dummett, etc.

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Can 3 valued logic (True, False, Indeterminate) be the basis of mathematics?

In mathematics, given a function f, the truth value of y = f(x) can informally be said to be to indeterminate if x is not in the domain of f. There is, however, no need invent a third possible truth value of "indeterminate." It is enough to simply say, in the language of set theory, that x is not an element of the domain of f -- a true-or-false statement.

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