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Gödel defined the min-max rule for the conjunction and disjunction in his multi-valued logic as $$ u\land v=\min\{u,v\} \quad \operatorname{and} \quad u\lor v=\max\{u,v\} $$ Łukasiewicz defined rules for the negation and implication in his multi-valued logic as $$ \neg u=1-u \quad \operatorname{and} \quad u\to v=\min\{1, 1-u+v\} $$ I'd like to know which sources (books or papers better in English) contain the above definitions. I know Gödel logics is in "Zum intuitionistischen Aussagenkalkül" (1932), but I do not have access to the paper.

Ihe min-max rule for the conjunction and disjunction is also in "Introduction to a General Theory of Elementary Propositions" by Post.

Eugene Zhang
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    The SEP article, including its bibliography, might be a good starting point. – Noah Schweber Feb 22 '21 at 21:14
  • I need a book or paper because I will add them to a paper. Online links normally are not included in a paper. (Maybe it is changed nowadays.) – Eugene Zhang Feb 22 '21 at 21:23
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    That's why I specifically referred to the bibliography (and gave the above as a comment as opposed to an answer). For example, just glancing quickly through the bibliography Gottwald's book seems like a good candidate (although I can't check since I don't have a copy). – Noah Schweber Feb 22 '21 at 21:25
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  • I want to have a specific paper/book containing the definitions so I can add them in the paper. It better avoids fuzzy logic-related sources though because they are not officially recognized. – Eugene Zhang Feb 22 '21 at 21:29
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    What do you mean by "not officially recognized"? – Timo Feb 22 '21 at 21:32
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    In what sense are fuzzy logic sources not recognized? – Noah Schweber Feb 22 '21 at 21:34
  • Let's say, if the paper is submitted to a (top) pure math journal, I am not sure if fuzzy related book/paper will be accepted by referees. Just my guess because fuzzy logic is not considered as a rigorous mathematical field by pure mathematicians now. – Eugene Zhang Feb 22 '21 at 21:35
  • More experience about it is welcome. I want to know more about this. – Eugene Zhang Feb 22 '21 at 21:36
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    "fuzzy logic is not considered as a rigorous mathematical field by pure mathematicians now" Huh? That's certainly not my impression. I definitely do get the impression that there is lots of bad work in fuzzy logic, but there is also plenty of good work; it's in no sense unrigorous, or - as far as I know - viewed as such. Certainly I can't imagine rejecting a reference to a text on it (I could by contrast imagine a paper on fuzzy logic being rejected by a specific journal on the grounds that fuzzy logic is considered out of focus for that journal, but that's a separate issue). – Noah Schweber Feb 22 '21 at 21:39
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    While it is true that the reputation of fuzzy logic has suffered due to a large body of bad research, fuzzy logic as developed by Petr Hajek and his followers is as rigorous as any other mathematical discipline (sometimes it is called "mathematical" fuzzy logic to distinguish it from less formal work). I guess also journal referees would know that, at least if they are logicians. – Timo Feb 22 '21 at 21:47
  • @ Noah Schweber and @Timo, thanks. I have found the right sources which are "Untersuchungen uber den Aussagenkalkul" (Lukasiewicz & Tarski), and "Axiomatization of Infinite Valued Logics" (Barkley Rosser). – Eugene Zhang Feb 22 '21 at 23:07

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