Abstract
This paper focuses on the problem of constructing of some standard Hilbert style proof systems for any version of many valued propositional logic. The generalization of Kalmar’s proof of deducibility for two valued tautologies inside classical propositional logic gives us a possibility to suggest some method for defining of two types axiomatic systems for any version of 3-valued logic, completeness of which is easy proved direct, without of loading into two valued logic. This method i) can be base for direct proving of completeness for all well-known axiomatic systems of k-valued (k≥3) logics and may be for fuzzy logic also, ii) can be base for constructing of new Hilbert-style axiomatic systems for all mentioned logics.
Highlights
Many-valued logic (MVL) as a separate subject was created and developed first by Łukasiewicz [4]
Parallel to the Łukasiewicz approach, Post [5] introduced the basic idea of additional truth degrees, and applied it to problems of there presentation of functions
In the earlier years of development, this caused some doubts about the use fullness of MVL
Summary
Many-valued logic (MVL) as a separate subject was created and developed first by Łukasiewicz [4] His intention was to use a third, additional truth value for “possible” (or “unknown”). First of constructed system based on the logic with one designated value and conjunction, disjunction, implication, defined by Gödel, and negation, defined by permuting the truth values cyclically. Axioms of this system_are generalizations of formulas, using in Kalmar’s proof of deducibility for two valued tautologies. Chubaryan Anahit and Khamisyan Artur: Generalization of Kalmar’s Proof of Deducibility in Two Valued
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