Truth Values and Connectives in Some Non-Classical Logics

Subject Areas : Statistics

P.  Safari ¹ (Department of Mathematics, Qazvin Branch, Islamic Azad University, Qazvin , Iran)
S.  Salehi ² (Basic Sciences Research Center, Tabriz University, Tabriz, Iran)

Received: 2018-05-14 Accepted : 2018-10-14 Published : 2019-08-01

Keywords: منطق فازی, مدل کریپکی, تعریف پذیری, منطق شهودی, منطق گودل,

Abstract :

The question as to whether the propositional logic of Heyting, which was a formalization of Brouwer's intuitionistic logic, is finitely many valued or not, was open for a while (the question was asked by Hahn). Kurt Gödel (1932) introduced an infinite decreasing chain of intermediate logics, which are known nowadays as Gödel logics, for showing that the intuitionistic logic is not finitely (many) valued. Now we know that the propositional intuitionistic logic is infinitely many valued (with a countably many logical values). In this paper we provide another proof for this result of Gödel, from the perspective of Kripke model theory. Švejadr and Bendova (2000) proved that in Gödel fuzzy logic the conjunction and implication are not definable by the rest of the propositional connectives (while disjunction is definable by conjunction and implication). In this paper, we show that disjunction is not definable by implication and negation in Gödel fuzzy logic; two proofs, one by Kripke models and one by fuzzy semantics, are provided for this new theorem.

References:

 

[1]      K. BENDOVA, A Note on Gödel Fuzzy Logic, Soft Computing 2 (1999) 167.

 

[2]      M. DUMMET, A Propositional Calculus with Denumerable Matrix, The Jour­nal of Symbolic Logic 24 (1959) 97—106.

 

[3] K. GÖDEL, Zum Intuitionistischen Aussagenkalkül, Anzeiger Akademie der Wissenschaften Wien 69 (1932) 65—66. Translated as ‘On the Intuitionistic Propositional Calculus', in: Kurt Cödel Collected Works: Volume I: Publications 1929-1936, eds.: S. Feferman et al. (Ox­ford University Press 1986).

 

[4] G. MINTS, A Short Introduction to Intuitionistic Logic (Kluwer 2002).

 

[5]      P. SAFARI & S. SALEHI, Kripke Semantics for Fuzzy Logics, Soft Computing 22 (2018) 839—844.

 

[6]      V.ŠVEJDAR &K.BENDOVA, On Inter—Expressibility of Logical Connectives in Gödel Fuzzy Logic, Soft Computing 4 (2000) 103—105.