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Studia Logica

, Volume 20, Issue 1, pp 144–144 | Cite as

A proof of axiomatizability of łukasiewicz’s three-valued implicational propositional calculus

  • T. Prucnal
Article

Summary

LetL 3 c be the smallest set of propositional formulas, which contains
  1. 1.

    CpCqp

     
  2. 2.

    CCCpqCrqCCqpCrp

     
  3. 3.

    CCCpqCCqrqCCCpqpp

     
and is closed with respect to substitution and detachment. Let\(\mathfrak{M}_3^c \) be Łukasiewicz’s three-valued implicational matrix defined as follows:cxy=min (1,1−x+y), where\(x,y \in \{ 0,\tfrac{1}{2},1\}\). In this paper the following theorem is proved:
$$L_3^c = E( \mathfrak{M}_3^c )$$

The idea used in the proof is derived from Asser’s proof of completeness of the two-valued propositional calculus. The proof given here is based on the Pogorzelski’s deduction theorem fork-valued propositional calculi and on Lindenbaum’s theorem.

Keywords

Mathematical Logic Computational Linguistic Propositional Calculus Propositional Formula Implicational Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Pañstwowe Wydawnictwo Naukowe 1967

Authors and Affiliations

  • T. Prucnal

There are no affiliations available

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