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


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


  2. 2.


  3. 3.


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.


Mathematical Logic Computational Linguistic Propositional Calculus Propositional Formula Implicational Matrix 
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© Pañstwowe Wydawnictwo Naukowe 1967

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  • T. Prucnal

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