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Characterization of prime numbers in Łukasiewicz's logical matrix

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Abstract

In this paper we define n+1-valued matrix logic Kn+1 whose class of tautologies is non-empty iff n is a prime number. This result amounts to a new definition of a prime number. We prove that if n is prime, then the functional properties of Kn+1 are the same as those of Łukasiewicz's n +1-valued matrix logic Łn+1. In an indirect way, the proof we provide reflects the complexity of the distribution of prime numbers in the natural series. Further, we introduce a generalization K *n+1 of Kn+1 such that the set of tautologies of Kn+1 is not empty iff n is of the form p β, where p is prime and β is natural. Also in this case we prove the equivalence of functional properties of the introduced logic and those of Łn+1. In the concluding part, we discuss briefly a partition of the natural series into equivalence classes such that each class contains exactly one prime number. We conjecture that for each prime number the corresponding equivalence class is finite.

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  1. Department of Non-Classical Logic, Institute of Philosophy Academy of Sciences of the USSR, 119842, Moscow

    Alexander S. Karpenko

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  1. Alexander S. Karpenko
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To the memory of Jerzy Słupecki

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Karpenko, A.S. Characterization of prime numbers in Łukasiewicz's logical matrix. Studia Logica 48, 465–478 (1989). https://doi.org/10.1007/BF00370201

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