David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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Studia Logica 48 (4):465 - 478 (1989)
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 ukasiewicz's n +1-valued matrix logic n+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 , where p is prime and is natural. Also in this case we prove the equivalence of functional properties of the introduced logic and those of n+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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References found in this work BETA
Robert McNaughton (1951). A Theorem About Infinite-Valued Sentential Logic. Journal of Symbolic Logic 16 (1):1-13.
Trevor Evans & P. B. Schwartz (1958). On Słupecki T-Functions. Journal of Symbolic Logic 23 (3):267-270.
Herbert E. Hendry (1983). Minimally Incomplete Sets of Ł Ukasiewiczian Truth Functions. Notre Dame Journal of Formal Logic 24 (1):146-150.
Herbert E. Hendry (1980). Functional Completeness and Non-Ł Ukasiewiczian Truth Functions. Notre Dame Journal of Formal Logic 21 (3):536-538.
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