Lukasiewicz’s Logics And Prime Numbers

Alexander S. Karpenko0955117038, 9780955117039

Is there any link between the doctrine of logical fatalism and prime numbers? What do logic and prime numbers have in common?The book adopts truth-functional approach to examine functional properties of finite-valued Łukasiewicz logics Łn+1. Prime numbers are defined in algebraic-logical terms (Finn’s theorem) and represented as rooted trees. The author designs an algorithm which for every prime number n constructs a rooted tree where nodes are natural numbers and n is a root. Finite-valued logics Kn+1 are specified that they have tautologies if and only if n is a prime number. It is discovered that Kn+1 have the same functional properties as Łn+1 whenever n is a prime number. Thus, Kn+1 are ‘logics’ of prime numbers. Amazingly, combination of logics of prime numbers led to uncovering a law of generation of classes of prime numbers. Along with characterization of prime numbers author also gives characterization, in terms of Łukasiewicz logical matrices, of powers of primes, odd numbers, and even numbers.

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Table of contents :
Front Cover……Page 1
ŁUKASIEWICZ’S LOGICS……Page 3
Content……Page 4
Introduction……Page 6
I. Two-Valued Classical Propositional Logic……Page 10
II. Łukasiewicz’s Three-Valued Logic……Page 19
III. Łukasiewicz’s finite-valued logics……Page 29
IV. Functional properties of Łukasiewucz’sn-valued logic……Page 40
V. Structuralization of prime numbers……Page 50
VI.classes ofLet’sa appeared as [Karpenko, 1982].A matrix logic for prime numbers and the law of generation ofprime numbers……Page 85
VII. CHARACTERIZATION OF CLASSESOF NATURAL NUMBERS BYLOGICAL MATRICES……Page 105
Numerical Tables……Page 114
Table 1……Page 115
Table 2……Page 125
Table 3……Page 135
Concluding remarks……Page 143
REFERENCES……Page 145
AUTHOR INDEX……Page 156
SUBJECT INDEX……Page 158
Back Cover
……Page 161