文档介绍：Alexander S. Karpenko
Russian Academy of Sciences,
Institute of Philosophy,
Department of Logic
ŁUKASIEWICZ'S LOGICS
AND
PRIME NUMBERS
CONTENTS
Introduction 4
I. Two-valued classical propositional logic C2 8
1. Logical connectives. Truth-tables .......................................... 8
2. The laws of classical propositional logic ................................. 10
3. pleteness ......................................................... 12
. Sheffer stroke ........................................................................ 13
4. Axiomatization. Adequacy …………………………………. 17
II. Łukasiewicz’s three-valued logic Ł3 17
1. Logical fatalism………………………………………………. 17
2. Truth-tables. Axiomatization ………………………………… 19
3. Differences between Ł3 and C2 ………………………………. 21
4. An embedding C2 into Ł3 and three-valued isomorphs of C2… 23
III. Łukasiewicz’s finite-valued logics 27
1. Łukasiwicz’s n-valued matrix ………………………………. 27
2. Matrix logic Łn ……………………………………………… 27
3. Axiomatization of Łn ………………………………………… 28
4. Cardinal degrees pleteness of Łn ….………………….. 30
5. Interpretations of Łn ………………………………………….. 31
. T-F-sequences as truth-values ……………………………… 33
. Factor-semantics for Łn …………………………………….. 35
IV. Functional properties of Łukasiewucz’s
n-valued logic Łn 38
1. Preliminary remarks …………………………………………. 38
2. Ji-functions …………………………………………………... 38
3. McNaughton’s criterion for definability of functions in Łn …. 39
4. Sheffer stroke for Łn …………………………………………. 39
5. Functional extensions of Łn ………………………………….. 40
6. Post logics Pn ……………………………………………….... 41
7. Logic as a functional system:
Closure operation, completeness and pleteness ………. 43
8. Ł3 between C2 and P3: the continuality of Ł3 ………………… 44
9. Maximal n+1-valued non-Postian logic ………………………. 45
10. pleteness and prime numbers ……………….……….. 47
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V. Structuralization of prime