FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2002, VOLUME 8, NUMBER 3, PAGES 783-828

V. E. Plisko

Abstract

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```
A generalized predicate is defined as a function from the natural
numbers
```$\mathbf N$ to $2^{\mathbf N}$ . The values of
a generalized predicate are treated
as ``the realizations'' of sentences. The logical operations on the
generalized predicates are based on the ideas of Kleene's recursive
realizability. A generalized algebraic system is defined on the ground of
the concept of a generalized predicate. The notions of constructive
truth in an enumerated system and in an arbitrary denumerable system are
defined. It is shown that the relations of logical consequence corresponding
to these semantics have not the compactness property and the set of logical
tautologies is $\Pi_1^1$ -complete. The problems of axiomatizing
the classes of algebraic systems in the languages with constructive
semantics are studied.

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Last modified: February 17, 2003