Abstract:
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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