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IMI/Publicaţii/BASM/Ediţii/BASM n.1 (50), 2006/

A test for completeness with respect to implicit reducibility in the chain super-intutionistic logics.

Authors: I.V. Cucu

Abstract

We examine chain logics C2, C3, ..., which are intermediary between classical and intuitionistic logics. They are also the logics of pseudo-Boolean algebras of type <Em, &, ∨, ⊃, ¬, >, where Em is the chain 0 < τ1 < τ2 < ... < τm-2 < 1 (m = 2, 3, ...). The formula F is called to be implicitly expressible in logic L by the system Σ of formulas if the relation

is true, where q do not appear in F, and formulas Gi and Hi, for i=1, ...,k, are explicitly expressible in L via Σ. The formula F is said to be implicitly reducible in logic L to formulas of Σ if there exists a finite sequence of formulas G1, G2, ..., Gl where Gl coincides with F and for j=1, ...,l the formula Gj is implicitly expressible in L by Σ The system is called complete relative to implicit reducibility in logic L if any formula is implicitly reducible in L to Σ. The paper contains the criterion for recognition of completeness with respect to implicit reducibility in the logic Cm, for any m=2, 3, .... The criterion is based on 13 closed pre-complete classes of formulas.

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