Authors: I.V. Cucu
Abstract
We examine chain logics C
2, C
3, ..., which are intermediary between classical and intuitionistic logics. They are also the logics of pseudo-Boolean algebras of type <E
m, &, ∨, ⊃, ¬, >, where E
m is the chain 0 < τ
1 < τ
2 < ... < τ
m-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 G
i and H
i, 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 G
1, G
2, ..., G
l where G
l coincides with F and for j=1, ...,l the formula G
j 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 C
m, for any m=2, 3, .... The criterion is based on 13 closed pre-complete classes of formulas.
E-mail:
Fulltext
–
0.13 Mb