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Chains of structurally complete predicate logics with the application of Prucnal’s substitution. (English) Zbl 1058.03013

The notion of structurally complete logic (logic in which structural and admissible rules are derivable) was introduced by W. A. Pogorzelski [Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 19, 349–351 (1971; Zbl 0214.00704)]. It was shown that classical propositional logic (with modus ponens as the only inference rule) is structurally complete. Several other propositional logics were shown to be structurally complete, but it was shown that intuitionistic logic is not structurally complete. In this paper it is shown that if some peculiar ‘omitting’ rules are neglected, then classical predicate logic is structurally complete, and its unique structurally complete extension is described. It is shown that, among the class of negation-free intermediate logics between intuitionistic logic and classical predicate logic, there exists a chain of type \(\omega^\omega+1\) of hereditarily structurally complete predicate logics which are not finitely axiomatizable, and infinitely many such logics which are Kripke incomplete. This result is in contrast with the analogous results regarding propositional logics.

MSC:

03B10 Classical first-order logic
03B55 Intermediate logics
03F03 Proof theory in general (including proof-theoretic semantics)

Citations:

Zbl 0214.00704
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