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A note on admissible rules and the disjunction property in intermediate logics. (English) Zbl 1248.03047
Summary: With any structural inference rule \(A/B\), we associate the rule \((A \lor p)/(B \lor p)\), providing that formulas \(A\) and \(B\) do not contain the variable \(p\). We call the latter rule a join-extension (\(\lor\)-extension, for short) of the former. Obviously, for any intermediate logic with disjunction property, a \(\lor\)-extension of any admissible rule is also admissible in this logic. We investigate intermediate logics in which the \(\lor\)-extension of each admissible rule is admissible. We prove that any structural finitary consequence operator (for intermediate logic) can be defined by a set of \(\lor\)-extended rules if and only if it can be defined through a set of well-connected Heyting algebras of a corresponding quasivariety. As we exemplify, the latter condition is satisfied for a broad class of algebraizable logics.

MSC:
03B55 Intermediate logics
06D20 Heyting algebras (lattice-theoretic aspects)
08C15 Quasivarieties
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