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The property (HD) in intermediate logics. A partial solution of a problem of H. Ono. (English) Zbl 0696.03009

A first-order formula in which every occurrence of disjunction and existential quantifier lays inside the antecedent of some implication or inside the scope of some negation symbol is called Harrop formula (H- formula). An intermediate predicate logic L has the HD-property if for every H-formula \(\alpha\) if \(\alpha\) \(\to (\beta \vee \gamma)\in L\) then \(\alpha\) \(\to \beta \in L\) or \(\alpha\) \(\to \gamma \in L\). It is clear that if an intermediate logic has the HD-property then it has the usual disjunction property (DP-property). The question of H. Ono is, if the DP- property implies the HD-property for every intermediate predicate logic. The authors give an answer for the propositional case: the HP-property and the DP-property are equivalent.
Reviewer: V.V.Rybakov

MSC:

03B55 Intermediate logics
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