New technique of constructive designing of counterexamples to the interpolation property and Beth property was developed. It was proved that the intuitionistic logic of finite domains has neither interpolation nor the Beth property by using this technique. Also was found the first example of the predicate intermediate logic without Beth property. It is established that for any propositional superintuitionistic logic L there exists a continuum of predicate superintuitionistic logic with equality, whose propositional fragment is L and which do not possess the Beth property and interpolation property. Also there exists a continuum of predicate superintuitionistic logic without equality that have not Beth property and interpolation property. It was shown that the fragment of predicate intuitionistic logic in the language without the disjunction and existential quantifier coincides with a similar fragment of logic of constant domains. It was proved that this fragment has interpolation property and Beth property. Although the intuitionistic logic of finite domains has neither interpolation nor the Beth property the fragment of this logic in the language without disjunction and existential quantifier enjoys both properties.
Main publications:
Schreiner P. A. Continua of superintuitionistic predicate logics without Beth's property.
Schreiner P.A. Fragment of Logic of Finite Constant Domains Without Disjunction and Existential Quantifier, Abstract of contributed papers, LC2000 and ELSS2000. Paris: La Sorbonne, 2000, 247.
