Информационная система "Конференции"

Abstracts

Non-classical logics, proof theory and universal algebra

The given work continues work "Automatical recognition of the interpolation property in some propositional superintuitionistic logics" (Vestnik of Novosibirsk State University, v.3, No.4 (2003), 74-84).

In the given work the algorithm of automatical recognition of the Craig interpolation property in extensions of the modal logic S5 is described.

Larisa Maksimova in the work "Interpolation theorems in modal logics and amalgamable varieties of pseudoboolean algebras " (Algebra and Logic, 18, 5 (1977), 556-586) has proved that number of extensions of the logic S4 with Craig interpolation property at most 38.

In the article "Interpolation theorems in modal logics: sufficient conditions" (Algebra and logic, 19, 5 (1977), 194-213) Larisa Maksimova has established that exist not less than 25 extensions of the logic S4 with Craig interpolation property.

This range has been defined more exactly afterward. It is known that the number of the extensions of the logic S4 no less than 31 and no more than 38 for the present moment. For six logics it remains unknown have its interpolation property whether or not still.

However for the extensions of the Lewis logic S5 it appears, that interpolation property equally four logic will possess:

• logic S5;
• the logic characterized by a single-element Kripke frame;
• the logic characterized by a two-element Kripke frame;
• inconsistent logic.

Denote by L+A the extension of logic L by an extra axiom scheme A. We shall notice that the given definition can be distributed to finite number of formulas by taken conjunction of given formulas as new scheme of the axioms.

The following proposition will help us to automate recognition of the interpolation property for the logic L=S5+A (formula A has N variables).

1. Logic L coincides with logic S5 if formula A is true on an N+1-elements cluster.

2. If formula A is denied on a three-element cluster then logic L coincides with one of above mentioned logics with Craig interpolation property excepting logic S5.

The author creates the computer program realizing above described algorithm of recognition of the interpolation property at modal logics which contains Lewis logic S5.

The work is supported by grant of Russian Foundation for Basic Research 03-06-80178 and Scientific Schools 2069.2003.1.

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