Model Theory for Modal Logic pp 71-74 | Cite as

# Joint Consistency and Interpolation

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## Abstract

If đŹ is a modal structure for the language ML and if MLâ is a sublanguage of ML, we define the *restriction of* đŹ *to* MLâ. đŹâ MLâ, as follows. Let đŹ = <*A* ^{k}, **K, L, O, N**> and let Lâ be the underlying classical language of MLâ. Then đŹâ, where *A*â^{k}|Lâ is the restriction of the classical structure *A* ^{k} to Lâ (cf. Shoenfield (1967), p. 43). We also say that A is an *expansion* of A| MLâ to ML. Let S be a modal system and let T and Tâ be S-theories with languages ML(T) and ML(Tâ), respectively. We define the *union of* T *and* Tâ, T âȘ Tâ, to be the S-theory with language Lââ whose nonlogical symbols consist of precisely those of ML(T) together with those of ML(Tâ) (at this point, if we have not already, we agree to the convention of Shoenfield (1967), p. 15, that if a given symbol is used in one manner in a certain language, it is used in all other languages in the identical manner), and whose nonlogical axioms are precisely those of T together with those of Tâ. The following proof of the Joint Consistency Theorem is a rather direct adaptation of the classical proof as given in Shoenfield (1967). An approach using somewhat different techniques is contained in Gabbey (1972a).

## Keywords

Modal SystemÂ Classical StructureÂ Predicate SymbolÂ Closed FormulaÂ Classical LanguageÂ## Preview

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