Three interpolation theorems for typeless logics
Tarek Sayed Ahmed · Logic Journal of IGPL · 2010
We prove three interpolation theorems for typless logics with equality. Typless logics, introduced by Henkin and Tarski, are algebraizable extensions of first-order logic allowing infinitary predicate symbols. Δα (α an infinite ordinal) denotes the language of a typless logic with α many variables. As a sample, we shall prove: THEOREM 1 Fmr denotes the set of all formulas in a language. Now assuming, that everything is countable, we show that there exists two unary connectives, such that if Fm denotes the formulas in this expanded language, then That is, in the countable case, we can add finitely many connectives to code the quantified variables in Theorem 1. Our last interpolation theorem addresses certain (natural) expansions of Fmr. We also give counterexamples showing that the scope of our results is the best possible. Our treatmet is algebraic via reducts of polyadic equality algebras. (2000 Mathematics Subject Classification. Primary 03G15. Secondary 03C05, 03C40.)