Forcing, Downward Löwenheim-Skolem and Omitting Types Theorems, Institutionally

Logica Universalis 8 (3-4):469-498 (2014)
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Abstract

In the context of proliferation of many logical systems in the area of mathematical logic and computer science, we present a generalization of forcing in institution-independent model theory which is used to prove two abstract results: Downward Löwenheim-Skolem Theorem and Omitting Types Theorem . We instantiate these general results to many first-order logics, which are, roughly speaking, logics whose sentences can be constructed from atomic formulas by means of Boolean connectives and classical first-order quantifiers. These include first-order logic , logic of order-sorted algebras , preorder algebras , as well as their infinitary variants \ , \ , \ . In addition to the first technique for proving OTT, we develop another one, in the spirit of institution-independent model theory, which consists of borrowing the Omitting Types Property from a simpler institution across an institution comorphism. As a result we export the OTP from FOL to partial algebras and higher-order logic with Henkin semantics , and from the institution of \ to \ and \ . The second technique successfully extends the domain of application of OTT to non conventional logical systems for which the standard methods may fail

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References found in this work

A formulation of the simple theory of types.Alonzo Church - 1940 - Journal of Symbolic Logic 5 (2):56-68.
Completeness in the theory of types.Leon Henkin - 1950 - Journal of Symbolic Logic 15 (2):81-91.
A Formulation of the Simple Theory of Types.Alonzo Church - 1940 - Journal of Symbolic Logic 5 (3):114-115.
A generalization of the concept of ω-consistency.Leon Henkin - 1954 - Journal of Symbolic Logic 19 (3):183-196.

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