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A finite analog to the löwenheim-skolem theorem

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Abstract

The traditional model theory of first-order logic assumes that the interpretation of a formula can be given without reference to its deductive context. This paper investigates an interpretation which depends on a formula's location within a derivation. The key step is to drop the assumption that all quantified variables must have the same range and to require only that the ranges of variables in a derivation must be related in such way as to preserve the soundness of the inference rules. With each (consistent) derivation there is associated a “Buridan-Volpin (orBV) structure” [M, {r(x)}] which is simply a Tarski structureM for the language and a map giving the ranger(x) of each variablex in the derivation. IfLK* is (approximately) the classical sequent calculusLK of Gentzen from which the structural contraction rules have been dropped, then our main result reads: If a set of first-ordered formulas Γ has a Tarski modelM, then from any normal derivationD inLK* of Γ ⇒ Δ can be constructed aBV modelM D=[M, {r(x)}] of Γ where each ranger(x) is finite.

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Isles, D. A finite analog to the löwenheim-skolem theorem. Stud Logica 53, 503–532 (1994). https://doi.org/10.1007/BF01057648

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