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Studia Logica

, Volume 53, Issue 4, pp 503–532 | Cite as

A finite analog to the löwenheim-skolem theorem

  • David Isles
Article

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 modelMD=[M, {r(x)}] of Γ where each ranger(x) is finite.

Keywords

Mathematical Logic Model Theory Inference Rule Computational Linguistic Traditional Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • David Isles
    • 1
  1. 1.Department of MathematicsTufts UniversityMedfordUSA

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