Studia Logica

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

A finite analog to the löwenheim-skolem theorem

  • David Isles


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.


Mathematical Logic Model Theory Inference Rule Computational Linguistic Traditional Model 
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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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