Bulletin of Symbolic Logic

Provability with Finitely Many Variables

Robin Hirsch, Ian Hodkinson, and Roger D. Maddux

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Abstract

For every finite $n \geq 4$ there is a logically valid sentence $\varphi_n$ with the following properties: $\varphi_n$ contains only 3 variables (each of which occurs many times); $\varphi_n$ contains exactly one nonlogical binary relation symbol (no function symbols, no constants, and no equality symbol): $\varphi_n$ has a proof in first-order logic with equality that contains exactly n variables, but no proof containing only n - 1 variables. This result was first proved using the machinery of algebraic logic developed in several research monographs and papers. Here we replicate the result and its proof entirely within the realm of (elementary) first-order binary predicate logic with equality. We need the usual syntax, axioms, and rules of inference to show that $\varphi_n$ has a proof with only n variables. To show that $\varphi_n$ has no proof with only n - 1 variables we use alternative semantics in place of the usual, standard, set-theoretical semantics of first-order logic.

Article information

Source
Bull. Symbolic Logic, Volume 8, Number 3 (2002), 348-379.

Dates
First available in Project Euclid: 20 June 2007

Permanent link to this document
https://projecteuclid.org/euclid.bsl/1182353893

Digital Object Identifier
doi:10.2178/bsl/1182353893

Mathematical Reviews number (MathSciNet)
MR1931348

Zentralblatt MATH identifier
1024.03010

JSTOR
links.jstor.org

Citation

Hirsch, Robin; Hodkinson, Ian; Maddux, Roger D. Provability with Finitely Many Variables. Bull. Symbolic Logic 8 (2002), no. 3, 348--379. doi:10.2178/bsl/1182353893. https://projecteuclid.org/euclid.bsl/1182353893


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