Geometrisation of First-Order Logic

Bulletin of Symbolic Logic 21 (2):123-163 (2015)
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Abstract

That every first-order theory has a coherent conservative extension is regarded by some as obvious, even trivial, and by others as not at all obvious, but instead remarkable and valuable; the result is in any case neither sufficiently well-known nor easily found in the literature. Various approaches to the result are presented and discussed in detail, including one inspired by a problem in the proof theory of intermediate logics that led us to the proof of the present paper. It can be seen as a modification of Skolem’s argument from 1920 for his “Normal Form” theorem. “Geometric” being the infinitary version of “coherent”, it is further shown that every infinitary first-order theory, suitably restricted, has a geometric conservative extension, hence the title. The results are applied to simplify methods used in reasoning in and about modal and intermediate logics. We include also a new algorithm to generate special coherent implications from an axiom, designed to preserve the structure of formulae with relatively little use of normal forms.

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References found in this work

Modal logic.Yde Venema - 2000 - Philosophical Review 109 (2):286-289.
From Frege to Gödel.Jean van Heijenoort - 1968 - Philosophy of Science 35 (1):72-72.
Proof Analysis in Modal Logic.Sara Negri - 2005 - Journal of Philosophical Logic 34 (5-6):507-544.
A formal system for euclid’s elements.Jeremy Avigad, Edward Dean & John Mumma - 2009 - Review of Symbolic Logic 2 (4):700--768.
Proof analysis in intermediate logics.Roy Dyckhoff & Sara Negri - 2012 - Archive for Mathematical Logic 51 (1):71-92.

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