Geometrisation of first-order logic
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.
Citation
Dyckhoff , R & Negri , S 2015 , ' Geometrisation of first-order logic ' , Bulletin of Symbolic Logic , vol. 21 , no. 2 , pp. 123–163 . https://doi.org/10.1017/bsl.2015.7
Publication
Bulletin of Symbolic Logic
Status
Peer reviewed
ISSN
1079-8986Type
Journal article
Collections
Items in the St Andrews Research Repository are protected by copyright, with all rights reserved, unless otherwise indicated.