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GEOMETRISATION OF FIRST-ORDER LOGIC

Published online by Cambridge University Press:  04 June 2015

ROY DYCKHOFF
Affiliation:
SCHOOL OF COMPUTER SCIENCE JACK COLE BUILDING, NORTH HAUGH UNIVERSITY OF ST ANDREWS ST ANDREWS, FIFE, KY 16 9SX, UKE-mail: rd@st-andrews.ac.uk
SARA NEGRI
Affiliation:
DEPARTMENT OF PHILOSOPHY P.O. BOX 24 (UNIONINKATU 40 A), 00014 UNIVERSITY OF HELSINKI, FINLANDE-mail: sara.negri@helsinki.fi

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.

Type
Articles
Copyright
Copyright © The Association for Symbolic Logic 2015 

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