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GEOMETRISATION OF FIRST-ORDER LOGIC

Published online by Cambridge University Press:  04 June 2015

ROY DYCKHOFF  and
SARA NEGRI
ROY DYCKHOFF
Affiliation:
SCHOOL OF COMPUTER SCIENCE JACK COLE BUILDING, NORTH HAUGH UNIVERSITY OF ST ANDREWS ST ANDREWS, FIFE, KY 16 9SX, UKE-mail: [email protected]
SARA NEGRI
Affiliation:
DEPARTMENT OF PHILOSOPHY P.O. BOX 24 (UNIONINKATU 40 A), 00014 UNIVERSITY OF HELSINKI, FINLANDE-mail: [email protected]
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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.

Keywords

coherent implicationcoherent logicgeometric logicconservative extensionweakly positive formula
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 21 , Issue 2 , June 2015 , pp. 123 - 163
Copyright
Copyright © The Association for Symbolic Logic 2015 

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