Geometrisation of first-order logic
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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.
Dyckhoff , R & Negri , S 2015 , ' Geometrisation of first-order logic ' Bulletin of Symbolic Logic , vol. 21 , no. 2 , pp. 123–163 . DOI: 10.1017/bsl.2015.7
Bulletin of Symbolic Logic
Copyright © The Association for Symbolic Logic 2015. Originally published in The Bulletin of Symbolic Logic. This work is made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at http://dx.doi.org/10.1017/bsl.2015.7
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