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SUBFORMULA AND SEPARATION PROPERTIES IN NATURAL DEDUCTION VIA SMALL KRIPKE MODELS

Published online by Cambridge University Press:  03 June 2010

PETER MILNE
PETER MILNE*
Affiliation:
Department of Philosophy, University of Stirling
*
*DEPARTMENT OF PHILOSOPHY, UNIVERSITY OF STIRLING, STIRLING FK9 4LA, SCOTLAND, UNITED KINGDOM E-mail:[email protected]
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Abstract

Various natural deduction formulations of classical, minimal, intuitionist, and intermediate propositional and first-order logics are presented and investigated with respect to satisfaction of the separation and subformula properties. The technique employed is, for the most part, semantic, based on general versions of the Lindenbaum and Lindenbaum–Henkin constructions. Careful attention is paid (i) to which properties of theories result in the presence of which rules of inference, and (ii) to restrictions on the sets of formulas to which the rules may be employed, restrictions determined by the formulas occurring as premises and conclusion of the invalid inference for which a counterexample is to be constructed. We obtain an elegant formulation of classical propositional logic with the subformula property and a singularly inelegant formulation of classical first-order logic with the subformula property, the latter, unfortunately, not a product of the strategy otherwise used throughout the article. Along the way, we arrive at an optimal strengthening of the subformula results for classical first-order logic obtained as consequences of normalization theorems by Dag Prawitz and Gunnar Stålmarck.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 3 , Issue 2 , June 2010 , pp. 175 - 227
Copyright
Copyright © Association for Symbolic Logic 2010

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