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ON FLATTENING ELIMINATION RULES

Published online by Cambridge University Press:  23 January 2014

GRIGORY K. OLKHOVIKOV  and
PETER SCHROEDER-HEISTER
GRIGORY K. OLKHOVIKOV*
Affiliation:
Department of Philosophy, Ural Federal University
PETER SCHROEDER-HEISTER*
Affiliation:
Department of Computer Science, Universität Tübingen
*
*DEPARTMENT OF PHILOSOPHY URAL FEDERAL UNIVERSITY 51 LENIN AVENUE 620083 YEKATERINBURG, RUSSIA E-mail:[email protected]
†DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF TÜBINGEN SAND 13 72076 TÜBINGEN, GERMANY E-mail:[email protected]
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Abstract

In proof-theoretic semantics of intuitionistic logic it is well known that elimination rules can be generated from introduction rules in a uniform way. If introduction rules discharge assumptions, the corresponding elimination rule is a rule of higher level, which allows one to discharge rules occurring as assumptions. In some cases, these uniformly generated elimination rules can be equivalently replaced with elimination rules that only discharge formulas or do not discharge any assumption at all—they can be flattened in a terminology proposed by Read. We show by an example from propositional logic that not all introduction rules have flat elimination rules. We translate the general form of flat elimination rules into a formula of second-order propositional logic and demonstrate that our example is not equivalent to any such formula. The proof uses elementary techniques from propositional logic and Kripke semantics.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 7 , Issue 1 , March 2014 , pp. 60 - 72
Copyright
Copyright © Association for Symbolic Logic 2013 

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