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ELIMINATING DISJUNCTIONS BY DISJUNCTION ELIMINATION

Published online by Cambridge University Press:  21 June 2017

DAVIDE RINALDI ,
PETER SCHUSTER  and
DANIEL WESSEL
DAVIDE RINALDI
Affiliation:
DIPARTIMENTO DI INFORMATICA UNIVERSITÀ DEGLI STUDI DI VERONA STRADA LE GRAZIE, 15 37134 VERONA, ITALY E-mail: davide.rinaldi@univr.it
PETER SCHUSTER
Affiliation:
DIPARTIMENTO DI INFORMATICA UNIVERSITÀ DEGLI STUDI DI VERONA STRADA LE GRAZIE, 15 37134 VERONA, ITALY E-mail: peter.schuster@univr.it
DANIEL WESSEL
Affiliation:
DIPARTIMENTO DI MATEMATICA UNIVERSITÀ DEGLI STUDI DI TRENTO VIA SOMMARIVE, 14 38123 POVO (TN), ITALY E-mail: daniel.wessel@unitn.it
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Abstract

Completeness and other forms of Zorn’s Lemma are sometimes invoked for semantic proofs of conservation in relatively elementary mathematical contexts in which the corresponding syntactical conservation would suffice. We now show how a fairly general syntactical conservation theorem that covers plenty of the semantic approaches follows from an utmost versatile criterion for conservation given by Scott in 1974.

To this end we work with multi-conclusion entailment relations as extending single-conclusion entailment relations. In a nutshell, the additional axioms with disjunctions in positive position can be eliminated by reducing them to the corresponding disjunction elimination rules, which in turn prove admissible in all known mathematical instances. In deduction terms this means to fold up branchings of proof trees by way of properties of the relevant mathematical structures.

Applications include the syntactical counterparts of the theorems or lemmas known under the names of Artin–Schreier, Krull–Lindenbaum, and Szpilrajn. Related work has been done before on individual instances, e.g., in locale theory, dynamical algebra, formal topology and proof analysis.

Keywords

03F0303E2503F65syntactical conservationaxiom of choiceHilbert’s Programmedisjunction eliminationentailment relation

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Type
Articles
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Bulletin of Symbolic Logic , Volume 23 , Issue 2 , June 2017 , pp. 181 - 200
Copyright
Copyright © The Association for Symbolic Logic 2017 

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