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ON THE WAY TO A WIDER MODEL THEORY: COMPLETENESS THEOREMS FOR FIRST-ORDER LOGICS OF FORMAL INCONSISTENCY

Published online by Cambridge University Press:  03 June 2014

WALTER CARNIELLI ,
MARCELO E. CONIGLIO ,
RODRIGO PODIACKI  and
TARCÍSIO RODRIGUES
WALTER CARNIELLI*
Affiliation:
Centre for Logic, Epistemology and the History of Science – CLE and Department of Philosophy – UNICAMP
MARCELO E. CONIGLIO*
Affiliation:
Centre for Logic, Epistemology and the History of Science – CLE and Department of Philosophy – UNICAMP
RODRIGO PODIACKI*
Affiliation:
Centre for Logic, Epistemology and the History of Science – CLE and Department of Philosophy – UNICAMP
TARCÍSIO RODRIGUES*
Affiliation:
Centre for Logic, Epistemology and the History of Science – CLE and Department of Philosophy – UNICAMP
*
*CENTRE FOR LOGIC, EPISTEMOLOGY AND THE HISTORY OF SCIENCE - CLE – UNICAMP, P.O. BOX 6133, 13083-970, CAMPINAS, SP, BRAZIL E-mail: walter.carnielli@cle.unicamp.br, coniglio@cle.unicamp.br, podiacki@gmail.com and tarcisio.genaro@gmail.com DEPARTMENT OF PHILOSOPHY–UNICAMP, P.O. BOX 6133, 13083-970, CAMPINAS, SP, BRAZIL
*CENTRE FOR LOGIC, EPISTEMOLOGY AND THE HISTORY OF SCIENCE - CLE – UNICAMP, P.O. BOX 6133, 13083-970, CAMPINAS, SP, BRAZIL E-mail: walter.carnielli@cle.unicamp.br, coniglio@cle.unicamp.br, podiacki@gmail.com and tarcisio.genaro@gmail.com DEPARTMENT OF PHILOSOPHY–UNICAMP, P.O. BOX 6133, 13083-970, CAMPINAS, SP, BRAZIL
*CENTRE FOR LOGIC, EPISTEMOLOGY AND THE HISTORY OF SCIENCE - CLE – UNICAMP, P.O. BOX 6133, 13083-970, CAMPINAS, SP, BRAZIL E-mail: walter.carnielli@cle.unicamp.br, coniglio@cle.unicamp.br, podiacki@gmail.com and tarcisio.genaro@gmail.com DEPARTMENT OF PHILOSOPHY–UNICAMP, P.O. BOX 6133, 13083-970, CAMPINAS, SP, BRAZIL
*CENTRE FOR LOGIC, EPISTEMOLOGY AND THE HISTORY OF SCIENCE - CLE – UNICAMP, P.O. BOX 6133, 13083-970, CAMPINAS, SP, BRAZIL E-mail: walter.carnielli@cle.unicamp.br, coniglio@cle.unicamp.br, podiacki@gmail.com and tarcisio.genaro@gmail.com DEPARTMENT OF PHILOSOPHY–UNICAMP, P.O. BOX 6133, 13083-970, CAMPINAS, SP, BRAZIL
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Abstract

This paper investigates the question of characterizing first-order LFIs (logics of formal inconsistency) by means of two-valued semantics. LFIs are powerful paraconsistent logics that encode classical logic and permit a finer distinction between contradictions and inconsistencies, with a deep involvement in philosophical and foundational questions. Although focused on just one particular case, namely, the quantified logic QmbC, the method proposed here is completely general for this kind of logics, and can be easily extended to a large family of quantified paraconsistent logics, supplying a sound and complete semantical interpretation for such logics. However, certain subtleties involving term substitution and replacement, that are hidden in classical structures, have to be taken into account when one ventures into the realm of nonclassical reasoning. This paper shows how such difficulties can be overcome, and offers detailed proofs showing that a smooth treatment of semantical characterization can be given to all such logics. Although the paper is well-endowed in technical details and results, it has a significant philosophical aside: it shows how slight extensions of classical methods can be used to construct the basic model theory of logics that are weaker than traditional logic due to the absence of certain rules present in classical logic. Several such logics, however, as in the case of the LFIs treated here, are notorious for their wealth of models precisely because they do not make indiscriminate use of certain rules; these models thus require new methods. In the case of this paper, by just appealing to a refined version of the Principle of Explosion, or Pseudo-Scotus, some new constructions and crafty solutions to certain nonobvious subtleties are proposed. The result is that a richer extension of model theory can be inaugurated, with interest not only for paraconsistency, but hopefully to other enlargements of traditional logic.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 7 , Issue 3 , September 2014 , pp. 548 - 578
Copyright
Copyright © Association for Symbolic Logic 2014 

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