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Analyzing completeness of axiomatic functional systems for temporal × modal logics
Author(s) -
Burrieza Alfredo,
de Guzmán Inmaculada P.,
MuñozVelasco Emilio
Publication year - 2010
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.200810038
Subject(s) - surjective function , completeness (order theory) , axiom , mathematical proof , injective function , mathematics , axiomatic system , property (philosophy) , modal , algebra over a field , calculus (dental) , discrete mathematics , pure mathematics , medicine , mathematical analysis , philosophy , chemistry , geometry , dentistry , epistemology , polymer chemistry
In previous works, we presented a modification of the usual possible world semantics by introducing an independent temporal structure in each world and using accessibility functions to represent the relation among them. Different properties ofthe accessibility functions (being injective, surjective, increasing, etc.) have been considered and axiomatic systems (called functional) which define these properties have been given. Only a few ofthese systems have been proved tobe complete. The aim ofthis paper is to make a progress in the study ofcompleteness for functional systems. For this end, we use indexes as names for temporal flows and give new proofs of completeness. Specifically, we focus our attention on the system which defines injectivity, because the system which defines this property without using indexes was proved to be incomplete in previous works. The only system considered which remains incomplete is the one which defines surjectivity, even ifwe consider a sequence ofnatural extensions ofthe previous one (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)