TY - JOUR
T1 - Distinguished algebraic semantics for t-norm based fuzzy logics
T2 - Methods and algebraic equivalencies
AU - Cintula, Petr
AU - Esteva, Francesc
AU - Gispert, Joan
AU - Godo, Lluís
AU - Montagna, Franco
AU - Noguera, Carles
N1 - Copyright 2009 Elsevier B.V., All rights reserved.
PY - 2009/7
Y1 - 2009/7
N2 - This paper is a contribution to Mathematical fuzzy logic, in particular to the algebraic study of t-norm based fuzzy logics. In the general framework of propositional core and ?-core fuzzy logics we consider three properties of completeness with respect to any semantics of linearly ordered algebras. Useful algebraic characterizations of these completeness properties are obtained and their relations are studied. Moreover, we concentrate on five kinds of distinguished semantics for these logics-namely the class of algebras defined over the real unit interval, the rational unit interval, the hyperreals (all ultrapowers of the real unit interval), the strict hyperreals (only ultrapowers giving a proper extension of the real unit interval) and finite chains, respectively-and we survey the known completeness methods and results for prominent logics. We also obtain new interesting relations between the real, rational and (strict) hyperreal semantics, and good characterizations for the completeness with respect to the semantics of finite chains. Finally, all completeness properties and distinguished semantics are also considered for the first-order versions of the logics where a number of new results are proved.
AB - This paper is a contribution to Mathematical fuzzy logic, in particular to the algebraic study of t-norm based fuzzy logics. In the general framework of propositional core and ?-core fuzzy logics we consider three properties of completeness with respect to any semantics of linearly ordered algebras. Useful algebraic characterizations of these completeness properties are obtained and their relations are studied. Moreover, we concentrate on five kinds of distinguished semantics for these logics-namely the class of algebras defined over the real unit interval, the rational unit interval, the hyperreals (all ultrapowers of the real unit interval), the strict hyperreals (only ultrapowers giving a proper extension of the real unit interval) and finite chains, respectively-and we survey the known completeness methods and results for prominent logics. We also obtain new interesting relations between the real, rational and (strict) hyperreal semantics, and good characterizations for the completeness with respect to the semantics of finite chains. Finally, all completeness properties and distinguished semantics are also considered for the first-order versions of the logics where a number of new results are proved.
UR - http://www.scopus.com/inward/record.url?partnerID=yv4JPVwI&eid=2-s2.0-67349142165&md5=ccbd8c33ffe56156a33bc51b4ca4590f
U2 - 10.1016/j.apal.2009.01.012
DO - 10.1016/j.apal.2009.01.012
M3 - Article
AN - SCOPUS:67349142165
VL - 160
SP - 53
EP - 81
JO - Annals of Pure and Applied Logic
JF - Annals of Pure and Applied Logic
SN - 0168-0072
IS - 1
ER -