Geometry of Robinson Consistency in Lukasiewicz Logic.

BUSANICHE, MANUELA

Barcelona, España

Congreso; Algebraic and Topological Methods in Non-Classical Logics II; 2005

Universidad de Barcelona

For all unexplained notions about  MV-algebras and Lukasiewicz(always propositional in the present talk) logic: we refer to [1].For X$ an arbitrary set of variables, L{X} denotesthe set of formulas  whose variables are in X.  Any such formula is said tobe an  L{X}-formula. The definition is the same for boolean logic and for many-valued logic.A proper subset S of  L{X} is called a  theory   if- S contains all L{X}-tautologies of  Lukasiewicz infinite-valued propositional logic, and- S is closed under modus ponens.Theories are in one-one correspondence with ideals of free MV-algebras.An L{X}-theory S is said to be  prime (also called ``complete''  in Hájek's monograph[2]) if  for any L{X}-formulas varphi and psi either varphi Implies psi orpsi implies varphi belongs to S.Prime theories are in one-one correspondence with prime ideals of free MV-algebras.Every prime theory  S has a unique maximally consistent completion  S'. By contrast with boolean logic S' generally does not coincide with S.Maximally consistent theories are in one-one correspondence withmaximal ideals of free MV-algebras. The  Robinson  consistency property for boolean,as well for Lukasiewicz logic, can be stated as follows: Suppose S is a prime L{X}-theory, and R  is a prime L_{Y}-theory. Let L{Z} be the intersection of  L{X} and  L{Y}, and L_{W} the joint of L{X}  and  L{Y}.If  S intersection  L_{Z}  equals  R intersection L{Z} then  there is a prime L{W}-theoryV  such that S is the intersection of V and L{X} and R is the intersection of V and L{Y}. We give a  proof of  the  Robinson  consistency property  for Lukasiewicz  propositional logic.As a corollary we  obtain a new proof of the amalgamation property for MV-algebras. For the proof of our main results we make no use of lattice-ordered groups and the Gamma  functor.Rather, we make use of geometric tools naturally arising from the rich theory of MV-algebras, such as McNaughton's representation of free MV-algebras via [0,1]-valued piecewise linear functions,  unimodular triangulations of the n-cube, and the classification of  spectral spacesof free MV-algebras via bases in euclidean space. [1] R. Cignoli, I. M. L. D'Ottaviano,  D. Mundici,  Algebraic Foundations of Many-Valued Reasoning,   Kluwer, Dordrecht, 2000.[2]  P. Hajek, Metamathematics of Fuzzy Logic, Kluwer, Dordrecht, 1998.