Bilattices with Implications

In a previous work we studied, from the perspective of Abstract Algebraic Logic, the implicationless fragment of a logic introduced by O. Arieli and A. Avron using a class of bilattice-based logical matrices called logical bilattices. Here we complete this study by considering the Arieli-Avron logic in the full language, obtained by adding two implication connectives to the standard bilattice language. We prove that this logic is algebraizable and investigate its algebraic models, which turn out to be distributive bilattices with additional implication operations. We axiomatize and state several results on these new classes of algebras, in particular representation theorems analogue to the well-known one for interlaced bilattices.

[1]  Sergei P. Odintsov.  On the Representation of N4-Lattices , 2004, Stud Logica.

[2]  Antoni Torrens Torrell,et al.  Free Algebras in Varieties of Glivenko MTL-algebras Satisfying the Equation 2(x2) = (2x)2 , 2006, Stud Logica.

[3]  David Nelson,et al.  Constructible falsity , 1949, Journal of Symbolic Logic.

[4]  Arnon Avron,et al.  Logical bilattices and inconsistent data , 1994, Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science.

[5]  Matthew L. Ginsberg,et al.  Multivalued logics: a uniform approach to reasoning in artificial intelligence , 1988, Comput. Intell..

[6]  Arnon Avron,et al.  The Value of the Four Values , 1998, Artif. Intell..

[7]  Arnon Avron,et al.  Reasoning with logical bilattices , 1996, J. Log. Lang. Inf..

[8]  Nuel D. Belnap,et al.  A Useful Four-Valued Logic , 1977 .

[9]  Josep Maria Font,et al.  Characterization of the reduced matrices for the {∧,∨}-fragment of classical logic , 1991 .

[10]  Yu. M. Movsisyan,et al.  SUPERPRODUCTS, HYPERIDENTITIES, AND ALGEBRAIC STRUCTURES OF LOGIC PROGRAMMING , 2006 .

[11]  Umberto Rivieccio,et al.  Implicative twist-structures , 2014 .

[12]  Sergei P. Odintsov,et al.  Algebraic Semantics for Paraconsistent Nelson's Logic , 2003, J. Log. Comput..

[13]  G. Slutzki,et al.  A duality theory for bilattices , 2000 .

[14]  Nuel D. Belnap,et al.  How a Computer Should Think , 2019, New Essays on Belnap-­Dunn Logic.

[15]  Umberto Rivieccio,et al.  The logic of distributive bilattices , 2011, Log. J. IGPL.

[16]  Umberto Rivieccio,et al.  An Algebraic Study of Bilattice-based Logics , 2010, 1010.2552.

[17]  Ryszard Wójcicki,et al.  Theory of Logical Calculi , 1988 .

[18]  Melvin Fitting,et al.  Bilattices in logic programming , 1990, Proceedings of the Twentieth International Symposium on Multiple-Valued Logic.

[19]  Brian A. Davey.  Dualities for equational classes of Brouwerian algebras and Heyting algebras , 1976 .

[20]  Haskell B. Curry,et al.  Foundations of Mathematical Logic , 1964 .

[21]  J. Dunn,et al.  The algebra of intensional logics , 2019 .

[22]  Arnon Avron.  The Structure of Interlaced Bilattices , 1996, Math. Struct. Comput. Sci..

[23]  Josep Maria Font,et al.  Belnap's Four-Valued Logic and De Morgan Lattices , 1997, Log. J. IGPL.

[24]  Alexej P. Pynko.  Functional completeness, axiomatizability within Belnap's four valued logic and its expansions , 1999, J. Appl. Non Class. Logics.

[25]  Michael R. Darnel,et al.  Generalized Boolean Algebras in Lattice-Ordered Groups , 1997 .

[26]  R. Wójcicki.  Theory of Logical Calculi: Basic Theory of Consequence Operations , 1988 .

[27]  Umberto Rivieccio,et al.  Varieties of interlaced bilattices , 2011 .

[28]  Arnon Avron.  A Note on the Structure of Bilattices , 1995, Math. Struct. Comput. Sci..