On Partial and Paraconsistent Logics

In this paper we consider the theory of predicate logics in which the principle of Bivalence or the principle of Non-Contradiction or both fail. Such logics are partial or paraconsistent or both. We consider sequent calculi for these logics and prove Model Existence. For L4, the most general logic under consideration, we also prove a version of the Craig-Lyndon Interpolation Theorem. The paper shows that many techniques used for classical predicate logic generalise to partial and paraconsistent logics once the right set-up is chosen. Our logic L4 has a semantics that also underlies Belnap’s [4] and is related to the logic of bilattices. L4 is in focus most of the time, but it is also shown how results obtained for L4 can be transferred to several variants.

[1]  Tore Langholm.  How different is partial logic , 1996 .

[2]  T. Langholm.  Partiality, truth and persistence , 1988 .

[3]  Albert Visser,et al.  Four valued semantics and the Liar , 1984, J. Philos. Log..

[4]  Reiner Hähnle,et al.  Automated deduction in multiple-valued logics , 1993, International series of monographs on computer science.

[5]  J. Barkley Rosser,et al.  Methoden zur Axiomatisierung Beliebiger Aussagen- und Pradikatenkalkule. , 1970 .

[6]  John P. Cleave,et al.  The Notion of Logical Consequence in the Logic of Inexact Predicates , 1974 .

[7]  Paul C. Gilmore,et al.  The Consistency of Partial Set Theory without Extensionality , 1974 .

[8]  N. Cocchiarella,et al.  Situations and Attitudes. , 1986 .

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

[10]  Marit Holden,et al.  Weak Logic Theory , 1991, Theor. Comput. Sci..

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

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

[13]  Melvin Fitting,et al.  Bilattices and the Semantics of Logic Programming , 1991, J. Log. Program..

[14]  Epistemologia e História de Ciência.  The journal of non-classical logic , 1982 .

[15]  Elias George Coenraad Thijsse,et al.  Partial logic and knowledge representation , 1992 .

[16]  J. Dunn,et al.  Intuitive semantics for first-degree entailments and ‘coupled trees’ , 1976 .

[17]  Alexander Bochman,et al.  A Logical Foundation for Logic Programming I: Biconsequence Relations and Nonmonotonic Completion , 1998, J. Log. Program..

[18]  G. Rousseau.  Sequents in many valued logic I , 1967 .

[19]  Johannes Ogier Maria Jaspars,et al.  Calculi for constructive communication : a study of the dynamics of partial states , 1994 .

[20]  Walter Alexandre Carnielli,et al.  Systematization of finite many-valued logics through the method of tableaux , 1987, Journal of Symbolic Logic.

[21]  Melvin Fitting,et al.  First-Order Logic and Automated Theorem Proving , 1990, Graduate Texts in Computer Science.

[22]  Reinhard Muskens.  Going partial in Montague grammar , 1989 .

[23]  Solomon Feferman,et al.  Toward useful type-free theories. I , 1984, Journal of Symbolic Logic.

[24]  S. C. Kleene,et al.  Introduction to Metamathematics , 1952 .

[25]  Richard Zach,et al.  Proof Theory of Finite-valued Logics , 1993 .

[26]  Marcello D'Agostino,et al.  Investigations into the complexity of some propositional calculi , 1990 .

[27]  Peter W. Woodruff.  Paradox, truth and logic part I: Paradox and truth , 1984, J. Philos. Log..

[28]  Christian G. Fermüller,et al.  Elimination of Cuts in First-order Finite-valued Logics , 1993, J. Inf. Process. Cybern..

[29]  R. Muskens.  Meaning and Partiality , 1995 .