David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Journal of Symbolic Logic 65 (2):756-766 (2000)
In the present paper we prove that the poset of all extensions of the logic defined by a class of matrices whose sets of distinguished values are equationally definable by their algebra reducts is the retract, under a Galois connection, of the poset of all subprevarieties of the prevariety generated by the class of the algebra reducts of the matrices involved. We apply this general result to the problem of finding and studying all extensions of the logic of paradox (viz., the implication-free fragment of any non-classical normal extension of the relevance-mingle logic). In order to solve this problem, we first study the structure of prevarieties of Kleene lattices. Then, we show that the poset of extensions of the logic of paradox forms a four-element chain, all the extensions being finitely many-valued and finitely-axiomatizable logics. There are just two proper consistent extensions of the logic of paradox. The first is the classical logic that is relatively axiomatized by the Modus ponens rule for the material implication. The second extension, being intermediate between the logic of paradox and the classical logic, is the one relatively axiomatized by the Ex Contradictione Quodlibet rule
|Keywords||Propositional Logic Extension Relative Axiomatization Matrix Prevariety Galois Retraction Kleene Lattice Paraconsisent Logic Logic of Paradox Paraconsistent Kleene Lattice Classical Logic Classical Kleene Lattice Boolean Lattice|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
No references found.
Citations of this work BETA
No citations found.
Similar books and articles
Marcus Kracht (1998). On Extensions of Intermediate Logics by Strong Negation. Journal of Philosophical Logic 27 (1):49-73.
Yutaka Miyazaki (2007). A Splitting Logic in NExt(KTB). Studia Logica 85 (3):381 - 394.
Frank Wolter (1998). On Logics with Coimplication. Journal of Philosophical Logic 27 (4):353-387.
Dominique Luzeaux, Jean Sallantin & Christopher Dartnell (2008). Logical Extensions of Aristotle's Square. Logica Universalis 2 (1):167-187.
W. J. Blok & Don Pigozzi (1986). Protoalgebraic Logics. Studia Logica 45 (4):337 - 369.
Walter Sinnott-Armstrong & Amit Malhotra (2002). How to Avoid Deviance (in Logic). History and Philosophy of Logic 23 (3):215--36.
Dimiter Vakarelov (2005). Nelson's Negation on the Base of Weaker Versions of Intuitionistic Negation. Studia Logica 80 (2-3):393 - 430.
Kazimierz Swirydowicz (1999). There Exist Exactly Two Maximal Strictly Relevant Extensions of the Relevant Logic R. Journal of Symbolic Logic 64 (3):1125-1154.
W. J. Blok (1980). The Lattice of Modal Logics: An Algebraic Investigation. Journal of Symbolic Logic 45 (2):221-236.
Marcus Kracht (1990). An Almost General Splitting Theorem for Modal Logic. Studia Logica 49 (4):455 - 470.
Added to index2009-01-28
Total downloads4 ( #258,330 of 1,102,698 )
Recent downloads (6 months)3 ( #120,304 of 1,102,698 )
How can I increase my downloads?