Three-valued Logics in Modal Logic Authors Article Open Access First Online: 21 August 2012 Received: 23 January 2012
Every truth-functional three-valued propositional logic can be conservatively translated into the modal logic
S5. We prove this claim constructively in two steps. First, we define a Translation Manual that converts any propositional formula of any three-valued logic into a modal formula. Second, we show that for every S5-model there is an equivalent three-valued valuation and vice versa. In general, our Translation Manual gives rise to translations that are exponentially longer than their originals. This fact raises the question whether there are three-valued logics for which there is a shorter translation into S5. The answer is affirmative: we present an elegant linear translation of the Logic of Paradox and of Strong Three-valued Logic into S5.
Batens, D., On some remarkable relations between paraconsistent logics, modal logics, and ambiguity logics, in W. A. Carnielli, M. E. Coniglio, and I. M. L. D’Ottaviano (eds.),
Paraconsistency. The Logical Way to the Inconsistent, Marcel Dekker, New York, 2002, pp. 275–293.
Brown B.: Yes, Virginia, there really are paraconsistent logics. Journal of Philosophical Logic
, 489–500 (1999)
Bush, D., Sequent formalizations of three-valued logic, in P. Doherty (ed.),
Partiality, Modality and Nonmonotonicity, CSLI Publications, Stanford, 1996, pp. 45–75.
Cadoli M., Schaerf M.: On the complexity of entailment in propositional multivalued logics. Annals of Mathematics and Artificial Intelligence
, 29–50 (1996)
Chellas B.F.: Modal Logic. Cambridge University Press, Cambridge (1980)
Cobreros P., Egré P., Ripley D., van Rooij R.: Tolerant, classical, strict. Journal of Philosophical Logic
, 347–385 (2012)
D’Ottaviano, I. M. L., and H. A. Feitosa, On Gödel’s modal interpretation of the intuitionistic logic, in J.-Y. Béziau (ed.),
Universal Logic: An Anthology, Birkhäuser, Basel, 2012, pp. 71–88.
Feitosa H.A., D’Ottaviano I.M.L.: Conservative translations. Annals of Pure and Applied Logic
, 205–227 (2001)
Gödel, K., Eine Interpretation des intuitionistischen Aussagenkalküls,
Ergebnisse eines mathematischen Kolloquiums 4:39–40, 1933. Reprinted in K. Gödel, Collected Works, Volume 1, Clarendon Press, Oxford, 1986, pp. 300–302.
Kleene S.C.: On notation for ordinal numbers. Journal of Symbolic Logic
, 150–155 (1938)
Kleene, S. C.,
Introduction to Metamathematics, P. Noordhoff N.V., Groningen, 1952.
Ladner R.E.: The computational complexity of provability in systems of modal propositional logic. SIAM Journal on Computing
, 467–480 (1977)
Lewis, D., Logic for equivocators,
Noûs 16:431–441, 1982. Reprinted with corrections in D. Lewis, Papers in Philosophical Logic, Cambridge University Press, Cambridge, 1998, pp. 97–110.
Priest G.: The logic of paradox. Journal of Philosophical Logic
, 219–241 (1979)
Priest G.: Boolean negation and all that. Journal of Philosophical Logic
, 201–215 (1990)
van Benthem J.: Partiality and nonmonotonicity in classical logic. Logique et Analyse
29, 225–247 (1986) Copyright information
© Springer Science+Business Media B.V. 2012