Notre Dame Journal of Formal Logic

Four-valued Logic

Katalin Bimbó and J. Michael Dunn


Four-valued semantics proved useful in many contexts from relevance logics to reasoning about computers. We extend this approach further. A sequent calculus is defined with logical connectives conjunction and disjunction that do not distribute over each other. We give a sound and complete semantics for this system and formulate the same logic as a tableaux system. Intensional conjunction (fusion) and its residuals (implications) can be added to the sequent calculus straightforwardly. We extend a simplified version of the earlier semantics for this system and prove soundness and completeness. Then, with some modifications to this semantics, we arrive at a mathematically elegant yet powerful semantics that we call generalized Kripke semantics.

Article information

Notre Dame J. Formal Logic, Volume 42, Number 3 (2001), 171-192.

First available in Project Euclid: 12 September 2003

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03B47: Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) {For proof-theoretic aspects see 03F52}

substructural logics Lambek calculi relevance logic lattice representation Kripke semantics


Bimbó, Katalin; Dunn, J. Michael. Four-valued Logic. Notre Dame J. Formal Logic 42 (2001), no. 3, 171--192. doi:10.1305/ndjfl/1063372199.

Export citation


  • Allwein, G., and J. M. Dunn, "Kripke models for linear logic", The Journal of Symbolic Logic, vol. 58 (1993), pp. 514–45.
  • Belnap, N., "Life in the undistributed middle", pp. 31–41 in Substructural Logics, edited by erseeditorsnames K. Došen and P. Schroeder-Heister, Oxford University Press, New York, 1993.
  • Belnap, N. D., Jr., "A useful four-valued logic", pp. 5–37 in Modern Uses of Multiple-Valued Logic, edited by erseeditorsnames J. M. Dunn and G. Epstein, Reidel, Dordrecht, 1977.
  • Bimbó, K., "Semantics for structurally free logics $\mathit{LC}+$", Logic Journal of the IGPL, vol. 9 (2001), pp. 525–39.
  • Birkhoff, G., Lattice Theory, American Mathematical Society, Providence, 1967.
  • Birkhoff, G., and O. Frink, Jr., "Representations of lattices by sets", Transactions of the American Mathematical Society, vol. 64 (1948), pp. 299–316.
  • Curry, H. B., Foundations of Mathematical Logic, McGraw-Hill Book Company, New York, 1963.
  • Dunn, J. M., "Relevance logic and entailment", pp. 117–229 in Handbook of Philosophical Logic, 2d edition, edited by D. M. Gabbay and F. Guenthner, Kluwer Academic Publishers, Dordrecht, 2001.
  • Dunn, J. M., "Intuitive semantics for first-degree entailments and `coupled trees'", Philosophical Studies, vol. 29 (1976), pp. 149–68.
  • Dunn, J. M., "Positive modal logic", Studia Logica, vol. 55 (1995), pp. 301–17.
  • Gentzen, G., Untersuchungen über das logische Schließ en, Wissenschaftliche Buchgesellschaft, Darmstadt, 1969.
  • Goldblatt, R., "Varieties of complex algebras", Annals of Pure and Applied Logic, vol. 44 (1989), pp. 173–242.
  • Hartonas, C., and J. M. Dunn, "Stone duality for lattices", Algebra Universalis, vol. 37 (1997), pp. 391–401.
  • Jónsson, B., and A. Tarski, "Boolean algebras with operators. I", American Journal of Mathematics, vol. 73 (1951), pp. 891–939.
  • Jónsson, B., and A. Tarski, "Boolean algebras with operators. II", American Journal of Mathematics, vol. 74 (1952), pp. 127–62.
  • Meyer, R. K., and R. Routley, "Algebraic analysis of entailment. I", Logique et Analyse (Nouvelle Série), vol. 15 (1972), pp. 407–28.
  • Routley, R., V. Plumwood, R. K. Meyer, and R. T. Brady, Relevant Logics and Their Rivals. Part I, Ridgeview Publishing Company, Atascadero, 1982.
  • Smullyan, R. M., First-Order Logic, Springer-Verlag, New York, 1968.
  • Urquhart, A., "A topological representation theory for lattices", Algebra Universalis, vol. 8 (1978), pp. 45–58.