Notre Dame Journal of Formal Logic

A Natural Deduction System for First Degree Entailment

Allard M. Tamminga and Koji Tanaka


This paper is concerned with a natural deduction system for First Degree Entailment (FDE). First, we exhibit a brief history of FDE and of combined systems whose underlying idea is used in developing the natural deduction system. Then, after presenting the language and a semantics of FDE, we develop a natural deduction system for FDE. We then prove soundness and completeness of the system with respect to the semantics. The system neatly represents the four-valued semantics for FDE.

Article information

Notre Dame J. Formal Logic Volume 40, Number 2 (1999), 258-272.

First available in Project Euclid: 3 December 2002

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}


Tamminga, Allard M.; Tanaka, Koji. A Natural Deduction System for First Degree Entailment. Notre Dame J. Formal Logic 40 (1999), no. 2, 258--272. doi:10.1305/ndjfl/1038949541.

Export citation


  • [1] Ackermann, W., ``Begründung einer strengen Implikation,'' The Journal of Symbolic Logic, vol. 21 (1956), pp. 113--28.
  • [2] Anderson, A. R., and N. D. Belnap, Jr., Entailment: the Logic of Relevance and Necessity, vol. 1, Princeton University Press, Princeton, 1975.
  • [3] Anderson, A. R., N. D. Belnap, Jr., and J. M. Dunn, Entailment: the Logic of Relevance and Necessity, vol. 2, Princeton University Press, Princeton, 1992.
  • [4] Brentano, F. C. H. H. J., Die Lehre vom richtigen Urteil, Francke Verlag, Bern, 1956.
  • [5] Dunn, J. M., ``Intuitive semantics for first-degree entailments and `coupled trees','' Philosophical Studies, vol. 29 (1976), pp. 149--68.
  • [6] Konikowska, B., ``A two-valued logic for reasoning about different types of consequence in Kleene's three-valued logic,'' Studia Logica, vol. 49 (1990), pp. 541--55.
  • [7] Ł ukasiewicz, J., ``Two-valued logic,'' pp. 89--109 in Jan Ł ukasiewicz. Selected Works, edited by L. Borkowski, North-Holland, Amsterdam, 1970.
  • [8] Ł ukasiewicz, J., Aristotle's Syllogistic from the Standpoint of Modern Formal Logic, 2d edition, Clarendon Press, Oxford, 1957.
  • [9] Priest, G., ``Paraconsistent logic,'' pp. 287--393 in Handbook of Philosophical Logic, 2d edition, vol. 8, edited by D. M. Gabbay and F. Guenthner, Kluwer Academic Publishers, Dordrecht, 2001.
  • [10] Priest, G., and R. Sylvan, ``Simplified semantics for basic relevant logics,'' Journal of Philosophical Logic, vol. 21 (1992), pp. 217--32.
  • [11] Routley, R., and V. Routley, ``The semantics of first degree entailment,'' Noûs, vol. 6 (1972), pp. 335--59.
  • [12] Smullyan, R. M., First-Order Logic, Springer-Verlag, Berlin, 1968.
  • [13] Tamminga, A. M., ``Logics of rejection: two systems of natural deduction,'' Logique & Analyse, vol. 146 (1994), pp. 169--208.
  • [14] Tennant, N., Natural Logic, Edinburgh University Press, Edinburgh, 1978.
  • [15] Troelstra, A. S., and H. Schwichtenberg, Basic Proof Theory, Cambridge University Press, Cambridge, 1996.