Studia Logica

, Volume 103, Issue 3, pp 545–558

From Bi-facial Truth to Bi-facial Proofs

Article

Abstract

In their recent paper Bi-facial truth: a case for generalized truth values Zaitsev and Shramko [7] distinguish between an ontological and an epistemic interpretation of classical truth values. By taking the Cartesian product of the two disjoint sets of values thus obtained, they arrive at four generalized truth values and consider two “semi-classical negations” on them. The resulting semantics is used to define three novel logics which are closely related to Belnap’s well-known four valued logic. A syntactic characterization of these logics is left for further work. In this paper, based on our previous work on a functionally complete extension of Belnap’s logic, we present a sound and complete tableau calculus for these logics. It crucially exploits the Cartesian nature of the four values, which is reflected in the fact that each proof consists of two tableaux. The bi-facial notion of truth of Z&S is thus augmented with a bi-facial notion of proof. We also provide translations between the logics for semi-classical negation and classical logic and show that an argument is valid in a logic for semi-classical negation just in case its translation is valid in classical logic.

Keywords

Four-valued logic Bifacial logic Analytic tableaux

Preview

References

1. 1.
Anderson A.R., Belnap N.D.: Entailment: the Logic of Relevance and Necessity, vol. I. Princeton University Press, Princeton (1975)Google Scholar
2. 2.
Baaz M., Fermüller C.G., Salzer G.: Automated deduction for many-valued logics. In: Robinson, A., Voronkov, A. (eds.), Handbook of Automated Reasoning, pp. 1355–1402. Elsevier Science Publishers, Amsterdam (2000)Google Scholar
3. 3.
Belnap N.D.: A useful four-valued logic. In: Dunn, J.M., Epstein, G. (eds.), Modern Uses of Multiple-Valued Logic., pp. 8–37. Reidel, Dordrecht (1977)Google Scholar
4. 4.
Muskens R.A.: Meaning and Partiality. CSLI, Stanford (1995)Google Scholar
5. 5.
Smullyan R.M.: First-Order Logic. Springer, Berlin (1968)
6. 6.
Wintein S., Muskens R.A.: A calculus for Belnap’s logic in which each proof consists of two trees. Logique & Analyse 220, 643–656 (2012)Google Scholar
7. 7.
Zaitsev D., Shramko Y.: Bi-facial truth: a case for generalized truth values. Studia Logica 101, 1299–1318 (2013)