Abstract
In this paper, we investigate an extended first-order Belnap-Dunn logic with classical negation. We introduce a Gentzen-type sequent calculus FBD+ for this logic and prove theorems for syntactically and semantically embedding FBD+ into a Gentzen-type sequent calculus for first-order classical logic. Moreover, we show the cut-elimination theorem for FBD+ and prove the completeness theorems with respect to both valuation and many-valued semantics for FBD+.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Another system which is equivalent to BD+ is PŁ4 of Méndez and Robles (cf. [11]).
- 2.
Belnap-Dunn logic with \(\triangle \) is equivalent to the expansion of Belnap-Dunn logic by what is sometimes called exclusion negation (cf. [5, p. 829]).
- 3.
By using the strong equivalence substitution property, we can show the Herbrand theorem for FBD+, although we omit the details due to space limitations.
- 4.
Another interesting property of BD+ is the maximality with respect to the set of theorems (but not with respect to the rules of inference) which is proved in [5, Sect. 3.3]. Maximality does not hold for Nelson logics (even for “classical” extensions) since there are extensions, obtained by adding some axioms, that are not classical logic.
- 5.
In FBD+, we can replace the multiplicative (context splitting) type inference rules (cut) and (\(\rightarrow \)left) with their additive (non context splitting) type modifications. But, we adopt the multiplicative type inference rules since these are compatible in the system LK (for the classical logic) presented in [17].
- 6.
Note that \(\varGamma \not \vdash _\mathrm{FBD+} \varPi \) is defined as \(\varGamma \not \vdash _\mathrm{FBD+} \alpha _1 \, \vee \dots \vee \alpha _n\) for some \(\alpha _1, \dots , \alpha _n\in \varPi \).
- 7.
For connexive logic in general, see [21].
References
Almukdad, A., Nelson, D.: Constructible falsity and inexact predicates. J. Symbol. Logic 49(1), 231–233 (1984)
Belnap, N.D.: A useful four-valued logic. In: Epstein, G., Dunn, J.M. (eds.) Modern Uses of Multiple-Valued Logic, pp. 5–37. Reidel, Dordrecht (1977)
Belnap, N.D.: How a computer should think. In: Ryle, G. (ed.) Contemporary Aspects of Philosophy, pp. 30–56. Oriel Press, Stocksfield (1977)
Béziau, J.Y.: A new four-valued approach to modal logic. Logique et Analyse 54(213), 109–121 (2011)
De, M., Omori, H.: Classical negation and expansions of Belnap-Dunn logic. Stud. Logica 103(4), 825–851 (2015)
Dunn, J.M.: Intuitive semantics for first-degree entailment and ‘coupled trees’. Philos. Stud. 29(3), 149–168 (1976)
Gentzen, G.: Collected papers of Gerhard Gentzen. In: Szabo, M.E. (ed.) Studies in Logic and the Foundations of Mathematics. North-Holland (English translation) (1969)
Gurevich, Y.: Intuitionistic logic with strong negation. Stud. Logica 36, 49–59 (1977)
Kamide, N.: Paraconsistent double negation that can simulate classical negation. In: Proceedings of the 46th IEEE International Symposium on Multiple-Valued Logic (ISMVL 2016), pp. 131–136 (2016)
Kamide, N., Shramko, Y.: Embedding from multilattice logic into classical logic and vice versa. J. Logic Comput. 27(5), 1549–1575 (2017)
Méndez, J.M., Robles, G.: A strong and rich 4-valued modal logic without Łukasiewicz-type paradoxes. Log. Univers. 9(4), 501–522 (2015)
Nelson, D.: Constructible falsity. J. Symbol. Logic 14, 16–26 (1949)
Omori, H.: From paraconsistent logic to dialetheic logic. In: Andreas, H., Verdée, P. (eds.) Logical Studies of Paraconsistent Reasoning in Science and Mathematics. TL, vol. 45, pp. 111–134. Springer, Cham (2016). doi:10.1007/978-3-319-40220-8_8
Omori, H., Sano, K.: Generalizing functional completeness in Belnap-Dunn logic. Stud. Logica 103(5), 883–917 (2015)
Rautenberg, W.: Klassische und nicht-klassische Aussagenlogik. Vieweg, Braunschweig (1979)
Sano, K., Omori, H.: An expansion of first-order Belnap-Dunn logic. Logic J. IGPL 22(3), 458–481 (2014)
Takeuti, G.: Proof Theory, 2nd edn. Dover Publications, Inc., Mineola (2013)
Vorob’ev, N.N.: A constructive propositional calculus with strong negation. Dokl. Akad. Nauk SSSR 85, 465–468 (1952). (in Russian)
Wansing, H.: The Logic of Information Structures. LNCS, vol. 681, 163 pages. Springer, Heidelberg (1993)
Wansing, H.: Informational interpretation of substructural propositional logics. J. Logic Lang. Inform. 2(4), 285–308 (1993)
Wansing, H.: Connexive logic, Stanford Encyclopedia of Philosophy (2014). http://plato.stanford.edu/entries/logic-connexive/
Zaitsev, D.: Generalized relevant logic and models of reasoning. Moscow State Lomonosov University doctoral dissertation (2012)
Acknowledgments
We would like to thank the anonymous referees for their valuable comments. Norihiro Kamide was partially supported by JSPS KAKENHI Grant Number JP26330263. Hitoshi Omori is a Postdoctoral Research Fellow of Japan Society for the Promotion of Science (JSPS), and was partially supported by JSPS KAKENHI Grant Number JP16K16684.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer-Verlag GmbH Germany
About this paper
Cite this paper
Kamide, N., Omori, H. (2017). An Extended First-Order Belnap-Dunn Logic with Classical Negation. In: Baltag, A., Seligman, J., Yamada, T. (eds) Logic, Rationality, and Interaction. LORI 2017. Lecture Notes in Computer Science(), vol 10455. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55665-8_6
Download citation
DOI: https://doi.org/10.1007/978-3-662-55665-8_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-55664-1
Online ISBN: 978-3-662-55665-8
eBook Packages: Computer ScienceComputer Science (R0)
