Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
A Logical Framework for Graded Predicates Read chapter Read first chapter

2017 | OriginalPaper | Chapter

An Extended First-Order Belnap-Dunn Logic with Classical Negation

Authors : Norihiro Kamide, Hitoshi Omori

Published in: Logic, Rationality, and Interaction

Publisher: Springer Berlin Heidelberg

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

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+.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Multi-Path vs. Single-Path Replies to Skepticism
next chapter A Characterization Theorem for Trackable Updates
Footnotes
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].
 
Literature
1.
2.
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)
3.
Belnap, N.D.: How a computer should think. In: Ryle, G. (ed.) Contemporary Aspects of Philosophy, pp. 30–56. Oriel Press, Stocksfield (1977)
4.
Béziau, J.Y.: A new four-valued approach to modal logic. Logique et Analyse 54(213), 109–121 (2011)MathSciNetMATH
5.
6.
Dunn, J.M.: Intuitive semantics for first-degree entailment and ‘coupled trees’. Philos. Stud. 29(3), 149–168 (1976)MathSciNetCrossRef
7.
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)
8.
9.
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)
10.
Kamide, N., Shramko, Y.: Embedding from multilattice logic into classical logic and vice versa. J. Logic Comput. 27(5), 1549–1575 (2017)
11.
Méndez, J.M., Robles, G.: A strong and rich 4-valued modal logic without Łukasiewicz-type paradoxes. Log. Univers. 9(4), 501–522 (2015)MathSciNetCrossRefMATH
12.
13.
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 CrossRef
14.
15.
16.
17.
Takeuti, G.: Proof Theory, 2nd edn. Dover Publications, Inc., Mineola (2013)MATH
18.
Vorob’ev, N.N.: A constructive propositional calculus with strong negation. Dokl. Akad. Nauk SSSR 85, 465–468 (1952). (in Russian)
19.
Wansing, H.: The Logic of Information Structures. LNCS, vol. 681, 163 pages. Springer, Heidelberg (1993)
20.
21.
22.
Zaitsev, D.: Generalized relevant logic and models of reasoning. Moscow State Lomonosov University doctoral dissertation (2012)
Metadata
Title
An Extended First-Order Belnap-Dunn Logic with Classical Negation
Authors
Norihiro Kamide
Hitoshi Omori
Copyright Year
2017
Publisher
Springer Berlin Heidelberg
Book
Logic, Rationality, and Interaction

Print ISBN: 978-3-662-55664-1

Electronic ISBN: 978-3-662-55665-8

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-3-662-55665-8_6

Premium Partner

    Image Credits