Skip to main content Skip to main navigation menu Skip to site footer
  • Register
  • Login
  • Language
    • English
    • Język Polski
  • Menu
  • Home
  • Current
  • Archives
  • Online First Articles
  • About
    • About the Journal
    • Submissions
    • Editorial Team
    • Advisory Board
    • Peer Review Process
    • Logic and Logical Philosophy Committee
    • Open Access Policy
    • Privacy Statement
    • Contact
  • Register
  • Login
  • Language:
  • English
  • Język Polski

Logic and Logical Philosophy

Relevant generalization starts here (and here = 2)
  • Home
  • /
  • Relevant generalization starts here (and here = 2)
  1. Home /
  2. Archives /
  3. Vol. 19 No. 4 (2010) /
  4. Articles

Relevant generalization starts here (and here = 2)

Authors

  • Dmitry Zaitsev Lomonosov Moscow State University
  • Oleg Grigoriev Lomonosov Moscow State University

DOI:

https://doi.org/10.12775/LLP.2010.012

Keywords

relevant logic, first-degree entailment, generalization, information

Abstract

There is a productive and suggestive approach in philosophical logic based on the idea of generalized truth values. This idea, which stems essentially from the pioneering works by J.M. Dunn, N. Belnap, and which has recently been developed further by Y. Shramko and H. Wansing, is closely connected to the power-setting formation on the base of some initial truth values. Having a set of generalized truth values, one can introduce fundamental logical notions, more specifically, the ones of logical operations and logical entailment. This can be done in two different ways. According to the first one, advanced by M. Dunn, N. Belnap, Y. Shramko and H. Wansing, one defines on the given set of generalized truth values a specific ordering relation (or even several such relations) called the logical order(s), and then interprets logical connectives as well as the entailment relation(s) via this ordering(s). In particular, the negation connective is determined then by the inversion of the logical order. But there is also another method grounded on the notion of a quasi-field of sets, considered by Białynicki-Birula and Rasiowa. The key point of this approach consists in defining an operation of quasi-complement via the very specific function g and then interpreting entailment just through the relation of set-inclusion between generalized truth values.

In this paper, we will give a constructive proof of the claim that, for any finite set V with cardinality greater or equal 2, there exists a representation of a quasi-field of sets <P(V ), ∪, ∩, −> isomorphic to de Morgan lattice. In particular, it means that we offer a special procedure, which allows to make our negation de Morgan and our logic relevant.

Author Biographies

Dmitry Zaitsev, Lomonosov Moscow State University

Department of Logic

Oleg Grigoriev, Lomonosov Moscow State University

Department of Logic

References

Anderson, A.R., and N.D. Belnap, Entailment: The Logic of Relevance and Necessity, Vol. I, Princeton University Press, Princeton, NJ, 1975.

Belnap, N.D., “A useful four-valued logic”, pp. 8–37 in: J.M. Dunn and G. Epstein (eds.), Modern Uses of Multiple-Valued Logic, D. Reidel Publishing Company, Dordrecht, 1977.

Belnap, N.D., “How a computer should think”, pp. 30–55 in: G. Ryle (ed.), Contemporary Aspects of Philosophy, Oriel Press, 1977.

Białynicki-Birula, A., and H. Rasiowa, “On the representation of quasi-Boolean algebras”, Bulletin de l’Academie Polonaise des Sciences 5 (1957): 259–261.

Dunn, J.M., The Algebra of Intensional Logics, Doctoral Dissertation, University of Pittsburgh, Ann Arbor, 1966 (University Microfilms).

Dunn, J.M., “Intuitive semantics for first-degree entailment and coupled trees”, Philosophical Studies 29 (1976): 149–168.

Dunn, J.M., “Partiality and its dual”, Studia Logica 66 (2000): 5–40.

Shramko, Y., and H. Wansing, “Some useful sixteen-valued logics: How a computer network should think”, Journal of Philosophical Logic 34 (2005): 121–153.

Shramko, Y., and H. Wansing, “Hyper-contradictions, generalized truth values and logics of truth and falsehood”, Journal of Logic, Language and Information 15 (2006): 403–424.

Logic and Logical Philosophy

Downloads

  • PDF

Published

2010-12-30

How to Cite

1.
ZAITSEV, Dmitry and GRIGORIEV, Oleg. Relevant generalization starts here (and here = 2). Logic and Logical Philosophy. Online. 30 December 2010. Vol. 19, no. 4, pp. 329-340. [Accessed 5 July 2025]. DOI 10.12775/LLP.2010.012.
  • ISO 690
  • ACM
  • ACS
  • APA
  • ABNT
  • Chicago
  • Harvard
  • IEEE
  • MLA
  • Turabian
  • Vancouver
Download Citation
  • Endnote/Zotero/Mendeley (RIS)
  • BibTeX

Issue

Vol. 19 No. 4 (2010)

Section

Articles

Stats

Number of views and downloads: 584
Number of citations: 0

Crossref
Scopus
Google Scholar
Europe PMC

Search

Search

Browse

  • Browse Author Index
  • Issue archive

User

User

Current Issue

  • Atom logo
  • RSS2 logo
  • RSS1 logo

Information

  • For Readers
  • For Authors
  • For Librarians

Newsletter

Subscribe Unsubscribe

Language

  • English
  • Język Polski

Tags

Search using one of provided tags:

relevant logic, first-degree entailment, generalization, information
Up

Akademicka Platforma Czasopism

Najlepsze czasopisma naukowe i akademickie w jednym miejscu

apcz.umk.pl

Partners

  • Akademia Ignatianum w Krakowie
  • Akademickie Towarzystwo Andragogiczne
  • Fundacja Copernicus na rzecz Rozwoju Badań Naukowych
  • Instytut Historii im. Tadeusza Manteuffla Polskiej Akademii Nauk
  • Instytut Kultur Śródziemnomorskich i Orientalnych PAN
  • Instytut Tomistyczny
  • Karmelitański Instytut Duchowości w Krakowie
  • Ministerstwo Kultury i Dziedzictwa Narodowego
  • Państwowa Akademia Nauk Stosowanych w Krośnie
  • Państwowa Akademia Nauk Stosowanych we Włocławku
  • Państwowa Wyższa Szkoła Zawodowa im. Stanisława Pigonia w Krośnie
  • Polska Fundacja Przemysłu Kosmicznego
  • Polskie Towarzystwo Ekonomiczne
  • Polskie Towarzystwo Ludoznawcze
  • Towarzystwo Miłośników Torunia
  • Towarzystwo Naukowe w Toruniu
  • Uniwersytet im. Adama Mickiewicza w Poznaniu
  • Uniwersytet Komisji Edukacji Narodowej w Krakowie
  • Uniwersytet Mikołaja Kopernika
  • Uniwersytet w Białymstoku
  • Uniwersytet Warszawski
  • Wojewódzka Biblioteka Publiczna - Książnica Kopernikańska
  • Wyższe Seminarium Duchowne w Pelplinie / Wydawnictwo Diecezjalne „Bernardinum" w Pelplinie

© 2021- Nicolaus Copernicus University Accessibility statement Shop