Skip to main content Accessibility help
×
Home
Hostname: page-component-568f69f84b-r4dm2 Total loading time: 0.299 Render date: 2021-09-22T05:30:19.661Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

BELNAP–DUNN MODAL LOGICS: TRUTH CONSTANTS VS. TRUTH VALUES

Published online by Cambridge University Press:  22 February 2019

SERGEI P. ODINTSOV*
Affiliation:
Sobolev Institute of Mathematics
STANISLAV O. SPERANSKI*
Affiliation:
St. Petersburg State University
*Corresponding
*SOBOLEV INSTITUTE OF MATHEMATICS 4 KOPTYUG AVENUE 630090 NOVOSIBIRSK, RUSSIA E-mail: odintsov@math.nsc.ru
ST. PETERSBURG STATE UNIVERSITY 29B LINE 14TH, VASILYEVSKY ISLAND 199178 SAINT PETERSBURG, RUSSIA E-mail: katze.tail@gmail.com

Abstract

We shall be concerned with the modal logic BK—which is based on the Belnap–Dunn four-valued matrix, and can be viewed as being obtained from the least normal modal logic K by adding ‘strong negation’. Though all four values ‘truth’, ‘falsity’, ‘neither’ and ‘both’ are employed in its Kripke semantics, only the first two are expressible as terms. We show that expanding the original language of BK to include constants for ‘neither’ or/and ‘both’ leads to quite unexpected results. To be more precise, adding one of these constants has the effect of eliminating the respective value at the level of BK-extensions. In particular, if one adds both of these, then the corresponding lattice of extensions turns out to be isomorphic to that of ordinary normal modal logics.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arieli, O. & Avron, A. (1996). Reasoning with logical bilattices. Journal of Logic, Language and Information, 5(1), 2563.CrossRefGoogle Scholar
Arieli, O. & Avron, A. (1996). The value of the four values. Artificial Intelligence, 102(1), 97141.CrossRefGoogle Scholar
Avron, A. (1999). On the expressive power of three-valued and four-valued languages. Journal of Logic and Computation, 9(6), 977994.CrossRefGoogle Scholar
Busaniche, M. & Cignoli, M. (2009). Residuated lattices as an algebraic semantics for paraconsistent Nelson’s logic. Journal of Logic and Computation, 19(6), 10191029.CrossRefGoogle Scholar
Busaniche, M. & Cignoli, M. (2010). Constructive logic with strong negation as a substructural logic. Journal of Logic and Computation, 20(4), 761793.CrossRefGoogle Scholar
Galatos, N., Jipsen, P., Kowalski, T., & Ono, H. (2007). Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Amsterdam: Elsevier.Google Scholar
Jung, A. & Rivieccio, U. (2013). Kripke semantics for modal bilattice logic. Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science. Piscataway, NJ: IEEE, pp. 438447.CrossRefGoogle Scholar
Kracht, M. (1999). Tools and Techniques in Modal Logic. Amsterdam: Elsevier.Google Scholar
Odintsov, S. P. (2008). Constructive Negations and Paraconsistency. Dordrecht: Springer.CrossRefGoogle Scholar
Odintsov, S. P. (2014). On the equivalence of paraconsistent and explosive versions of Nelson logic. In Brattka, V., Diener, H., and Spreen, D., editors. Logic, Computation, Hierarchies. Berlin: De Gruyter, pp. 259272.Google Scholar
Odintsov, S. P. (2015). Belnap constants and Nelson logic. In Koslow, A. and Buchsbaum, A., editors. The Road to Universal Logic. Cham: Birkhäuser/Springer, pp. 521538.CrossRefGoogle Scholar
Odintsov, S. P. & Latkin, E. I. (2012). BK-lattices. Algebraic semantics for Belnapian modal logics. Studia Logica, 100(1–2), 319338.CrossRefGoogle Scholar
Odintsov, S. P. & Speranski, S. O. (2016). The lattice of Belnapian modal logics: Special extensions and counterparts. Logic and Logical Philosophy, 25(1), 333.Google Scholar
Odintsov, S. P. & Wansing, H. (2010). Modal logics with Belnapian truth values. Journal of Applied Non-Classical Logics, 20(3), 279301.CrossRefGoogle Scholar
Odintsov, S. P. & Wansing, H. (2017). Disentangling FDE-based paraconsistent modal logics. Studia Logica, 105(6), 12211254.CrossRefGoogle Scholar
Omori, H. (2016). From paraconsistent logic to dialetheic logic. In Andreas, H. and Verdée, P., editors. Logical Studies of Paraconsistent Reasoning in Science and Mathematics. Cham: Springer, pp. 111134.CrossRefGoogle Scholar
Rivieccio, U. (2011). Paraconsistent modal logics. Electronic Notes in Theoretical Computer Science, 278, 173186.CrossRefGoogle Scholar
Speranski, S. O. (2013). On Belnapian modal algebras: Representations, homomorphisms, congruences, and so on. Siberian Electronic Mathematical Reports, 10, 517534.Google Scholar
Spinks, M. & Veroff, R. (2008). Constructive logic with strong negation is a substructural logic. I. Studia Logica, 88(3), 325348.CrossRefGoogle Scholar
Vakarelov, D. (1977). Notes on ${\cal N}$-lattices and constructive logic with strong negation. Studia Logica 36(1–2), 109125.CrossRefGoogle Scholar
Wansing, H. (2016). Connexive logic. In Zalta, E. N., editor. Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/archives/spr2016/entries/logic-connexive/.Google Scholar
3
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

BELNAP–DUNN MODAL LOGICS: TRUTH CONSTANTS VS. TRUTH VALUES
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

BELNAP–DUNN MODAL LOGICS: TRUTH CONSTANTS VS. TRUTH VALUES
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

BELNAP–DUNN MODAL LOGICS: TRUTH CONSTANTS VS. TRUTH VALUES
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *