Skip to main content Accessibility help
×
Home
Hostname: page-component-59b7f5684b-npccv Total loading time: 0.417 Render date: 2022-10-01T23:40:25.538Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": false, "useSa": true } hasContentIssue true

Substructural fuzzy logics

Published online by Cambridge University Press:  12 March 2014

George Metcalfe
Affiliation:
Department of Mathematics, 1326 Stevenson Center, Vanderbilt University, Nashville TN 37240, USA. E-mail: metcalfe@logic.at
Franco Montagna
Affiliation:
Department of Mathematics, University of Siena, Via Del Capitano 15, 53100 Siena, Italy. E-mail: montagna@unisi.it

Abstract

Substructural fuzzy logics are substructural logics that are complete with respect to algebras whose lattice reduct is the real unit interval [0, 1]. In this paper, we introduce Uninorm logic UL as Multiplicative additive intuitionistic linear logic MAILL extended with the prelinearity axiom ((A → B) ∧ t) V ((B → A)∧ t). Axiomatic extensions of UL include known fuzzy logics such as Monoidal t-norm logic MIX and Gödel logic G, and new weakening-free logics. Algebraic semantics for these logics are provided by subvarieties of (representable) pointed bounded commutative residuated lattices. Gentzen systems admitting cut-elimination are given in the framework of hypersequents. Completeness with respect to algebras with lattice reduct [0, 1] is established for UL and several extensions using a two-part strategy. First, completeness is proved for the logic extended with Takeuti and Titani's density rule. A syntactic elimination of the rule is then given using a hypersequent calculus. As an algebraic corollary, it follows that certain varieties of residuated lattices are generated by their members with lattice reduct [0, 1].

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2007

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

[1] Avron, A., A constructive analysis of RM, this Journal, vol. 52 (1987), no. 4, pp. 939–951.Google Scholar
[2] Avron, A., Hypersequents. logical consequence and intermediate logics for concurrency, Annals of Mathematics and Artificial Intelligence, vol. 4 (1991), no. 3–4, pp. 225–248.CrossRefGoogle Scholar
[3] Baaz, M., Ciabattoni, A., and Montagna, F., Analytic calculi for monoidal t-norm based logic, Fundamenta Informaticae, vol. 59 (2004), no. 4, pp. 315–332.Google Scholar
[4] Baaz, M. and Zach, R., Hypersequents and the proof theory of intuitionistic fuzzy logic, Proceedings of CSL 2000, LNCS, Springer-Verlag, 2000, pp. 187–201.Google Scholar
[5] Ciabattoni, A., Esteva, F., and Godo, L., T-norm based logics with n-contraction, Neural Network World, vol. 12 (2002), no. 5, pp. 441–453.Google Scholar
[6] Cintula, P., Weakly implicative (fuzzy) logics I: Basic properties, Archive for Mathematical Logic, vol.45 (2006), pp. 673–704.CrossRefGoogle Scholar
[7] De Baets, B., Idempotent uninorms, European Journal of Operational Research, vol. 118 (1999), pp. 631–642.CrossRefGoogle Scholar
[8] Esteva, F., Gispert, J., Godo, L., and Montagna, F., On the standard and rational completeness of some axiomatic extensions of the monoidal t-norm logic, Studio Logica, vol. 71 (2002), no. 2, pp. 199–226.CrossRefGoogle Scholar
[9] Esteva, F. and Godo, L., Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy Sets and Systems, vol. 124 (2001), pp. 271–288.CrossRefGoogle Scholar
[10] Gabbay, D. and Metcalfe, G., Logics based on [0, 1)-continuous uninorms, Archive for Mathematical Logic, vol. 46 (2007), pp. 425–449.CrossRefGoogle Scholar
[11] Hájek, P., Metamathematics of fuzzy logic, Kluwer, Dordrecht, 1998.CrossRefGoogle Scholar
[12] Jenei, S. and Montagna, F., A proof of standard completeness for Esteva and Godo's MTL logic, Studia Logica, vol. 70 (2002), no. 2, pp. 183–192.CrossRefGoogle Scholar
[13] Metcalfe, G., Olivetti, N., and Gabbay, D., Analytic proof calculi for product logics, Archive for Mathematical Logic, vol. 43 (2004), no. 7, pp. 859–889.CrossRefGoogle Scholar
[14] Metcalfe, G., Sequent and hypersequent calculi for abelian and Łukasiewicz logics, ACM Transactions on Computational Logic, vol. 6 (2005), no. 3, pp. 578–613.CrossRefGoogle Scholar
[15] Ono, H. and Komori, Y., Logics without the contraction rule, this Journal, vol. 50 (1985), pp. 169–201.Google Scholar
[16] Pottinger, G., Uniform, cut-free formulations of T, S4 and S5 (abstract), this Journal, vol. 48 (1983), no. 3, p. 900.Google Scholar
[17] Restall, G., An introduction to substructural logics, Routledge, London, 1999.Google Scholar
[18] Takeuti, G. and Titani, T., Intuitionistic fuzzy logic and intuitionistic fuzzy set theory, this Journal, vol. 49 (1984), no. 3, pp. 851–866.Google Scholar
[19] Tsinakis, C. and Blount, K., The structure of residuated lattices, International Journal of Algebra and Computation, vol. 13 (2003), no, 4, pp. 437–461.Google Scholar
[20] Yager, R. and Rybalov, A., Uninorm aggregation operators. Fuzzy Sets and Systems, vol. 80 (1996), pp. 111–120.CrossRefGoogle Scholar
99
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@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 saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved 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.

Substructural fuzzy logics
Available formats
×

Save article to Dropbox

To save 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 used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Substructural fuzzy logics
Available formats
×

Save article to Google Drive

To save 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 used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Substructural fuzzy logics
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? *