×

An inferentially many-valued two-dimensional notion of entailment. (English) Zbl 1423.03068

Summary: Starting from the notions of \(q\)-entailment and \(p\)-entailment, a two-dimensional notion of entailment is developed with respect to certain generalized \(q\)-matrices referred to as \(\mathsf B\)-matrices. After showing that every purely monotonic single-conclusion consequence relation is characterized by a class of \(\mathsf B\)-matrices with respect to \(q\)-entailment as well as with respect to \(p\)-entailment, it is observed that, as a result, every such consequence relation has an inferentially four-valued characterization. Next, the canonical form of \(\mathsf B\)-entailment, a two-dimensional multiple-conclusion notion of entailment based on \(\mathsf B\)-matrices, is introduced, providing a uniform framework for studying several different notions of entailment based on designation, antidesignation, and their complements. Moreover, the two-dimensional concept of a \(\mathsf B\)-consequence relation is defined, and an abstract characterization of such relations by classes of \(\mathsf B\)-matrices is obtained. Finally, a contribution to the study of inferential many-valuedness is made by generalizing Suszko’s Thesis and the corresponding reduction to show that any \(\mathsf B\)-consequence relation is, in general, inferentially four-valued.

MSC:

03B50 Many-valued logic
03B22 Abstract deductive systems
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] C. Blasio, Revisitando a l´ogica de Dunn-Belnap, Manuscrito 40 (2017), pp. 99-126.
[2] A. Bochman, Biconsequence relations: A general formalism of reasoning with inconsistency and incompleteness, Notre Dame Journal of Formal Logic 39 (1998), pp. 47-73. · Zbl 0967.03019
[3] C. Caleiro, W. Carnielli, M. Coniglio and J. Marcos, Suszko’s Thesis and dyadic semantics,Research Report. 1049-001 Lisbon,PT: CLC, Department of Mathematics, Instituto Superior T´ecnico, 2003. http://sqig.math.ist.utl.pt/pub/CaleiroC/03-CCCM-dyadic1.pdf.
[4] C. Caleiro, W. Carnielli, M. Coniglio and J. Marcos, Two’s company: “The humbug of many logical values”, [in:] J.-Y. B´eziau (ed.), Logica Universalis, Birkh¨auser, Basel, 2005, pp. 169-189. · Zbl 1076.03006
[5] C. Caleiro, J. Marcos and M. Volpe, Bivalent semantics, generalized compositionality and analytic classic-like tableaux for finite-valued logics, Theoretical Computer Science 603 (2015), pp. 84-110. · Zbl 1331.03024
[6] J. M. Dunn and G. M. Hardegree, Algebraic Methods in Philosophical Logic, Oxford Logic Guides, Vol. 41, Oxford Science Publications, Oxford, 2001. · Zbl 1014.03002
[7] S. Frankowski, Formalization of a plausible inference, Bulletin of the Section of Logic 33 (2004), pp. 41-52. · Zbl 1060.03027
[8] S. Frankowski, p-consequence versus q-consequence operations, Bulletin of the Section of Logic 33 (2004), pp. 41-52. · Zbl 1060.03027
[9] S. Frankowski, Plausible reasoning expressed by p-consequence, Bulletin of the Section of Logic 37 (2008), pp. 161-170. · Zbl 1286.03108
[10] R. French and D. Ripley, Valuations: bi, tri and tetra, Under submission (2017).
[11] L. Humberstone, Heterogeneous logic, Erkenntnis 29 (1988), pp. 395-435.
[12] T. Langholm, How different is partial logic?, [in:] P. Doherty (ed.), Partiality, Modality, and Nonmonotonicity, CSLI, Stanford, 1996, pp. 3-43. · Zbl 0904.03012
[13] G. Malinowski, q-consequence operation, Reports on Mathematical Logic 24 (1990), pp. 49-59. · Zbl 0759.03008
[14] G. Malinowski, Towards the concept of logical many-valuedness, Folia Philosophica 7 (1990), pp. 97-103.
[15] G. Malinowski, Many-Valued Logics, Oxford Logic Guides, Vol. 25, Clarendon Press, Oxford, 1993.
[16] G. Malinowski, Inferential many-valuedness, [in:]Jan Wole´nski (ed.), Philosophical Logic in Poland, Kluwer Academic Publishers, Dordrecht, 1994, pp. 75-84.
[17] G. Malinowski, Inferential paraconsistency, Logic and Logical Philosophy 8 (2001), pp. 83-89. · Zbl 1005.03031
[18] G. Malinowski, Inferential intensionality, Studia Logica 76 (2004), pp. 3-16. · Zbl 1045.03011
[19] G. Malinowski, That p + q = c(onsequence), Bulletin of the Section of Logic 36 (2007), pp. 7-19.
[20] G. Malinowski, Beyond three inferential values, Studia Logica 92 (2009), pp. 203-213. · Zbl 1182.03030
[21] G. Malinowski, Multiplying logical values, Logical Investigations 18 (2012), Moscow-St. Petersburg, pp. 292-308. · Zbl 1311.03049
[22] J. Marcos, What is a non-truth-functional logic, Studia Logica 92 (2009), pp. 215-240. · Zbl 1182.03031
[23] D. J. Shoesmith and T. J. Smiley, Multiple-Conclusion Logic, Cambridge University Press, 1978. · Zbl 0381.03001
[24] Y. Shramko and H. Wansing, Truth and Falsehood. An Inquiry into Generalized Logical Values, Trends in Logic, Vol. 36, Springer, Berlin, 2011. · Zbl 1251.03002
[25] R. Suszko, The Fregean axiom and Polish mathematical logic in the 1920’s, Studia Logica 36 (1977), pp. 373-380. · Zbl 0404.03004
[26] A. Urquhart, Basic many-valued logic, [in:] D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Vol. 2 (2nd edition), Kluwer, Dordrecht, 2001, pp. 249-295. · Zbl 1003.03523
[27] H. Wansing and Y. Shramko, Suszko’s Thesis, inferential many-valuedness, and the notion of a logical system, Studia Logica 88 (2008), pp. 405-429, 89 (2008), p. 147. · Zbl 1189.03026
[28] R. W´ojcicki, Some remarks on the consequence operation in sentential logics, Fundamenta Mathematicae 68 (1970), pp. 269-279. · Zbl 0206.27401
[29] R. W´ojcicki, Theory of Logical Calculi. Basic Theory of Consequence Operations, Kluwer, Dordrecht, 1988. · Zbl 0682.03001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.