An Epimorphism between Fine and Ferguson’s Matrices for Angell’s AC
DOI:
https://doi.org/10.12775/LLP.2022.025Keywords
analytic containment, many-valued logic, matrix congruence, tableau calculus, computational algebraAbstract
Angell’s logic of analytic containment AC has been shown to be characterized by a 9-valued matrix NC by Ferguson, and by a 16-valued matrix by Fine. It is shown that the former is the image of a surjective homomorphism from the latter, i.e., an epimorphic image. Some candidate 7-valued matrices are ruled out as characteristic of AC. Whether matrices with fewer than 9 values exist remains an open question. The results were obtained with the help of the MUltlog system for investigating finite-valued logics; the results serve as an example of the usefulness of techniques from computational algebra in logic. A tableau proof system for NC is also provided.
.
References
Angell, Richard Bradshaw, 1977, “Three systems of first degree entailment (abstract)”, The Journal of Symbolic Logic 42 (1): 147. DOI: http://dx.doi.org/10.2307/2272332
Angell, Richard Bradshaw, 1989, “Deducibility, entailment and analytic containment”, pages 119–143 in J. Norman and R. Sylvan (eds.), Directions in Relevant Logic, Dordrecht: Kluwer. DOI: http://dx.doi.org/10.1007/978-94-009-1005-8_8
Baaz, Matthias, Christian G. Fermüller, and Richard Zach, 1993, “Elimination of cuts in first-order finite-valued logics”, Journal of Information Processing and Cybernetics EIK 29 (6): 333–355. DOI: http://dx.doi.org/10.11575/PRISM/38801
Ferguson, Thomas Macaulay, 2016, “Faulty Belnap computers and subsystems of FDE”, Journal of Logic and Computation 26 (5): 1617–1636. DOI: http://dx.doi.org/10.1093/logcom/exu048
Ferguson, Thomas Macaulay, 2021, “Tableaux and restricted quantification for systems related to weak Kleene logic”, pages 3–19 in A. Das and S. Negri (eds.), Automated Reasoning with Analytic Tableaux and Related Methods, no. 12842 in Lecture Notes in Computer Science, Cham: Springer. DOI: http://dx.doi.org/10.1007/978-3-030-86059-2_1
Fine, Kit, 2016, “Angellic content”, Journal of Philosophical Logic 45 (2): 199–226. DOI: http://dx.doi.org/10.1007/s10992-015-9371-9
Freese, Ralph, 2008, “Computing congruences efficiently”, Algebra Universalis 59 (3): 337–343. DOI: http://dx.doi.org/10.1007/s00012-008-2073-1
Grätzer, George, 1968, Universal Algebra, Princeton, NJ: van Nostrand.
Hähnle, Reiner, 1993, Automated Deduction in Multiple-Valued Logics, Oxford: Oxford University Press.
Salzer, Gernot, 1996, “MUltlog: An expert system for multiple-valued logics”, pages 50–55 in Collegium Logicum, Collegium Logicum, Vienna: Springer. DOI: http://dx.doi.org/10.1007/978-3-7091-9461-4_3
Salzer, Gernot, 2000, “Optimal axiomatizations of finitely valued logics”, Information and Computation 162 (1–2): 185–205. DOI: http://dx.doi.org/10.1006/inco.1999.2862
Salzer, Gernot et al., 2022, MUltlog v1.16a. Software program. https://github.com/rzach/multlog. DOI: http://dx.doi.org/10.5281/zenodo.6893267
Wójcicki, Ryszard, 1988, Theory of Logical Calculi: Basic Theory of Consequence Operations, No. 147 in Synthese Library, Dordrecht: Kluwer.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 Richard Zach
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 1346
Number of citations: 0
