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Logic and Logical Philosophy

A simple Henkin-style completeness proof for Gödel 3-valued logic G3
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A simple Henkin-style completeness proof for Gödel 3-valued logic G3

Authors

  • Gemma Robles Universidad de León

DOI:

https://doi.org/10.12775/LLP.2014.001

Keywords

many-valued logic, Gödel 3-valued logic, bivalent under-determined and over-determined semantics

Abstract

A simple Henkin-style completeness proof for Gödel 3-valued propositional logic G3 is provided. The idea is to endow G3 with an under-determined semantics (u-semantics) of the type defined by Dunn. The key concept in u-semantics is that of “under-determined interpretation” (u-interpretation). It is shown that consistent prime theories built upon G3 can be understood as (canonical) u-interpretations. In order to prove this fact we follow Brady by defining G3 as an extension of Anderson and Belnap’s positive fragment of First Degree Entailment Logic.

Author Biography

Gemma Robles, Universidad de León

Dpto. de Psicología, Sociología y Filosofía

References

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

Baaz, M., N. Preining, and R. Zach, “First-Order Gödel Logics”, Annals of Pure and Applied Logic, 147 (2007): 23–47. DOI: 10.1016/j.apal.2007.03.001

Brady, R., “Completeness Proofs for the Systems RM3 and BN4”, Logique et Analyse, 25 (1982): 9–32.

Dunn, J.M., “The algebra of intensional logics” (1966). Doctoral dissertation, University of Pittsburgh (Ann Arbor, University Microfilms).

Dunn, J.M., “Intuitive semantics for first-degree entailments and ‘coupled trees’”, Philosophical Studies, 29 (1976): 149–168. DOI: 10.1007/BF00373152

Dunn, J.M. “A Kripke-style semantics for R-Mingle using a binary accessibility relation”, Studia Logica, 35 (1976): 163–172. DOI: 10.1007/BF02120878

Dunn, J.M., “Partiality and its dual”, Studia Logica, 66 (2000), 5–40. DOI: 10.1023/A:1026740726955

Dunn, J.M., and R.K. Meyer, “Algebraic completeness results for Dummett’s LC and its extensions”, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 17 (1971), 225–230. DOI: 10.1002/malq.19710170126

Gödel, K., “Zum intuitionistischen Aussagenkalkül”, Anzeiger Akademie der Wissenschaffen Wien, Math.-Naturwissensch, Klasse, 69 (1933): 65–66.

González, C., “MaTest” (2012), available at Link (Last access 10/10/2013).

Łukasiewicz, J., “Die Logik und das Grundlagenproblem”, Les Entretiens de Zürich sur les Fondaments et la Méthode des Sciences Mathématiques, 6–9 (1938), 12: 82–100.

Robles. G., “A Routley-Meyer semantics for Gödel 3-valued logic and its paraconsistent counterpart”, Logica Universalis (forthcoming). DOI: 10.1007/s11787-013-0088-7

Robles, G., and J.M. Méndez,“A paraconsistent 3-valued logic related to Gödel logic G3”(manuscript).

Robles, G., F. Salto, and J.M. Méndez, “Dual equivalent two-valued under-determined and over-determined interpretations for Łukasiewicz’s 3-valued Logic Ł3”, Journal of Philosophical Logic (2013). DOI: 10.1007/s10992-012-9264-0

Routley, R., V. Routley, “Semantics of first-degree entailment”, Noûs, 1(1972): 335–359. DOI: 10.2307/2214309

Routley, R., R.K. Meyer, V. Plumwood, and R.T. Brady, Relevant Logics and their Rivals, vol. 1, Atascadero, CA: Ridgeview Publishing Co., 1982.

Slaney, J., MaGIC, Matrix Generator for Implication Connectives: Version 2.1, Notes and Guide, Canberra: Australian National University, 1995. Link

Van Fraasen, B., “Facts and tautological entailments”, The Journal of Philosophy, 67 (1969): 477–487. DOI: 10.2307/2024563

Yang, E., “(Star-based) three-valued Kripke-style semantics for pseudo-and weak-Boolean logics”, Logic Journal of the IGPL, 20 (2012): 187–206. DOI: 10.1093/jigpal/jzr030

Logic and Logical Philosophy

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Published

2014-01-07

How to Cite

1.
ROBLES, Gemma. A simple Henkin-style completeness proof for Gödel 3-valued logic G3. Logic and Logical Philosophy [online]. 7 January 2014, T. 23, nr 4, s. 371–390. [accessed 21.3.2023]. DOI 10.12775/LLP.2014.001.
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Vol. 23 No. 4 (2014): December

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