Journal of Philosophical Logic

, Volume 43, Issue 2–3, pp 303–332

# Dual Equivalent Two-valued Under-determined and Over-determined Interpretations for Łukasiewicz’s 3-valued Logic Ł3

Article

## Abstract

Łukasiewicz three-valued logic Ł3 is often understood as the set of all 3-valued valid formulas according to Łukasiewicz’s 3-valued matrices. Following Wojcicki, in addition, we shall consider two alternative interpretations of Ł3: “well-determined” Ł3a and “truth-preserving” Ł3b defined by two different consequence relations on the 3-valued matrices. The aim of this paper is to provide (by using Dunn semantics) dual equivalent two-valued under-determined and over-determined interpretations for Ł3, Ł3a and Ł3b. The logic Ł3 is axiomatized as an extension of Routley and Meyer’s basic positive logic following Brady’s strategy for axiomatizing many-valued logics by employing two-valued under-determined or over-determined interpretations. Finally, it is proved that “well determined” Łukasiewicz logics are paraconsistent.

## Keywords

Many-valued logic Łukasiewicz 3-valued logic Two-valued under-determined and over-determined interpretations Paraconsistent logics

## References

1. 1.
Avron, A. (1991). Natural 3-valued logics. characterization and proof theory. Journal of Symbolic Logic, 56, 276–294.
2. 2.
Baaz, M., & Zach, R. (1998). Compact propositional Gӧdel logics. In 28th international symposium on multiple-valued logic proceedings, (pp. 108–113). Fukuoka, Japan.Google Scholar
3. 3.
Baaz, M., Preining, N., Zach, R. (2007). First-order Gödel logics. Annals of Pure and Applied Logic, 147, 23–47.
4. 4.
Bimbó, K., & Dunn, J.M. (2008). Generalized Galois logics. Relational semantics of non-classical logic calculi. Stanford, CA: CSLI Publications.Google Scholar
5. 5.
Brady, R. (1982). Completeness proofs for the systems RM3 and BN4. Logique et Analyse, 25, 9–32.Google Scholar
6. 6.
Brady, R. (Ed.) (2003). Relevant logics and their rivals, Vol. II. Ashgate.Google Scholar
7. 7.
Carnielli, W., Coniglio, M., Marcos, J. (2007). Logics of formal inconsistency In D. Gabbay, & F. Guenthner (Eds.), Handbook of philosophical logic (Vol. 14, pp. 1–93). Dordrecht: Kluwer Academic Publishers.
8. 8.
Dunn, J.M. (1966). The algebra of intensional logics. Doctoral dissertation, University of Pittsburg (Ann Arbor, University Microfilms).Google Scholar
9. 9.
Dunn, J.M. (1976). Intuitive semantics for first-degree entailments and ‘coupled trees’. Philosophical Studies, 29, 149–168.
10. 10.
Dunn, J.M. (2000). Partiality and its dual. Studia Logica, 65, 5–40.
11. 11.
Gabbay, D., & Guenthner, F. (2001). Handbook of philosophical logic, 2nd edn. Dordrecht: Kluwer Academic Publishers.Google Scholar
12. 12.
González, C. (2012). MaTest. Available at http://ceguel.es/matest. Last access 26 November 2012.
13. 13.
Hähnle, R. (2002). Advanced many-valued logics. In D. Gabbay, & F. Guenthner (Eds.), Handbook of philosophical logic (Vol. 2, pp. 297–395). Dordrecht: Kluwer Academic Publishers.Google Scholar
14. 14.
Karpenko, A.S. (2006). Łukasiewicz logics and prime numbers. Beckington: Luniver Press.Google Scholar
15. 15.
Łukasiewicz, J. (1920). On three-valued logic. In J. Łukasiewicz (1970) (pp. 87–88).Google Scholar
16. 16.
Łukasiewicz, J. (1970). Selected works. Amsterdam: North-Holland.Google Scholar
17. 17.
Łukasiewicz, J., & Tarski, A. (1930). Investigations into the sentential calculus. In Łukasiewicz (1970) (pp. 131–152).Google Scholar
18. 18.
Malinowski, G. (1993). Many-valued logics. Oxford: Clarendon Press.Google Scholar
19. 19.
McCall, S. (1967). Polish logic: 1920–1939. London, UK: Oxford University Press.Google Scholar
20. 20.
Meyer, R.K., & Routley, R. (1972). Algebraic analysis of entailment I. Logique et Analyse, 15, 407–428.Google Scholar
21. 21.
Priest, G. (2002). Paraconsistent logic. In D. Gabbay, & F. Guenthner (Eds.), Handbook of philosophical logic (Vol. 6, pp. 287–393). Dordrecht: Kluwer Academic Publishers.
22. 22.
Robles, G., & Méndez, J.M. (2008). The basic constructive logic for a weak sense of consistency. Journal of Logic Language and Information, 17(1), 89–107.
23. 23.
Robles, G., & Méndez, J.M. A Routley–Meyer semantics for truth-preserving and well-determined Łukasiewicz’s 3-valued logics (Manuscript).Google Scholar
24. 24.
Rose, A., & Rosser, J.B. (1958). Fragments of many-valued statement calculi. Transactions of the American Mathematical Society, 87, 1–53.
25. 25.
Routley, R., & Meyer, R.K. (1972). Semantics of entailment III. Journal of Philosophical Logic, 1, 192–208.
26. 26.
Routley, R., & Routley, V. (1972). Semantics of first-degree entailment. Noûs, 1, 335–359.
27. 27.
Routley, R., Meyer, R.K., Plumwood, V., Brady R.T. (1982). Relevant logics and their rivals (Vol. 1). Atascadero, CA: Ridgeview Publishing Co.Google Scholar
28. 28.
Slaney, J. (1995). MaGIC, matrix generator for implication connectives: version 2.1, notes and guide. Canberra: Australian National University. URL: http://users.rsise.anu.edu.au/~jks.
29. 29.
Suszko, R. (1975). Remarks on Łukasiewicz’s three-valued logics. Bulletin of the Section of Logic, 4, 87–90.Google Scholar
30. 30.
Tokarz, M. (1974). A method of axiomatization of Łukasiewicz logics. Studia Logica, 33, 333–338.
31. 31.
Tsuji, M. (1998). Many-valued logic and Suszko thesis revisited. Studia Logica, 60, 299–309.
32. 32.
Tuziak, R. (1988). An axiomatization of the finite-valued Łukasiewicz calculus. Studia Logica, 47, 49–55.
33. 33.
Urquhart, A. (2002). Basic many-valued logic. In D. Gabbay, & F. Guenthner (Eds.), Handbook of philosophical logic (Vol. 2, pp. 249–295). Dordrecht: Kluwer Academic Publishers.Google Scholar
34. 34.
Van Fraasen, B. (1969). Facts and tautological entailments. The Journal of Philosophy, 67, 477–487.
35. 35.
Wajsberg, M. (1931). Axiomatization of the 3-valued propositional calculus. In S. McCall (1967) (pp. 264–284)..Google Scholar
36. 36.
Wojcicki, R. (1974). The logics stronger than Łukasiewicz’z three-valued sentential calculus—the notion of degree of maximality versus the notion of degree of completeness. Studia Logica, 33, 201–214.
37. 37.
Wojcicki, R. (1984). Lectures on Propositional Calculi. Ossolineum.Google Scholar