Advertisement

Synthese

pp 1–23 | Cite as

Two traditions in abstract valuational model theory

  • Rohan FrenchEmail author
  • David Ripley
S.I.: Varieties of Entailment
  • 27 Downloads

Abstract

We investigate two different broad traditions in the abstract valuational model theory for nontransitive and nonreflexive logics. The first of these traditions makes heavy use of the natural Galois connection between sets of (many-valued) valuations and sets of arguments. The other, originating with work by Grzegorz Malinowski on nonreflexive logics, and best systematized in Blasio et al. (Bull Sect Log 46(3/4): 233–262, 2017), lets sets of arguments determine a more restricted set of valuations. After giving a systematic discussion of these two different traditions in the valuational model theory for substructural logics, we turn to looking at the ways in which we might try to compare two sets of valuations determining the same set of arguments.

Keywords

Valuational model theory p-consequence q-consequence Many-valued logic 

Notes

Acknowledgements

We would like to thank the audience at the 2018 Australasian Association for Logic meeting, and two anonymous referees for their helpful discussion and comments on this material. David Ripley’s contribution was partially supported by the project “Logic and Substructurality”, grant number FFI2017-84805-P, Ministerio de Economía, Industria y Competitividad, Government of Spain.

References

  1. Bimbó, K., & Dunn, J. M. (2008). Generalized Galois logics: Relational semantics of nonclassical logical calculi. Number 188 in CSLI lecture notes. Stanford: CSLI Publications.Google Scholar
  2. Birkhoff, G. (1967). Lattice theory (3rd ed.). New York: American Mathematical Society.Google Scholar
  3. Blasio, C., Marcos, J., & Wansing, H. (2017). An inferentially many-valued two-dimensional notion of entailment. Bulletin of the Section of Logic, 46(3/4), 233–262.Google Scholar
  4. Brink, C. (1993). Power structures. Algebra Uniersalis, 30, 177–216.CrossRefGoogle Scholar
  5. Carnap, R. (1943). Formalization of logic. Cambridge, Massachusetts: Harvard University Press.Google Scholar
  6. Davey, B. A., & Priestley, H. A. (2002). Introduction to lattices and order. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  7. Dunn, J. M. (1991). Gaggle theory: An abstraction of Galois connections and residuation with applications to negations and various logical operators. In Logics in AI, Proceedings of European workshop JELIA 1990 (pp. 31–51). Berlin: LNCS.Google Scholar
  8. Dunn, J. M., & Hardegree, G. M. (2001). Algebraic methods in philosophical logic. Oxford: Oxford University Press.Google Scholar
  9. Erné, M., Koslowski, J., Melton, A., & Strecker, G. (1993). A primer on Galois connections. Annals of the New York Academy of Sciences, 704(1), 103–125.CrossRefGoogle Scholar
  10. Frankowski, S. (2004). Formalization of a plausible inference. Bulletin of the Section of Logic, 33, 41–52.Google Scholar
  11. Frankowski, S. (2008). Plausible reasoning expressed by p-consequence. Bulletin of the Section of Logic, 37(3–4), 161–170.Google Scholar
  12. French, R., & Ripley, D. (2019). Valuations: Bi, tri, and tetra. Studia Logica, 107(6), 1313–1346. CrossRefGoogle Scholar
  13. Hardegree, G. M. (2005). Completeness and super-valuations. Journal of Philosophical Logic, 34(1), 81–95.CrossRefGoogle Scholar
  14. Humberstone, L. (1988). Heterogeneous logic. Erkenntnis, 29, 395–435.CrossRefGoogle Scholar
  15. Humberstone, L. (2012). The connectives. Cambridge, Massachusetts: MIT Press.Google Scholar
  16. Malinowski, G. (1990). Q-consequence operation. Reports on Mathematical Logic, 24, 49–59.Google Scholar
  17. Ore, O. (1944). Galois connexions. Transactions of the American Mathematical Society, 55, 493–513.CrossRefGoogle Scholar
  18. Ripley, D. (2018). Blurring: An approach to conflation. Notre Dame Journal of Formal Logic, 59(2), 171–188.CrossRefGoogle Scholar
  19. Scott, D. (1974). Completeness and axiomatizability in many-valued logic. In L. Henkin (Ed.), Proceedings of the Tarski symposium (pp. 411–436). Providence: American Mathematical Society.CrossRefGoogle Scholar
  20. Shoesmith, D. J., & Smiley, T. J. (1978). Multiple-conclusion logic. Cambridge: Cambridge University Press.CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of PhilosophyUniversity of California, DavisDavisUSA
  2. 2.Philosophy Department, SoPHISBuilding 11, Monash UniversityVICAustralia

Personalised recommendations