Abstract
Fine (J Philos Log 43:549–577, 2014) developed a truthmaker semantics for intuitionistic logic, which is also called exact semantics, since it is based on a relation of exact verification between states and formulas. A natural question arises as to what are the limits of Fine’s approach and whether an exact semantics of similar kind can be constructed for other important non-classical logics. In our paper, we will generalize Fine’s approach and develop an exact semantics for some substructural logics. In particular, we will provide a truthmaker semantics for the Non-associative Lambek calculus and some of its extensions. This generalization will reveal some interesting connections between Fine’s recent work on truthmaker semantics and his early work on relevant logic.
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References
Anglberger, A., Korbmacher, J., & Faroldi, F. (2016). An exact truthmaker semantics for permission and obligation. In O. Roy, A. Tamminga, & M. Willer (Eds.), Deontic logic and normative systems (pp. 16–31). College Publications.
Beth, E. W. (1956). Semantic construction of intuitionistic logic. Mededelingen der Koninklijke Nederlandse Akademie van Wetenschappen, 19, 357–388.
Bezhanishvili, G., & Holliday, W. (2019). A semantic hierarchy for intuitionistic logic. Indagationes Mathematicae, 30, 403–469.
Buszkowski, W. (1996). Categorial grammars with negative information. In H. Wansing (Ed.), Negation: A notion in focus (pp. 107–126). de Gruyter.
Chartwood, G. (1981). An axiomatic version of positive semilattice relevance logic. Journal of Symbolic Logic, 46, 233–239.
Church, A. (1951). The weak theory of implication. In A. Menne, A. Wilhelmy, & H. Angsil (Eds.), KontrolIiertes Denken: Untersuchungen zum Logikkalkül und der Logik der Einzelwissenschaften (pp. 22–37). Kommissions-Verlag Karl Alber.
Ciardelli, I. Inquisitive semantics as a truth-maker semantics (Unpublished manuscript).
Došen, K. (1992). A brief survey of frames for the Lambek calculus. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 38, 179–187.
Fine, K. (1976). Completeness for the semilattice semantics with disjunction and conjunction. Abstract in Journal of Symbolic Logic, 41, 560.
Fine, K. (2014). Truth-maker semantics for intuitionistic logic. Journal of Philosophical Logic, 43, 549–577.
Fine, K. (2017). Truthmaker semantics. In B. Hale, C. Wright, & A. Miller (Eds.), A companion to the philosophy of language (pp. 556–577). Blackwell.
Hale, B. (2018). What makes true universal statements true? In P. Arazim & T. Lávička (Eds.), The logica yearbook 2017 (pp. 89–108). College Publications.
Jago, M. (2020). Truthmaker semantics for relevant logic. Journal of Philosophical Logic, 49, 681–702.
Klev, A. (2017). Truthmaker semantics. Fine versus Martin-Löf. In P. Arazim & T. Lávička (Eds.), The logica yearbook 2016 (pp. 87–108). College Publications.
Kripke, S. (1965). Semantical analysis of intuitionistic logic I. In J. N. Crossley & M. A. E. Dummett (Eds.), Formal systems and recursive functions (pp. 92–130). North-Holland.
Kurtonina, N. (1994). Frames and labels. A modal analysis of categorial inference (Ph.D. thesis). Utrecht University.
Lambek, J. (1958). The mathematics of sentence structure. The American Mathematical Monthly, 65, 154–170.
Lambek, J. (1961). On the calculus of syntactic types. In R. Jacobson (Ed.), Proceedings of the Twelfth Symposium in Applied Mathematics (Vol. XII, pp. 166–178).
Moltmann, F. (2018). An object-based truthmaker semantics for modals. Philosophical Issues, 28, 255–288.
Moortgat, M. J. (1998). Multimodal linguistic inference. Journal of Logic, Language and Information, 5, 349–385.
Punčochář, V. (2017). Algebras of information states. Journal of Logic and Computation, 27, 1643–1675.
Tarski, A. (1938). Der Aussagenkalkül und die Topologie. Fundamenta Mathematicae, 31, 103–134.
Urquhart, A. (1971). Completeness of weak implication. Theoria, 37, 274–282.
Urquhart, A. (1972). Semantics for relevant logics. Journal of Symbolic Logic, 37, 159–169.
Van Fraassen, B. (1969). Facts and tautological entailments. The Journal of Philosophy, 66, 477–487.
Venema, Y. (1997). A crash course in arrow logic. In M. Marx, L. Pólos, & M. Masuch (Eds.), Arrow logic and multi-modal logic (pp. 3–34). Center for the Study of Language and Information.
Wansing, H. (2007). A note on negation in categorial grammar. Logic Journal of the IGPL, 15, 271–286.
Acknowledgements
The work of Ondrej Majer on this paper was supported by grant no. 16-07954J of the Czech Science Foundation.
The work of Vít Punčochář on this paper was supported by grant no. 17-15645S of the Czech Science Foundation.
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Majer, O., Punčochář, V., Sedlár, I. (2023). Truth-Maker Semantics for Some Substructural Logics. In: Faroldi, F.L.G., Van De Putte, F. (eds) Kit Fine on Truthmakers, Relevance, and Non-classical Logic. Outstanding Contributions to Logic, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-031-29415-0_11
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