Powers and limitations of Urquhart-style semantics I : basic substructural logics

• Autorzy: Yang Eunsuk

Identyfikatory

DOI

10.4467/20842589RM.24.001.20698

• Języki publikacji: EN

Abstrakty

EN

This paper addresses three kinds of binary operational semantics, called here Urquhart-style semantics, for basic substructural logics. First, we discuss the most basic substructural logic GL introduced by Galatos and Ono and its expansions with structural axioms and their algebraic semantics. Next, we provide one kind of Urquhart-style semantics, whose frames form the same structures as algebraic semantics, for those substructural logics and consider powers and limitations of this kind of semantics in substructural logic. We then introduce another kind of Urquhart-style semantics, whose canonical frames are based on prime theories, for DL, the GL with distributivity, and some of its non-associative expansions and extend it to the semantics with star operations for negations. Similarly, we consider powers and limitations of these two kinds of semantics in substructural logic.

Słowa kluczowe

EN

operational semantics; Urquhart-style semantics; star operations for negations; algebraic semantics; substructural logics

• Wydawca: Wydawnictwo Uniwersytetu Jagiellońskiego
• Czasopismo: Reports on Mathematical Logic
• Rocznik: 2024
• Tom: Vol. 59
• Strony: 49--78
• Opis fizyczny: Bibliogr. 53 poz., rys.

Twórcy

autor

Yang Eunsuk

• Department of Philosophy & Institute of Critical Thinking and Writing, Jeonbuk National University, Rm 417, Center for Humanities & Social Sciences, Jeonju, 54896, KOREA

Bibliografia

• [1] Beth, E. W.: Semantic construction of intuitionistic logic. MKNAW 19, 357–388 (1956)
• [2] Beth, E. W.: The Foundations of Mathematics. Harper & Row, New York (1964)
• [3] Bezhanishvili, G., Holliday, W. H.: A semantic hierarchy for intuitionistic logic. Indag. Math. 30, 403–469 (2019)
• [4] Charlwood, G.: Representations of Semilattice Relevance Logics. PhD thesis, Uni versity of Toronto, 1978.
• [5] Charlwood, G.: An axiomatic version of positive semilattice relevance logic G. J. Symb. Log. 46, 233–239 (1981)
• [6] Cintula, P., Horčík, R., Noguera, C.: Non-associative substructural logics and their semilinear extensions: Axiomatization and completeness properties. Rev. Symb. Log. 6, 394–423 (2013)
• [7] Cintula, P., Horčík, R., Noguera, C.: The quest for the basic fuzzy logic. In: Montagna, F. (ed.) Petr H´ajek on Mathematical Fuzzy Logic, pp. 245–290. Springer, Dordrecht (2015)
• [8] Cintula, P., Noguera, C.: A general framework for mathematical fuzzy logic. In: Cintula, P., H´ajek, P., Noguera, C. (eds.) Handbook of Mathematical Fuzzy Logic, Vol 1, pp. 103–207. College Publications, London (2011)
• [9] Došen, K.: Sequent systems and groupoid models I. Stud. Log. 47, 353–385 (1988)
• [10] Došen, K.: Sequent systems and groupoid models II. Stud. Log. 48, 41–65 (1989)
• [11] Dragalin, A. G.: Matematicheskii Intuitsionizm: Vvedenie v Teoriyu Dokazatelstv. In: Matematicheskaya Logika i Osnovaniya Matematiki. “Nauka”, Moscow (1979)
• [12] Dragalin, A. G.: Mathematical Intuitionism: Introduction to Proof Theory. In: Translations of Mathematical Monographs, vol 67. American Mathematical Society, Providence, RI. (1988)
• [13] Dunn, J. M.: Ternary relational semantics and beyond: programs as data and programs as instructions. Log. Stud. 7, 1–20 (2001)
• [14] Dunn, J. M.: The relevance of relevance to relevance logic. In: Banerjee, M., Krishna, S. N. (eds.) ICLA 2015: Logic and Its Applications, pp. 11–29. Springer, Berlin (2015)
• [15] Dunn, J. M.: Logics as Tools, or Humans as Rational Tool Making Animals. Manuscript presented at Jeonbuk National University (2018).
• [16] Dunn, J. M., Restall, G: Relevance logic. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, Vol 6, pp. 1–128. Reidel, Dordrecht (2002)
• [17] Fine, K.: Models for entailment. J. Phil. Log. 3, 347–372 (1974)
• [18] Fine, K.: Semantics for quantified relevance logic. J. Phil. Log. 17, 27–59 (1988)
• [19] Galatos, N., Jipsen, P.: Residuated frames with applications to decidability. Trans. Am. Math. Soc. 365, 1219–1249 (2012)
• [20] Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Elsevier, Amsterdam (2007)
• [21] Galatos, N., Ono, H. Cut elimination and strong separation for substructural logics. Ann. Pure Appl. Log. 161, 1097–1133 (2010).
• [22] Girard, J.-Y.: Linear logic. Theor. Comput. Sci. 50, 1–102 (1987)
• [23] Hartonas, C.: Order-dual relational semantics for non-distributive propositional logics. Log. J. IGPL 25, 145–182 (2017)
• [24] Hartonas, C.: Order-dual relational semantics for non-distributive propositional logics: A general framework. J. Phil. Log. 47, 67–94 (2018)
• [25] Humberstone, I. L.: Operational Semantics for Positive R. Notre Dame J. Form. Log. 29, 61–80 (1988)
• [26] Kripke, S.: Semantic analysis of modal logic I: Normal modal propositional calculi. Z. Math. Logik 9, 67–96 (1963)
• [27] Kripke, S.: Semantic analysis of intuitionistic logic I. In: Crossley, J. Dummett, M. (eds.) Formal Systems and Recursive Functions, pp. 92–129. North-Holland Publ. Co, Amsterdam (1965)
• [28] Kripke, S.: Semantic analysis of modal logic II. In: Addison, J., Henkin, L., Tarski, A. (eds.) The Theory of Models, pp. 206–220. North-Holland Publ. Co, Amsterdam (1965)
• [29] Montagna, F., Ono, H.: Kripke semantics, undecidability and standard completeness for Esteva and Godo’s Logic MTL∀. Stud. Log. 71, 227–245 (2002)
• [30] Montagna, F., Sacchetti, L.: Kripke–style semantics for many-valued logics. Math. Log. Q. 49, 629–641 (2003)
• [31] Montagna, F., Sacchetti, L.: Corrigendum to Kripke–style semantics for many-valued logics. Math. Log. Q. 50, 104–107 (2004)
• [32] Ono, H.: Completions of algebras and completeness of modal and substructural logics. In: Balbiani, P. et al. (eds.) Advances in Modal Logic, Vol 4, pp. 335–353. College publications, London (2003)
• [33] Restall, G.: An Introduction to Substructural Logics. Routledge, New York (2000)
• [34] Routley, R., Meyer, R. K.: The semantics of entailment – II. J. Phil. Log. 1, 53–73 (1972)
• [35] Routley, R., Meyer, R. K.: The semantics of entailment – III. J. Phil. Log. 1, 192–208 (1972)
• [36] Routley, R., Meyer, R. K.: The semantics of entailment. In: LeBranc, H. (ed.) Truth, Syntax, and Modality, pp. 199–243. North-Holland Publ. Co, Amsterdam (1973)
• [37] Standefer, S.: Revisiting semilattice semantics. In: D¨untsch, I. and Mares, E. (eds.) Alasdair Urquhart on Nonclassical and Algebraic Logic and Complexity of Proofs, pp. 243–259. Springer (2022).
• [38] Urquhart, A.: The completeness of weak implication. Theoria 37, 274–282 (1972)
• [39] Urquhart, A.: A general theory of implication. J. Symb. Log. 37, 443 (1972)
• [40] Urquhart, A.: Semantics for relevant logics. J. Symb. Log. 37, 159–169 (1972)
• [41] Urquhart, A.: The Semantics of Entailment. PhD thesis, University of Pittsburgh (1972d)
• [42] Venema, A.: Meeting strength in substructural logics. Stud. Log. 54, 3–32 (1995)
• [43] Yang, E.: Kripke–style semantics for UL. Korean J. Log. 15, 1–15 (2012)
• [44] Yang, E.: Algebraic Kripke–style semantics for relevance logics. J. Phil. Log. 43, 803–826 (2014)
• [45] Yang, E.: Algebraic Kripke-style semantics for weakening-free fuzzy logics. Korean J. Log. 17, 181–195 (2014)
• [46] Yang, E.: Weakening-free, non-associative fuzzy logics: Micanorm-based logics. Fuzzy Sets Syst. 276, 43–58 (2015)
• [47] Yang, E.: Basic substructural core fuzzy logics and their extensions: Mianorm-based logics. Fuzzy Sets Syst. 301, 1–18 (2016)
• [48] Yang, E.: Algebraic Kripke-style semantics for substructural fuzzy logics. Korean J. Log. 19, 295–322 (2016)
• [49] Yang, E.: Algebraic relational semantics for basic substructural logics. Log. et Anal. 252, 415–441 (2020)
• [50] Yang, E.: (Involutive) basic substructural fuzzy logics and Urquhart-style semantics. J. Mult.-Valued Log. 41(1-2), 187–207 (2023)
• [51] Yang, E.: Substructural nuclear (image-Based) logics and operational Kripke-style semantics. Stud. Log. 112(4), 805–833 (2024)
• [52] Yang, E., Dunn, J. M.: Implicational tonoid logics: algebraic and relational semantics. Log. Univers. 15, 435–456 (2021)
• [53] Yetter, D. N.: Quantales and (noncommutative) linear logic. J. Symb. Log. 55, 41–64 (1990)

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025).