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Denotational semantics for modal systems S3–S5 extended by axioms for propositional quantifiers and identity. (English) Zbl 1371.03026
The author focuses on a non-Fregean identity relation among propositions. That is, an identity relation for propositions that satisfies reflexivity and Leibniz’s law but not an identity of propositions on the ground of their being equivalent. In modal contexts, a non-Fregean approach will reject assuming an identity of propositions on the basis of their equivalence being necessary. The author will formulate certain second-order extensions of the modal systems S3, S4 and S5 that will follow the approach in question. These extensions are systems whose logical syntax includes as primitive symbols second-order quantifiers over propositions as well as an identity relation for propositions. In addition to the proper modal axioms of the S3–S5 modal systems, their axiomatic basis includes logical principles for the second-order quantifiers, non-Fregean principles for the identity sign, the Barcan formulas and their converse. Closures of all of these axioms under universal quantification as well as necessitation will also constitute axioms. Modus ponens is their only primitive rule. The modal systems in question are shown to be sound and complete with respect to an algebraic semantics formulated by the author. In the last section of the paper, the author characterizes another semantic system. Relative to this additional semantics, soundness and completeness theorems are proved for the second-order modal systems in question but without the Barcan formulas, their converse and certain extensional schema. The author also shows that his non-Fregean modal systems are conservative extensions of S3–S5.

MSC:
03B45 Modal logic (including the logic of norms)
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[1] Blackburn P., de Rijke M., Venema Y.: Modal Logic, . Cambridge University Press, Cambridge (2001) · Zbl 0988.03006
[2] Bloom, S.L., A completeness theorem for “theories of kind \(W\)”, Studia Logica, 27, 43-55, (1971) · Zbl 0249.02014
[3] Bloom, S.L.; Suszko, R., Semantics for the sentential calculus with identity, Studia Logica, 28, 77-81, (1971) · Zbl 0243.02016
[4] Bloom, S.L.; Suszko, R., Investigation into the sentential calculus with identity, Notre Dame Journal of Formal Logic, 13, 289-308, (1972) · Zbl 0188.01203
[5] Bull, R.A., On modal logic with propositional quantifiers, The Journal of Symbolic Logic, 34, 257-263, (1969) · Zbl 0184.28101
[6] Cresswell, M.J., Another basis of S4, Logique et Analyse, 31, 191-195, (1965) · Zbl 0147.24902
[7] Cresswell, M.J., Propositional identity, Logique et Analyse, 39, 283-292, (1967) · Zbl 0166.25104
[8] Fine, K., Quantifiers in modal logic, Theoria, 36, 336-346, (1970) · Zbl 0302.02005
[9] Fox C., Lappin S.: Foundations of Intensional Semantics. Blackwell Publishing, Hoboken (2005)
[10] Hermes, H., Term Logic with Choice Operator, Springer, Berlin, 1970, English version of Eine Termlogik mit Auswahloperator, Springer, Berlin, 1965.
[11] Hughes G.E., Cresswell M.J.: A new Introduction to Modal Logic. Routledge, London (1996) · Zbl 0855.03002
[12] Ishii, T., Propositional calculus with identity. Bulletin of the Section of Logic27/3:96-104, 1998. · Zbl 0577.03033
[13] Ishii, T., Propositional Calculus with Identity, Dissertation, Japan Advanced Institute of Science and Technology, Nomi, 2000.
[14] Lewitzka, S., ∈_{\(I\)}: an intuitionistic logic without Fregean axiom and with predicates for truth and falsity, Notre Dame Journal of Formal Logic, 50, 275-301, (2009) · Zbl 1190.03016
[15] Lewitzka, S., ∈_{\(K\)}: a non-Fregean logic of explicit knowledge, Studia Logica, 97, 233-264, (2011) · Zbl 1231.03016
[16] Lewitzka, S., Construction of a canonical model for a first-order non-Fregean logic with a connective for reference and a total truth predicate, The Logic Journal of the IGPL, 20, 1083-1109, (2012) · Zbl 1283.03058
[17] Pollard, C., Hyperintensions, Journal of Logic and Computation, 18, 257-282, (2008) · Zbl 1138.03026
[18] Rautenberg, W., Einführung in die Mathematische Logik, 3rd edn., Vieweg+Teubner, 2008, (English version A Concise Introduction to Mathematical Logic, 3rd edn., Springer, New York, 2009). · Zbl 1152.03002
[19] Sträter, W., ∈_{\(T\)}Eine Logik erster Stufe mit Selbstreferenz und totalem Wahrheitsprädikat, Dissertation, KIT-Report 98, Technische Universität Berlin, 1992. · Zbl 1138.03026
[20] Suszko, R., Ontology in the tractatus of L, Wittgenstein, Notre Dame Journal of Formal Logic, 9, 7-33, (1968) · Zbl 0198.32001
[21] Suszko, R., Connective and modality, Studia Logica, 27, 7-39, (1971) · Zbl 0263.02015
[22] Suszko, R., Abolition of the Fregean axiom, in R. Parikh (ed.), Logic Colloquium, vol. 453 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 169-239,1975. · Zbl 0308.02026
[23] Wójcicki, R.; Suszko’s, R., Situational semantics., Studia Logica, 43, 323-340, (1984) · Zbl 0577.03033
[24] Zeitz, P., Parametrisierte ∈_{T}-Logik—eine Theorie der Erweiterung abstrakter Logiken um die Konzepte Wahrheit, Referenz und klassische Negation, Dissertation, Logos Verlag Berlin, 2000. · Zbl 0992.03013
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