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WHAT IS THE CORRECT LOGIC OF NECESSITY, ACTUALITY AND APRIORITY?

Published online by Cambridge University Press:  12 May 2014

PETER FRITZ
PETER FRITZ*
Affiliation:
University of Oxford
*
*JESUS COLLEGE TURL STREET OXFORD, OX1 3DW, UK E-mail: [email protected]
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Abstract

This paper is concerned with a propositional modal logic with operators for necessity, actuality and apriority. The logic is characterized by a class of relational structures defined according to ideas of epistemic two-dimensional semantics, and can therefore be seen as formalizing the relations between necessity, actuality and apriority according to epistemic two-dimensional semantics. We can ask whether this logic is correct, in the sense that its theorems are all and only the informally valid formulas. This paper gives outlines of two arguments that jointly show that this is the case. The first is intended to show that the logic is informally sound, in the sense that all of its theorems are informally valid. The second is intended to show that it is informally complete, in the sense that all informal validities are among its theorems. In order to give these arguments, a number of independently interesting results concerning the logic are proven. In particular, the soundness and completeness of two proof systems with respect to the semantics is proven (Theorems 2.11 and 2.15), as well as a normal form theorem (Theorem 3.2), an elimination theorem for the actuality operator (Corollary 3.6), and the decidability of the logic (Corollary 3.7). It turns out that the logic invalidates a plausible principle concerning the interaction of apriority and necessity; consequently, a variant semantics is briefly explored on which this principle is valid. The paper concludes by assessing the implications of these results for epistemic two-dimensional semantics.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 7 , Issue 3 , September 2014 , pp. 385 - 414
Copyright
Copyright © Association for Symbolic Logic 2014 

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References

BIBLIOGRAPHY

Bezhanishvili, N. (2002). Varieties of two-dimensional cylindric algebras. Part I: Diagonal-free case. Algebra Universalis, 48, 11–42.CrossRefGoogle Scholar
Bezhanishvili, N., & Marx, M. (2003). All proper normal extensions of S5-square have the polynomial size property. Studia Logica, 73, 367–382.CrossRefGoogle Scholar
Blackburn, P., de Rijke, M., & Venema, Y. (2001). Modal Logic, Vol. 53, Cambridge Tracts in Theoretical Computer Science. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Boolos, G. (1993). The Logic of Provability. Cambridge: Cambridge University Press.Google Scholar
Bull, R. A. (1966). That all normal extensions of S4.3 have the finite model property. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 12, 341–344.CrossRefGoogle Scholar
Burgess, J. P. (1999). Which modal logic is the right one? Notre Dame Journal of Formal Logic, 40, 81–93.CrossRefGoogle Scholar
Chalmers, D. J. (1996). The Conscious Mind. Oxford: Oxford University Press.Google Scholar
Chalmers, D. J. (2004). Epistemic two-dimensional semantics. Philosophical Studies, 118, 153–226.Google Scholar
Chalmers, D. J. (2006). The foundations of two-dimensional semantics. In García- Carpintero, M., & Macià, J., editors. Two-Dimensional Semantics, Oxford: Oxford University Press, pp. 55–140.CrossRefGoogle Scholar
Chalmers, D. J. (2011). Propositions and attitude ascriptions: A Fregean account. Noûs, 45, 595–639.CrossRefGoogle Scholar
Chalmers, D. J., & Rabern, B. (forthcoming). Two-dimensional semantics and the nesting problem. Analysis.Google Scholar
Cresswell, M. J. (2009). Non-denumerable infinitary modal logic. Journal of Universal Computer Science, 14, 63–71.Google Scholar
Crossley, J. N., & Humberstone, L. (1977). The logic of “actually”. Reports on Mathematical Logic, 8, 11–29.Google Scholar
Davies, M., & Humberstone, L. (1980). Two notions of necessity. Philosophical Studies, 38, 1–30.CrossRefGoogle Scholar
Dugundji, J. (1940). Note on a property of matrices for Lewis and Langford’s calculi of propositions. Journal of Symbolic Logic, 5, 150–151.CrossRefGoogle Scholar
Fritz, P. (2011). Matrices and modalities: On the logic of two-dimensional semantics. Master’s thesis, Institute for Logic, Language and Computation, University of Amsterdam. Available from: http://www.illc.uva.nl/Research/Reports/MoL-2011-02.text.pdf.Google Scholar
Fritz, P. (2013). A logic for epistemic two-dimensional semantics. Synthese, 190, 1753–1770.CrossRefGoogle Scholar
Gabbay, D. M., Kurucz, A., Wolter, F., & Zakharyaschev, M. (2003). Many-Dimensional Modal Logics: Theory and Applications, Vol. 148, Studies in Logic and the Foundations of Mathematics. Amsterdam: Elsevier.Google Scholar
Gabbay, D. M., & Shehtman, V. B. (1998). Products of modal logics, part 1. Logic Journal of the IGPL, 6, 73–146.CrossRefGoogle Scholar
Gregory, D. (2011). Iterated modalities, meaning and a priori knowledge. Philosophers’ Imprint, 11, 1–11.Google Scholar
Gärdenfors, P. (1973). On the extensions of S5. Notre Dame Journal of Formal Logic, 14, 277–280.Google Scholar
Hanson, W. H. (2006). Actuality, necessity, and logical truth. Philosophical Studies, 130, 437–459.CrossRefGoogle Scholar
Hazen, A. (1978). The eliminability of the actuality operator in propositional modal logic. Notre Dame Journal of Formal Logic, 19, 617–622.CrossRefGoogle Scholar
Hazen, A. P., Rin, B. G., & Wehmeier, K. F. (2013). Actuality in propositional modal logic. Studia Logica, 101, 487–503.CrossRefGoogle Scholar
Hughes, G. E., & Cresswell, M. J. (1996). A New Introduction to Modal Logic. London: Routledge.CrossRefGoogle Scholar
Humberstone, L. (2004). Two-dimensional adventures. Philosophical Studies, 118, 17–65.CrossRefGoogle Scholar
Humberstone, L. (2011). The Connectives. Cambridge, MA: MIT Press.CrossRefGoogle Scholar
Kamp, H. (1971). Formal properties of ‘now’. Theoria, 37, 227–273.CrossRefGoogle Scholar
Kaplan, D. (1989). Demonstratives. In Almog, J., Perry, J., & Wettstein, H., editors. Themes from Kaplan. Oxford: Oxford University Press, pp. 481–563.Google Scholar
Kreisel, G. (1967). Informal rigour and completeness proofs. In Lakatos, I., editor. Problems in the Philosophy of Mathematics. Amsterdam: North-Holland, pp. 138–186.CrossRefGoogle Scholar
Kripke, S. A. (1965). Semantical analysis of modal logic II. Non-normal modal propositional calculi. In Addison, J. W., Henkin, L., & Tarski, A., editors. The Theory of Models. Amsterdam: North-Holland, pp. 206–220.Google Scholar
Kripke, S. A. (1972). Naming and necessity. In Davidson, D., & Harman, G., editors. Semantics of Natural Language. Dordrecht: D. Reidel, pp. 253–355, 763–769.CrossRefGoogle Scholar
Prior, A. N. (1961). On a family of paradoxes. Notre Dame Journal of Formal Logic, 2, 16–32.CrossRefGoogle Scholar
Prior, A. N. (1968). “Now”. Noûs, 2, 101–119.CrossRefGoogle Scholar
Restall, G. (2012). A cut-free sequent system for two-dimensional modal logic, and why it matters. Annals of Pure and Applied Logic, 163, 1611–1623.CrossRefGoogle Scholar
Salmon, N. (1989). The logic of what might have been. Philosophical Review, 98, 3–34.CrossRefGoogle Scholar
Scroggs, S. J. (1951). Extensions of the Lewis system S5. Journal of Symbolic Logic, 16, 112–120.CrossRefGoogle Scholar
Segerberg, K. (1971). An Essay in Classical Modal Logic. vol. 13, Filosofiska Studier. Uppsala: Uppsala Universitet.Google Scholar
Smith, P. (2011). Squeezing arguments. Analysis, 71, 22–30.CrossRefGoogle Scholar
Soames, S. (2005). Reference and Description: The Case Against Two-Dimensionalism. Princeton: Princeton University Press.Google Scholar
Tarski, A. (2002). On the concept of following logically. History and Philosophy of Logic, 23, 155–196. Originally published in Polish and German, 1936.CrossRefGoogle Scholar
Tucker, D., & Thomason, R. H. (2011). Paradoxes of intensionality. Review of Symbolic Logic, 4, 394–411.CrossRefGoogle Scholar
Vlach, F. (1973). ‘Now’ and ‘Then’: A Formal Study in the Logic of Tense Anaphora. Ph. D. thesis, University of California, Los Angeles.Google Scholar
Williamson, T. (2013). Modal Logic as Metaphysics. Oxford: Oxford University Press.CrossRefGoogle Scholar