Zolin and Pizzi: Defining Necessity from Noncontingency Authors
First Online: 28 November 2012 Received: 06 May 2012 Accepted: 04 November 2012 DOI:
Cite this article as: Humberstone, L. Erkenn (2013) 78: 1275. doi:10.1007/s10670-012-9412-5 Abstract
The point of the present paper is to draw attention to some interesting similarities, as well as differences, between the approaches to the logic of noncontingency of Evgeni Zolin and of Claudio Pizzi. Though neither of them refers to the work of the other, each is concerned with the definability of a (normally behaving, though not in general truth-implying) notion of necessity in terms of noncontingency, standard boolean connectives and additional but non-modal expressive resources. The notion of definability involved is different in the two cases (‘external’ for Zolin, ‘internal’ for Pizzi), as are the additional resources: infinitary conjunction in the case of Zolin, and for Pizzi, first, propositional quantification, and then, later, most ingeniously, the use of a propositional constant. As well as surveying and comparing of the work of these authors, the discussion includes some some novelties, such as the confirmation of a conjecture of Zolin’s (Theorem 2.7).
Anderson, A. R. (1967). The formal analysis of normative systems. In N. Rescher (Ed.),
The logic of decision and action (pp. 147–213). Pittsburgh, PA: University of Pittsburgh Press.
Bellissima, F., & Bucalo, A. (1995). A distinguishable model theorem for the minimal US-tense logic’
Notre Dame Journal of Formal Logic, 36
Chagrov, A. V., & Zakharyaschev, M. (1997).
Modal logic. Oxford: Clarendon Press.
Chellas, B. F. (1980). Modal logic: An introduction. Cambridge: Cambridge University Press.
Cresswell, M. J. (1988). Necessity and contingency.
Studia Logica, 47
Fine, K. (1970). Propositional quantifiers in modal logic.
Hiż, H. (1958). A warning about translating axioms.
American Mathematical Monthly, 65, 613–614.
Humberstone, L. (1995). The logic of non-contingency.
Notre Dame Journal of Formal Logic, 36
Humberstone, L. (2002). The modal logic of agreement and noncontingency.
Notre Dame Journal of Formal Logic, 43
Humberstone, L. (2004). Two-dimensional adventures.
Philosophical Studies, 118
Humberstone, L. (2005). Béziau’s translation paradox.
Kuhn, S. T. (1995). Minimal non-contingency logic.
Note Dame Journal of Formal Logic, 36
Lewis, C. I., & Langford, C. H. (1959).
Symbolic logic. NY: Dover (first printing 1932).
Meredith, C. A., & Prior, A. N. (1965). Modal logic with functorial variables and a contingent constant.
Notre Dame Journal of Formal Logic, 6
Meyer, R. K. (1974). Entailment is not strict implication.
Australasian Journal of Philosophy, 52
Pizzi C. (1999). Contingency logics and propositional quantification.
Manuscrito (UNICAMP, Campinas University), 22, 283–303.
Pizzi, C. (2006). A logic of contingency with a propositional constant. In E. Ballo, M. Franchella (Ed.),
Logic and philosophy in Italy: Some trends and perspectives (pp. 141–153). Milan: Polimetrica.
Pizzi, C. (2007). Necessity and relative contingency.
Studia Logica, 85
Pizzi, C. Relative contingency and multimodal logics, to appear in a forthcoming special issue of
Logica Universalis on Multimodal Logics.
Prior, A. N. (1957).
Time and modality. Oxford: Oxford University Press.
Saito, S. (1962). Circular definitions and analyticity.
Inquiry, 5, 158–162.
Saito, S. (1966). On the completeness of the Leibnizian modal system with a reduction.
Proceedings of Japan Academy, 42
Saito, S. (1968). On the Leibnizian modal system.
Notre Dame Journal of Formal Logic, 9
Segerberg, K. (1971).
An essay in classical modal logic. Uppsala: Filosofiska Studier.
Smiley, T. (1963). Relative necessity.
Journal of Symbolic Logic, 28
Ṡwirydowicz, K. (1992). Regular modal logics inconsistent if
\(\square \top\) is added: A contribution to the topography of the lattice of modal logics. Bulletin of the Section of Logic 21, 47–54.
Thomason, R. H., & Leblanc, H. (1967). All or none: A novel choice of primitives for elementary logic.
Journal of Symbolic Logic, 32
Wójcicki, R. (1988).
Theory of logical Calculi
. Dordrecht: Kluwer.
Zolin, E. E. (1999). Completeness and definability in the logic of noncontingency.
Notre Dame Journal of Formal Logic, 40
Zolin, E. E. (2000). Embeddings of propositional monomodal logics.
Logic Journal of the IGPL, 8
Zolin, E. E. (2001). Infinitary expressibility of necessity in terms of contingency. In K. Striegnitz (Ed.),
Proceedings of the sixth ESSLLI student session, pp. 325–334.
Zolin, E. E. (2002). Sequential reflexive logics with noncontingency operator.
Mathematical Notes, 72, 784–798 (translation from Matematicheskie Zametki, Russian Academy of Sciences). Copyright information
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