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  • Cited by 1
  • Online publication date: March 2017

Possible worlds semantics for predicates

Summary

Abstract. We develop possible worlds semantics for as a predicate rather than as an operator of sentences. The unary predicate symbol is added to the language of arithmetic (or an extension thereof); this yields the language. Every world in our possible worlds semantics is the standard model of arithmetic plus an interpretation of. We investigate possible–worlds models where is true at a world w if and only if A is true in all worlds seen by w. The paradoxes exclude certain frames from being frames foras a predicate. We provide some sufficient and also some necessary conditions on frames that are allowed to act as frames for the predicate approach. Completeness results for certain infinitary systems corresponding to well known modal operator systems are established. We draw some conclusions concerning the current state of the predicate approach to modalities.

Modalities as predicates. Modalities like necessity and possibility, may be analysed logically in essentially two ways: either as predicates, or as operators. In the first case they are applied to singular terms, whereas in the second case they are applied to formula, but in both cases the application gives us new formula. Thus the distinction between the operator and the predicate conception of necessity is made on the syntactical level at first. Both conceptions are tied to certain semantics respectively. If “necessary” and “possible” are regarded as predicates, they are interpreted as properties of objects and a decision has to be made concerning what precisely they should be predicates of: syntactical entities like sentences, or contents of syntactical entities like propositions (let us ignore further options like utterances or mental objects). In either case, necessity and possibility are properties of such entities, or, perhaps, relations between such entities and further objects. If “necessary” and “possible” are regarded as operators, they do not express properties or relations like predicate and relation expressions; necessity does not apply to anything— much like the logical connectives or the quantifiers. In this sense the operator conception of necessity is radically deflationary. Similar considerations apply not only to necessity but also to the notions of knowledge, belief, future and past truth, obligation and so on, which have been treated in analogous fashions as necessity.

[1] Peter, Aczel and Wayne, Richter, Inductive definitions and reflecting properties of admissible ordinals, Generalized recursion theory (Jens E., Fenstad and Peter, Hinman, editors), North-Holland, 1973, pp. 301–381.
[2] Nicholas, Asher and Hans, Kamp, Self-reference, attitudes, and paradox, Properties, types and meaning (Gennaro, Chierchia, Barbara H., Partee, and Raymond, Turner, editors), vol. 1, Kluwer, Dordrecht, 1989, pp. 85–158.
[3] Jon, Barwise, Admissible sets and structures, Perspectives in Mathematical Logic, Springer- Verlag, Berlin, 1975.
[4] George, Bealer, Quality and concept, Clarendon Press, Oxford, 1982.
[5] Nuel, Belnap and Anil, Gupta, The revision theory of truth,MIT Press, Cambridge, 1993.
[6] George, Boolos, The logic of provability, Cambridge University Press, Cambridge, 1993.
[7] Andrea, Cantini, A theory of formal truth arithmetically equivalent to ID1, The Journal of Symbolic Logic, vol. 55 (1990), pp. 244–259.
[8] Andrea, Cantini, Logical frameworks for truth and abstraction. an axiomatic study, Studies in Logic and the Foundations of Mathematics, vol. 135, Elsevier, Amsterdam, 1996.
[9] Alexander, Chagrov and Michael, Zakharyaschev,Modal logic, Oxford Logic Guides, Oxford University Press, Oxford, 1997.
[10] Jim des, Rivi`eres and Hector J., Levesque, The consistency of syntactical treatments of knowledge, Theoretical aspects of reasoning about knowledge: Proceedings of the 1986 conference (Joseph Y. Halpern, editor), Morgan Kaufmann, Los Altos, 1986, pp. 115–130.
[11] Solomon, Feferman, Reflecting on incompleteness, The Journal of Symbolic Logic, vol. 56 (1991), pp. 1–49.
[12] Harvey, Friedman and Michael, Sheard, An axiomatic approach to self-referential truth, Annals of Pure and Applied Logic, vol. 33 (1987), pp. 1–21.
[13] Patrick, Grim, The incomplete universe: Totality, knowledge, and truth, MIT Press, Cambridge,Mass., 1991.
[14] Volker, Halbach, A system of complete and consistent truth, Notre Dame Journal of Formal Logic, vol. 35 (1994), pp. 311–327.
[15] Volker, Halbach, Truth and reduction, Erkenntnis, vol. 53 (2000), pp. 97–126.
[16] Volker, Halbach, Semantics and deflationism, Habilitationsschrift, Universität Konstanz, 2001.
[17] Volker, Halbach, Hannes, Leitgeb, and Philip, Welch, Possible worlds semantics for modal notions conceived as predicates, Journal of Philosophical Logic, vol. 32 (2003), pp. 179–223.
[18] G. E., Hughes and M. J., Cresswell, A new introduction to modal logic, Routledge, London and New York, 1996.
[19] Hannes, Leitgeb, Theories of truth which have no standard models, Studia Logica, vol. 21 (2001), pp. 69–87.
[20] Vann, McGee, How truthlike can a predicate be?, Journal of Philosophical Logic, vol. 14 (1985), pp. 399–410.
[21] Richard, Montague, Syntactical treatments of modality, with corollaries on reflexion principles and finite axiomatizability, Acta Philosophica Fennica, vol. 16 (1963), pp. 153–167.
[22] Yiannis, Moschovakis, Elementary induction on abstract structures, North-Holland,Amsterdam, 1974.
[23] Willard Van Orman, Quine, Three grades of modal involvement, The ways of paradox, Harvard University Press, Cambridge,Mass., revised and enlarged ed., 1976, pp. 158–176.
[24] Willard Van Orman, Quine, Intensions revisited, Contemporary perspectives in the philosophy of language (French, Uehling, and Wettstein, editors), University of Minnesota Press, Minneapolis, 1977, pp. 5–11.
[25] Michael, Rathjen, Proof theory of reflection, Annals of Pure and Applied Logic, vol. 68 (1994), pp. 181–224.
[26] Hartley, Rogers, Theory of recursive functions and effective computability, McGraw-Hill Book Company, New York, 1967.
[27] Kurt, Schütte, Proof theory, Springer, Berlin, 1970.
[28] Michael, Sheard, A guide to truth predicates in the modern era, The Journal of Symbolic Logic, vol. 59 (1994), pp. 1032–1054.
[29] Barry Hartley, Slater, Natural language's semantic closure, forthcoming.
[30] Richmond H., Thomason, A note on syntactical treatments of modality, Synthese, vol. 44 (1980), pp. 391–396.
[31] Albert, Visser, Semantics and the liar paradox, Handbook of philosophical logic, vol. IV, Reidel, Dordrecht, 1989, pp. 617–706.