Abstract
If □ is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely restricted when compared to a language where □ is conceived as a predicate, i.e., an expression that yields a formula if it is applied to a term. This consideration favours the predicate approach. The predicate view, however, is threatened mainly by two problems: Some obvious predicate systems are inconsistent, and possible-worlds semantics for predicates of sentences has not been developed very far. By introducing possible-worlds semantics for the language of arithmetic plus the unary predicate □, we tackle both problems. Given a frame <W,R> consisting of a set W of worlds and a binary relation R on W, we investigate whether we can interpret □ at every world in such a way that □⌜A⌝ holds at a world w∈W if and only if A holds at every world v∈W such that wRv. The arithmetical vocabulary is interpreted by the standard model at every world. Several ‘paradoxes’ (like Montague's Theorem, Gödel's Second Incompleteness Theorem, McGee's Theorem on the ω-inconsistency of certain truth theories, etc.) show that many frames, e.g., reflexive frames, do not allow for such an interpretation. We present sufficient and necessary conditions for the existence of a suitable interpretation of □ at any world. Sound and complete semi-formal systems, corresponding to the modal systems K and K4, for the class of all possible-worlds models for predicates and all transitive possible-worlds models are presented. We apply our account also to nonstandard models of arithmetic and other languages than the language of arithmetic.
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REFERENCES
Aczel, P. and Richter, W.: Inductive definitions and reflecting properties of admissible ordinals, in J. E. Fenstad and P. Hinman (eds), Generalized Recursion Theory, North-Holland, 1973, pp. 301–381.
Asher, N. and Kamp, H.: Self-reference, attitudes, and paradox, in G. Chierchia, B. H. Partee and R. Turner (eds), Properties, Types and Meaning, Vol. 1, Kluwer, Dordrecht, 1989, pp. 85–158.
Barwise, J. Admissible Sets and Structures, Perspect. Math. Logic, Springer-Verlag, Berlin, 1975.
Bealer, G.: Quality and Concept, Clarendon Press, Oxford, 1982.
Belnap, N. and Gupta, A.: The Revision Theory of Truth, MIT Press, Cambridge, 1993.
Boolos, G.: The Logic of Provability, Cambridge University Press, Cambridge, 1993.
Burgess, J. P.: The truth is never simple, J. Symbolic Logic 51 (1986), 663–681.
Burgess, J. P.: Addendum to 'The truth is never simple', J. Symbolic Logic 53 (1988), 390–392.
Cantini, A.: Logical Frameworks for Truth and Abstraction. An Axiomatic Study, Stud. Logic Found. Math. 135, Elsevier, Amsterdam, 1996.
Chagrov, A. and Zakharyaschev, M.: Modal Logic, Oxford Logic Guides, Oxford University Press, Oxford, 1997.
Copeland, J.: The genesis of possible worlds semantics, J. Philos. Logic 31 (2002), 99–137.
Feferman, S.: Reflecting on incompleteness, J. Symbolic Logic 56 (1991), 1–49.
Feferman, S. and Spector, C.: Incompleteness along paths in progressions of theories, J. Symbolic Logic bd27 (1962), 383–390.
Field, H.: Disquotational truth and factually defective discourse, Philos. Review 103 (1994), 405–452.
Friedman, H. and Sheard, M.: An axiomatic approach to self-referential truth, Ann. Pure Appl. Logic 33 (1987), 1–21.
Germano, G.: Metamathematische Begriffe in Standardtheorien, Arch. Math. Logik 13 (1970), 22–38.
Grover, D., Camp, J. and Belnap, N.: A prosentential theory of truth, Philos. Stud. 27 (1975), 73–125.
Gupta, A.: Truth and paradox, J. Philos. Logic 11 (1982), 1–60.
Halbach, V.: A system of complete and consistent truth, Notre Dame J. Formal Logic 35 (1994), 311–327.
Halbach, V., Leitgeb, H. and Welch, P.: Possible worlds semantics for predicates, in R. Kahle (ed.), Intensionality, Association for Symbolic Logic, Los Angeles (in press).
Harrison, J.: Recursive pseudo-well-orderings, Trans. Amer. Math. Soc. 131 (1968), 526–543.
Herzberger, H. G.: Notes on naive semantics, J. Philos. Logic 11 (1982), 61–102.
Hinman, P.: Recursion Theoretic Hierarchies, Springer, Berlin, 1978.
Jockusch, C. and Simpson, S.: A degree theoretic definition of the ramified analytical hierarchy, Ann. Math. Logic 10 (1975), 1–32.
Kaye, R.: Models of Peano Arithmetic, Oxford Logic Guides, Oxford University Press, 1991.
Kotlarski, H., Krajewski, S. and Lachlan, A.: Construction of satisfaction classes for nonstandard models, Canad. Math. Bull. 24 (1981), 283–293.
Kripke, S.: Outline of a theory of truth, J. Philos. 72 (1975), 690–712.
Kripke, S.: Is there a problem about substitutional quantification?, in G. Evans and J. McDowell (eds), Truth and Meaning: Essays in Semantics, Clarendon Press, Oxford, 1976, pp. 325–419.
Lachlan, A.: Full satisfaction classes and recursive saturation, Canad. Math. Bull. 24 (1981), 295–297.
Leitgeb, H.: Theories of truth which have no standard models, Studia Logica 21 (2001), 69–87.
Leitgeb, H.: Truth as translation - part B, J. Philos. Logic 30 (2001), 309–328.
McGee, V.: How truthlike can a predicate be? A negative result, J. Philos. Logic 14 (1985), 399–410.
McGee, V.: Truth, Vagueness, and Paradox: An Essay on the Logic of Truth, Hackett Publishing, Indianapolis and Cambridge, 1991.
Montague, R.: Syntactical treatments of modality, with corollaries on reflexion principles and finite axiomatizability, Acta Philosophica Fennica 16 (1963), 153–167. Reprinted in [35, 286-302].
Montague, R.: Formal Philosophy: Selected Papers of Richard Montague, Yale University Press, New Haven and London, 1974, Edited and with an introduction by Richmond H. Thomason.
Moschovakis, Y. N.: Elementary Induction on Abstract Structures, Studies in Logic and the Foundations of Mathematics 77, North-Holland, Amsterdam, 1974.
Pap, A.: Analytische Erkenntnistheorie, Springer, Wien, 1955.
Rathjen, M.: Proof theory of reflection, Ann. Pure Appl. Logic 68 (1994), 181–224.
des Rivières, J. and Levesque, H. J.: The consistency of syntactical treatments of knowledge, in J. Y. Halpern (ed.), Theoretical Aspects of Reasoning about Knowledge: Proceedings of the 1986 Conference, Los Altos, Morgan Kaufmann, 1986, pp. 115–130.
Rogers, H.: Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York, 1967.
G. E. Sacks, Countable admissible ordinals and hyperdegrees, Adv. Math. 99 (1976), 213–262.
Schweizer, P.: A syntactical approach to modality, J. Philos. Logic 21 (1992), 1–31.
Skyrms, B.: An immaculate conception of modality, J. Philos. 75 (1978), 368–387.
Slater, B. H.: Paraconsistent logics?, J. Philos. Logic 24 (1995), 451–454.
Strawson, P.: Truth, Analysis 9 (1949), 83–97.
Visser, A.: Semantics and the Liar paradox, in D. Gabbay and F. Günthner (eds), Handbook of Philosophical Logic, Vol. 4, Reidel, Dordrecht, 1989, pp. 617–706.
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Halbach, V., Leitgeb, H. & Welch, P. Possible-Worlds Semantics for Modal Notions Conceived as Predicates. Journal of Philosophical Logic 32, 179–223 (2003). https://doi.org/10.1023/A:1023080715357
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DOI: https://doi.org/10.1023/A:1023080715357