- Hans Leitgeb

University of Bristol

We are interested in developing a type-free theory of modality and truth in which modalities are treated syntactically in the same way as truth, i.e., as predicates of sentences. As Tarski has shown, given a first-order language that both contains the truth predicate and the arithmetic language, the unrestricted truth scheme is inconsistent with arithmetic. Accordingly, Montague (1963) proved that if necessity is expressed by a predicate of sentences, the unrestricted predicate counterparts of the axioms and rules of the standard modal system T turn out to be inconsistent with a sufficiently strong background theory of syntax. Similar theorems have been derived for the predicate versions of other modal systems, including the minimal system of temporal logic (Horsten&Leitgeb 2001). On the semantic side, Halbach et al. (2003) show that certain modal frames do not allow for possible worlds evaluations if the standard satisfaction clause for modal operators is translated to the predicate case unrestrictedly. In contrast to Tarski and Montague, we do not conclude from these results that truth predicates have to be typed or that modal predicates are to be replaced by sentential operators. Instead we suggest to hold on to an unrestricted syntax but rather to restrict the admissible instances of the truth scheme and the modal axiom schemes and rules. We are going to present a possible worlds semantics and corresponding axiomatic systems for modal predicates in which these restrictions are taken care of. The possible worlds semantics for the reflexive frame that contains exactly one world coincides with the theory of truth in Leitgeb (2005).

REFERENCES- [1] Halbach, V., H. Leitgeb, P. Welch, 2003: "Possible Worlds Semantics for Modal Notions Conceived as Predicates", Journal of Philosophical Logic 32[2], 179-223.
- [2] Horsten, L., H. Leitgeb, 2001: "No Future", Journal of Philosophical Logic 30[3], 259-265.
- [3] Leitgeb, H., 2005: "What Truth Depends on", Journal of Philosophical Logic 34[2] (2005), 155-192.
- [4] Montague, R., 1963: "Syntactical treatments of modality, with corollaries on reflexion