A Theorem Concerning Syntactical Treatments Of Nonidealized Belief

In Syntactical Treatments of Modality, with Corollaries on Reflexion Principles and Finite Axiomatizability, Acta Philosophica Fennica16 (1963), 153–167, Richard Montague shows that the use of a single syntactic predicate (with a context-independent semantic value) to represent modalities of alethic necessity and idealized knowledge leads to inconsistency. In A Note on Syntactical Treatments of Modality, Synthese44 (1980), 391–395, Richmond Thomason obtains a similar impossibility result for idealized belief: under a syntactical treatment of belief, the assumption that idealized belief is deductively closed, together with certain other plausible conditions on idealized belief, imply that an ideal believer with consistent beliefs cannot believe the truth of Robinson's Arithmetic. In this essay I show that an impossibility result similar to Thomason's can be obtained which does not assume that belief is deductively closed or ideal in any other way.