Recently, it has been observed that the usual type-theoretic restrictions are not enough to block certain paradoxes involving two or more predicates. In particular, when we have a self-referential language containing modal predicates, new paradoxes might appear even if there are type restrictions for the principles governing those predicates. In this article we consider two type-theoretic solutions to multimodal paradoxes. The first one adds types for each of the modal predicates. We argue that there are a number of problems with (...) most versions of this approach. The second one, which we favour, represents modal notions by using the truth predicate together with the corresponding modal operator. This way of doing things is not only useful because it avoids multimodal paradoxes, but also because it preserves the expressive capacity of the language. As an example of the sort of theory we have in mind, we provide a type-theoretic axiomatization that combines truth with necessity and knowledge. (shrink)

This essay corrects an error in the presentation of the Paradox of the Knowledge-Plus Knower, which is the variant of Kaplan and Montague’s Knower Paradox presented in C. Cross 2001: ‘The Paradox of the Knower without Epistemic Closure,’ MIND, 110, pp. 319–33. The correction adds a universally quantified transitivity principle for derivability as an additional assumption leading to paradox. This correction does not affect the status of the Knowledge-Plus paradox as a rebuttal to an argument against epistemic closure, since the (...) quantified transitivity principle is true in the standard model of arithmetic and therefore innocuous. (shrink)

In this essay I present a new version of the Paradox of the Knower and show that this new paradox vitiates a certain argument against epistemic closure. I then prove a theorem that relates the new paradox to epistemological scepticism. I conclude by assessing the use of the Knower in arguments against syntactical treatments of knowledge.

[IMPORTANT CORRECTION - See end of abstract.] In Syntactical Treatments of Modality, with Corollaries on Reflexion Principles and Finite Axiomatizability, Acta Philosophica Fennica 16 (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, Synthese 44 (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 assert an impossibility result similar to Thomason's but which does not assume that belief is deductively closed or ideal in any other way. There are technical mistakes in the formulations of Lemmas 1-4 and Theorem 2, however. Corrections can be found in the Erratum section of the author's website. (shrink)

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. (shrink)