This collection assembles Church's referee reports, Fitch's 1963 paper, and nineteen new papers on the knowability paradox.

Knowledge and its Limits presents a systematic new conception of knowledge as a kind of mental stage sensitive to the knower's environment. It makes a major contribution to the debate between externalist and internalist philosophies of mind, and breaks radically with the epistemological tradition of analyzing knowledge in terms of true belief. The theory casts new light on such philosophical problems as scepticism, evidence, probability and assertion, realism and anti-realism, and the limits of what can be known. The arguments are (...) illustrated by rigorous models based on epistemic logic and probability theory. The result is a new way of doing epistemology and a notable contribution to the philosophy of mind. (shrink)

It will be an essential resource for philosophers, mathematicians, computer scientists, linguists, or any scholar who finds connectives, and the conceptual issues surrounding them, to be a source of interest.This landmark work offers both ...

J. R. Lucas; IV*—Mathematical Tennis, Proceedings of the Aristotelian Society, Volume 85, Issue 1, 1 June 1985, Pages 63–72, https://doi.org/10.1093/aristotelia.

Sorensen here offers a unified solution to a large family of philosophical puzzles and paradoxes through a study of "blindspots": consistent propositions that cannot be rationally accepted by certain individuals even though they might by true.

An account of desire as a species of belief may owe its appeal to the details of its proposal as to precisely what sort of beliefs desires are to be identified with, and its downfall may be due to those details it does provide. For example, it may be proposed that the desire that α is in fact the belief that it ought to be that α, or is morally good or desirable that it should be the case that α. (...) Here the appeal might be that of forging a link between the holding of a moral belief and the acknowledgment that one has a reason for acting a certain way; and the shortcoming of the suggestion is its evident implausibility: even if the ‘necessity’ direction could be established, having a desire hardly seems sufficient for the holding of any such belief. Again: it might be proposed, perhaps simply to bring some order into the realm of propositional attitudes by reducing some to others, that the belief with which we should identify a’s desire that α is a’s belief that he will or would be happy if α. This proposed identification can be seen to be incorrect by consideration of examples such as the following, due to J. Gosling. An aging and ailing parent might forego numerous pleasures in order that his children should reap the benefits of his saving and have a good start in life – perhaps by receiving an expensive education – after his impending death. Oearly he may want that they should so benefit even if he does not believe in an after-death existence in which he might come to know of, and so take pleasure in, his children’s subsequent well-being. So his wanting that they should prosper cannot consist in his believing that he will be happy if/when they do, since he has no expectation of even being in existence in that eventuality. As Gosling puts it, there is a clear difference between thinking that something’s coming about will bring one pleasure, on the one hand, and viewing the prospect of its coming about with pleasure, on the other. (shrink)

J. R. Lucas; IV*—Mathematical Tennis, Proceedings of the Aristotelian Society, Volume 85, Issue 1, 1 June 1985, Pages 63–72, https://doi.org/10.1093/aristotelia.

A well-known open problem in epistemic logic is to give a syntactic characterization of the successful formulas. Semantically, a formula is successful if and only if for any pointed model where it is true, it remains true after deleting all points where the formula was false. The classic example of a formula that is not successful in this sense is the “Moore sentence” p ∧ ¬BOXp, read as “p is true but you do not know p.” Not only is the (...) Moore sentence unsuccessful, it is self-refuting, for it never remains true as described. We show that in logics of knowledge and belief for a single agent (extended by S5), Moorean phenomena are the source of all self-refutation; moreover, in logics for an introspective agent (extending KD45), Moorean phenomena are the source of all unsuccessfulness as well. This is a distinctive feature of such logics, for with a non-introspective agent or multiple agents, non-Moorean unsuccessful formulas appear. We also consider how successful and self-refuting formulas relate to the Cartesian and learnable formulas, which have been discussed in connection with Fitch’s “paradox of knowability.” We show that the Cartesian formulas are exactly the formulas that are not eventually self-refuting and that not all learnable formulas are successful. In an appendix, we give syntactic characterizations of the successful and the self-refuting formulas. (shrink)

The appropriateness of S5 as a logic of knowledge has been attacked at some length in the philosophical literature. Here one particular attack based on the interplay between knowledge and belief is considered: Suppose that knowledge satisfies S5, belief satisfies KD45, and both the entailment property (knowledge implies belief) and positive certainty (if the agent believes something, she believes she knows it) hold. Then it can be shown that belief reduces to knowledge: it is impossible to have false beliefs. While (...) the entailment property has typically been viewed as perhaps the least controversial of these assumptions, an argument is presented that it can plausibly be viewed as the culprit. More precisely, it is shown that this attack fails if we weaken the entailment property so that it applies only to objective (nonmodal) formulas, rather than to arbitrary formulas. Since the standard arguments in favor of the entailment property are typically given only for objective formulas, this observation suggests that care must be taken in applying intuitions that seem reasonable in the case of objective formulas to arbitrary formulas. (shrink)

A textbook on modal logic, intended for readers already acquainted with the elements of formal logic, containing nearly 500 exercises. Brian F. Chellas provides a systematic introduction to the principal ideas and results in contemporary treatments of modality, including theorems on completeness and decidability. Illustrative chapters focus on deontic logic and conditionality. Modality is a rapidly expanding branch of logic, and familiarity with the subject is now regarded as a necessary part of every philosopher's technical equipment. Chellas here offers an (...) up-to-date and reliable guide essential for the student. (shrink)

In a paper on the logical work of the Jains, Graham Priest considers a consequence relation, semantically characterized, which has a natural analogue in modal logic. Here we give a syntactic/axiomatic description of the modal formulas which are consequences of the empty set by this relation, which is to say: those formulas which are, for every model, true at some point in that model.

Sections 1 and 2 respectively raise and settle the question of whether, if an affirmative modality collapses (reduces to the null modality, that is) in a normal modal logic, then all modalities of the same length collapse in that logic, while Section 3 considers some special cases of an analogous phenomenon for congruential modal logics, closing with a general question about collapsing modalities in this broader range of logics.

We construct a normal extension of K4 with the finite model property whose minimal tense extension is not complete with respect to Kripke semantics.

Given a 1-ary sentence operator , we describe L - another 1-ary operator - as as a left inverse of in a given logic if in that logic every formula is provably equivalent to L. Similarly R is a right inverse of if is always provably equivalent to R. We investigate the behaviour of left and right inverses for taken as the operator of various normal modal logics, paying particular attention to the conditions under which these logics are conservatively extended (...) by the addition of such inverses, as well as to the question of when, in such extensions, the inverses behave as normal modal operators in their own right. (shrink)

Global supervenience of beliefs about physical states of affairs on such states has strongly counter-intuitive consequences about what beliefs we can nomologically hold. This is an argument against a global supervenience of all mental properties on physical ones, and, since that is implied by strong supervenience, also against that as the preferred materialist thesis.

We give in this paper a sufficient condition, cast in semantic terms, for Hallden-completeness in normal modal logics, a modal logic being said to be Hallden-complete (or Ήallden-reasonable') just in case for any disjunctive formula provable in the logic, where the disjuncts have no propositional variables in common, one or other of those disjuncts is provable in the logic.

A sentence mentioning an object can be regarded as saying any one of several things about that object, without thereby being ambiguous. Some of the (logical) repercussions of this commonplace observation are recorded, and some critical discussion is provided of views which would appear to go against it.

The formula A (it is noncontingent whether A) is true at a point in a Kripke model just in case all points accessible to that point agree on the truth-value of A. We can think of -based modal logic as a special case of what we call the general modal logic of agreement, interpreted with the aid of models supporting a ternary relation, S, say, with OA (which we write instead of A to emphasize the generalization involved) true at a (...) point w just in case for all points x, y, with Swxy, x and y agree on the truth-value of A. The noncontingency interpretation is the special case in which Swxy if and only if Rwx and Rwy, where R is a traditional binary accessibility relation. Another application, related to work of Lewis and von Kutschera, allows us to think of OA as saying that A is entirely about a certain subject matter. (shrink)

The aim of this paper is to investigate the expressibility of classical propositional monomodal logics. To this end, a notion of embedding of one logic into another is introduced, which is a translation preserving theoremhood. Each translation trF is induced by a formula F of one variable p; it respects boolean connectives and translates a formula of a form □A into F). This notion enables to measure the expressibility of a logic by a number of logics embeddable into it. This (...) measure is calculated here for a large family of modal logics including K, K4, KB, K5, GL, GL.3, T, S4, B, S5, Grz, Grz.3, and provability logics. It is also shown that some of these logics are not embeddable into some others. (shrink)