The system obtained by adding full propositional quantification to S5 is known to be decidable, while that obtained by doing so for T is known to be recursively intertranslatable with full second-order logic. Recently it was shown that the system with two S5 operators and full propositional quantification is also recursively intertranslatable with second-order logic. This note establishes that the map assigning [1][2]p to \squarep provides a validity and satisfaction preserving translation between the T system and the double S5 system, (...) thus providing an easier proof of the recent result. (shrink)

Patrick Grim has presented arguments supporting the intuition that any notion of a totality of truths is incoherent. We suggest a natural semantics for various logics of belief which reflect Grim’s intuition. The semantics is a topological semantics, and we suggest that the condition can be interpreted to reflect Grim’s intuition. Beyond this, we present a natural canonical topological model for K4 and KD4.

We prove the following surprising property of Heyting's intuitionistic propositional calculus, IpC. Consider the collection of formulas, φ, built up from propositional variables (p,q,r,...) and falsity $(\perp)$ using conjunction $(\wedge)$ , disjunction (∨) and implication (→). Write $\vdash\phi$ to indicate that such a formula is intuitionistically valid. We show that for each variable p and formula φ there exists a formula Apφ (effectively computable from φ), containing only variables not equal to p which occur in φ, and such that for (...) all formulas ψ not involving $p, \vdash \psi \rightarrow A_p\phi$ if and only if $\vdash \psi \rightarrow \phi$ . Consequently quantification over propositional variables can be modelled in IpC, and there is an interpretation of the second order propositional calculus, IpC2, in IpC which restricts to the identity on first order propositions. An immediate corollary is the strengthening of the usual interpolation theorem for IpC to the statement that there are least and greatest interpolant formulas for any given pair of formulas. The result also has a number of interesting consequences for the algebraic counterpart of IpC, the theory of Heyting algebras. In particular we show that a model of IpC2 can be constructed whose algebra of truth-values is equal to any given Heyting algebra. (shrink)

Examples cited by Feldman, Lehrer and others of true beliefs that are justified, but not by false lemmas, turn out under scrutiny to involve false lemmas after all. In each case there is an EG inference whose conclusion is unwarranted unless its base instance is false. A shift to non-deductive justification does not avert the difficulty. The relation of this result to non-inferential Gettier cases is suggested.

We show that (contrary to the parallel case of intuitionistic logic, see [7], [4]) there does not exist a translation fromS42 (the propositional modal systemS4 enriched with propositional quantifiers) intoS4 that preserves provability and reduces to identity for Boolean connectives and.

In this paper I shall present some of the results I have obtained on modal theories which contain quantifiers for propositions. The paper is in two parts: in the first part I consider theories whose non-quantificational part is S5; in the second part I consider theories whose non-quantificational part is weaker than or not contained in S5. Unless otherwise stated, each theory has the same language L. This consists of a countable set V of propositional variables pl, pa, ... , (...) the operators v (or), ~ (not) and □ (necessarily), the universal quantifier (p), p a propositional variable, and brackets ( and ). The formulas of L are then defined in the usual way. (shrink)

I argue that it is not ideally rational to believe that some of one's current beliefs are false, despite the impressive inductive evidence concerning others and our former selves. One's own current beliefs represent a commitment which would be undermined by taking some of them to be false. The nature of this commitment is examined in the light of Nagel's distinction between subjective and objective points of view. Finally, I suggest how we might acknowledge our fallibility consistently with this special (...) attitude to our own beliefs. (shrink)

I am interested in extending modal calculi by adding propositional quantifiers, given by the rules for quantifier introduction: provided that p does not occur free in A.

A propositional system of modal logic is second-order if it contains quantifiers ∀p and ∃p, which, in the standard interpretation, are construed as ranging over sets of possible worlds (propositions). Most second-order systems of modal logic are highly intractable; for instance, when augmented with propositional quantifiers, K, B, T, K4 and S4 all become effectively equivalent to full second-order logic. An exception is S5, which, being interpretable in monadic second-order logic, is decidable.

This is an exploration of a cluster of related logical results. Taken together these seem to have something philosophically important to teach us: something about knowledge and truth and something about the logical impossibility of totalities of knowledge and truth. The book includes explorations of new forms of the ancient and venerable paradox of the :Liar, applications and extensions of Kaplan and Montague's paradox of the Knower, generalizations of Godel's work on incompleteness, and new uses of Cantorian diagonalization. Throughout, the (...) concern is with metaphysical and epistemological implications of these results: what such results may have to teach us about knowledge, truth, and possibility. (shrink)

In The Virtues of Ignorance the author demonstrates that classical theories of virtue are flawed and developes a consequentialist theory of virtue. ;Virtues are excellences of character. They are traits which are considered to be valuable in some way. A person who is virtuous is one who has a tendency to act well. Classical philosophers, such as Plato and Aristotle, believed that virtues, as human excellences, could not involve ignorance in any way. On their view, the virtuous agent, when acting (...) well, knows what she is doing under the description which makes the action conform to virtue. For example, the generous person must know that she is, in fact, helping others in need. Yet, there is a class of virtues, the "virtues of ignorance," which essentially involve ignorance. For example, modesty requires that the agent underestimate self worth. In the first portion of the dissertation, this class of virtues is discussed, and problems raised for the traditional theories of virtue. ;In the latter portion of the dissertation, the author outlines a consequentialist theory of virtue. It is argued that the only hope of developing a unified theory of virtue is along consequentialist lines, because attempts at identifying the psychological state which is necessary for virtue have failed. On a consequentialist theory, a virtue is defined in terms of its consequences. Thus, the moral quality of the character trait is determined by how much good it brings about . The virtues of ignorance can easily be accommodated in such a theory because these traits do bring about good, even if the agent possessing the trait acts out of ignorance. (shrink)

In this highly original text, Christopher Steinsvold explores an alternative semantics for logics of rational belief. Topologies, as mathematical objects, are typically interpreted in terms of space; here topologies are re-interpreted in terms of an agent with rational beliefs. The topological semantics tells us that the agent can never, in principle, know everything; that the agent's beliefs can never be complete. -/- A number of completeness proofs are given for a variety of logics of rational belief. Beyond this, the author (...) explores the philosophical question of why our beliefs can never be complete, and considers the possibility that a totality of truths is a dialethia. -/- This work will be of interest to all philosophers interested in epistemology, and modal logicians as well. (shrink)