In their simplest form, hybrid languages are propositional modal languages which can refer to states. They were introduced by Arthur Prior, the inventor of tense logic, and played an important role in his work: because they make reference to specific times possible, they remove the most serious obstacle to developing modal approaches to temporal representation and reasoning. However very little is known about the computational complexity of hybrid temporal logics.In this paper we analyze the complexity of the satisfiability problem of (...) a number of hybrid temporal logics: the basic hybrid language over transitive trees; nominal Until logic; and referential interval logic. We discuss the effects of including nominals, the @ operator, the somewhere modality E, and the difference operator D. Adding nominals to tense logic leads for several frame-classes to an increase in complexity of the satisfiability problem from PSPACE to EXPTIME. On transitive trees, however, the satisfiability problem for this language can be decided in PSPACE. Along the way we make a detour through hybrid propositional dynamic logic: we establish upper bounds for a number of temporal logics by generalizing results due to Passy and Tinchev [29] and De Giacomo [16]. We conclude with some remarks on the relevance of our results to description logic, and draw attention to the utility of the spypoint technique for proving upper and lower bounds. (shrink)
Hybridization is a method invented by Arthur Prior for extending the expressive power of modal languages. Although developed in interesting ways by Robert Bull, and by the Sofia school , the method remains little known. In our view this has deprived temporal logic of a valuable tool.The aim of the paper is to explain why hybridization is useful in temporal logic. We make two major points, the first technical, the second conceptual. First, we show that hybridization gives rise to well-behaved (...) logics that exhibit an interesting synergy between modal and classical ideas. This synergy, obvious for hybrid languages with full first-order expressive strength, is demonstrated for a weaker local language capable of defining the Until operator; we provide a minimal axiomatization, and show that in a wide range of temporally interesting cases extended completeness results can be obtained automatically. Second, we argue that the idea of sorted atomic symbols which underpins the hybrid enterprise can be developed further. To illustrate this, we discuss the advantages and disadvantages of a simple hybrid language which can quantify over paths. (shrink)
In this paper we discuss two hybrid languages, ℒ and ℒ, and provide them with complete axiomatizations. Both languages combine features of modal and classical logic. Like modal languages, they contain modal operators and have a Kripke semantics. Unlike modal languages, in these systems it is possible to 'label' states by using A and ↓ to bind special state variables.This paper explores the consequences of hybridization for completeness. As we shall show, the challenge is to blend the modal idea of (...) canonical models with the classical idea of witnessed maximal consistent sets. The languages ℒ and ℒ provide us with two extreme examples of the issues involved. In the case of ℒ, we can combine these ideas relatively straightforwardly with the aid of analogs of the Barcan axioms coupled with a modal theory of labeling. In the case of ℒ, on the other hand, although we can still formulate a theory of labeling, the Barcan analogs are not valid. We show how to overcome this difficulty by using COV*, an infinite collection of additional rules of proof which has been used in a number of investigations of extended modal logic. (shrink)
Traditional theistic proofs are often understood as evidence intended to compel belief in a divinity. John Clayton explores the surprisingly varied applications of such proofs in the work of philosophers and theologians from several periods and traditions, thinkers as varied as Ramanuja, al-Ghazali, Anselm, and Jefferson. He shows how the gradual disembedding of theistic proofs from their diverse and local religious contexts is concurrent with the development of natural theologies and atheism as social and intellectual options in early modern Europe (...) and America. Clayton offers a fresh reading of the early modern history of philosophy and theology, arguing that awareness of such history, and the local uses of theistic argument, offer important ways of managing religious and cultural difference in the public sphere. He argues for the importance of historically grounded philosophy of religion to the field of religious studies and public debate on religious pluralism and cultural diversity. (shrink)
Cambridge professor Simon Blackburn is best known to the general public as the author of several books of popular philosophy such as ink, Being Good andTruth: a Guide for the Perplexed. Academic philosophers also know him as the author of one of the most important books of contemporary moral philosophy, Ruling Passions, and as a former editor of the leading journal Mind.
Cambridge professor Simon Blackburn is best known to the general public as the author of several books of popular philosophy such as ink, Being Good andTruth: a Guide for the Perplexed. Academic philosophers also know him as the author of one of the most important books of contemporary moral philosophy, Ruling Passions, and as a former editor of the leading journal Mind.
Simon Blackburn has not shied away from the use of vivid imagery in developing, over a long and prolific career, a large-scale philosophical vision. Here one might think, for instance, of ‘Practical Tortoise Raising’ or ‘Ramsey's Ladder’ or ‘Frege's Abyss’. Blackburn develops a ‘quasi-realist’ account of many of our philosophical and everyday commitments, both theoretical (e.g., modality and causation) and practical (e.g., moral judgement and normative reasons). Quasi-realism aims to provide a naturalistic treatment of its targeted phenomena while (...) earning the right to deploy all of the ‘trappings’ of realism—i.e., while eschewing any idea that our normal thought and talk about such phenomena are pervasively in error. The quasi-realist project is that of explaining how (as Huw Price puts it here) ‘the folk come to “talk the realist talk” without committing ourselves—us theorists, as it were—to “walking the metaphysical walk”’ (p. 136). Quasi-realism, too, can speak of truth, facts, properties, belief, knowledge, and so on. The imagery in this collection also abounds, though, in capturing a different view of quasi-realism: No fewer than three of the contributors picture Blackburn as wanting to have his cake and eat it too (Louise Antony asking, in addition, ‘Who doesn't? It's cake’ [p. 19]). (shrink)
