Online research in philosophy

Modal logic is an enormous subject, and so any discussion of it must limit itself according to some set of principles. Modal logic is of interest to mathematicians, philosophers, linguists and computer scientists, for somewhat different reasons. Typically a philosopher may be interested in capturing some aspect of necessary truth, while a mathematician may be interested in characterizing a class of models having special structural features. For a computer scientist there is another criterion that is not as relevant for the other disciplines: a logic should be ‘well-behaved.’ This is, admittedly, a vague notion, but some things are clear enough. A logic that can be axiomatized is better than one that can’t be; a logic with a simple axiomatization is better yet; and a logic with a reasonably implementable proof procedure is best of all. My current interests are largely centered in computer science, and so I will only discuss well-behaved modal logics. My talk is organized into three sections, depending on the expressiveness of the modal logics considered: propositional; first-order with rigid designators; first-order with non-rigid designators. Even limited as I have said, the subject is a big one, and this talk can be no more than an outline. For a general discussion of propositional modal logic see [2]; for this and first-order modal logic see [9], [10] and [8]; for tableau systems in modal logic see [6].

Avicenna (d. 1037) and T?s? (d. 1274) have different doctrines on the contradiction and conversion of the absolute proposition. Following Avicenna's presentation of the doctrine in Pointers and reminders, and comparing it with what is given in T?s?'s commentary, allow us to pinpoint a major reason why Avicenna and T?s? have different treatments of the modal syllogistic. Further comparison shows that the syllogistic system Rescher described in his research on Arabic logic more nearly fits T?s? than Avicenna. This in turn has consequences for analysing Avicenna's logic, and for writing the history of a fascinating period of change and diversity in the discipline in the medieval Islamic world.

This long-awaited book replaces not one but both of Hughes and Cresswell's two previous classic studies of modal logic: An Introduction to Modal Logic and A Companion to Modal Logic . A New Introduction to Modal Logic has been completely rewritten by the authors to incorporate all the developments that have taken place since 1968 both in modal propositional logical and modal predicate logic, but without sacrificing the clarity of exposition and approachability that were essential features of the earlier works. The book takes readers through the most basic systems of modal prepositional logic right up to systems of modal predicate with identity. It deals with both technical developments such as completeness and incompleteness, and finite and infinite models, and discusses philosophical applications, especially, in the area of modal predicate logic.

Due to the influence of Nathan Salmon’s views, endorsement of the “flexibility of origins” thesis is often thought to carry a commitment to the denial of S4. This paper rejects the existence of this commitment and examines how Peacocke’s theory of the modal may accommodate flexibility of origins without denying S4. One of the essential features of Peacocke’s account is the identification of the Principles of Possibility, which include the Modal Extension Principle (MEP), and a set of Constitutive Principles. Regarding their modal status, Peacocke argues for the necessity of MEP, but leaves open the possibility that some of the Constitutive Principles be only contingently true. Here, I show that the contingency of the Constitutive Principles is inconsistent with the recursivity of MEP, and this makes the account validate S4. It is also shown that, compatibly with the necessity of the Constitutive Principles, the account can still accommodate intuitions about flexibility of origins. However, the account we end up with once those intuitions are consistently accommodated may not be satisfactory, and this opens up the debate about whether or not artefacts allow for some variation in their origins.

The goal of this paper is an interpretation of Aristotle's modal syllogistics closely oriented on the text using the resources of modern modal predicate logic. Modern predicate logic was successfully able to interpret Aristotle's assertoric syllogistics uniformly , that is, with one formula for universal premises. A corresponding uniform interpretation of modal syllogistics by means of modal predicate logic is not possible. This thesis does not imply that a uniform view is abandoned. However, it replaces the simple unity of the assertoric by the complex unity of the modal. The complexity results from the fact that though one formula for universal premises is used as the basis, it must be moderated if the text requires . Aristotle introduces his modal syllogistics by expanding his assertoric syllogistics with an axiom that links two apodictic premises to yield a single apodictic sentence . He thus defines a regular modern modal logic. By means of the regular modal logic that is thus defined, he is able to reduce the purely apodictic syllogistics to assertoric syllogistics. However, he goes beyond this simple structure when he looks at complicated inferences. In order to be able to link not only premises of the same modality, but also premises with different modalities, he introduces a second axiom, the T-axiom, which infers from necessity to reality or - equivalently - from reality to possibility. Together, the two axioms, the axiom of regularity and the T-axiom, define a regular T-logic. It plays an important role in modern logic. In order to be able to account for modal syllogistics adequately as a whole, another modern axiom is also required, the so-called B-axiom. It is very difficult to decide whether Aristotle had the B-axiom. The two last named axioms are sufficient to achieve the required contextual moderation of the basic formula for universal propositions.

Modern logicians have sought to unlock the modal secrets of Aristotle's Syllogistic by assuming a version of essentialism and treating it as a primitive within the semantics. These attempts ultimately distort Aristotle's ontology. None of these approaches make full use of tests found throughout Aristotle's corpus and ancient Greek philosophy. I base a system on Aristotle's tests for things that can never combine (polarity) and things that can never separate (inseparability). The resulting system not only reproduces Aristotle's recorded results for the apodictic syllogistic in the Prior Analytics but it also generates rather than assumes Aristotle's distinctions among 'necessary', 'essential' and 'accidental'. By developing a system around tests that are in Aristotle and basic to ancient Greek philosophy, the system is linked to a history of practices, providing a platform for future work on the origins of logic.

The principle of modal ubiquity - that every truth is necessary or contingent - and the validity of possibility introduction, are principles that any modal theory suffers for failing to accommodate. Advanced modal claims are modal claims about entities other than spatiotemporally unified individuals (perhaps, then, spatiotemporally disunified individuals, sets, numbers, properties, propositions and events). I show that genuine modal realism, as it has thus far been explicitly developed, and in so far as it deals with advanced modal claims, cannot accommodate the principles in question. On behalf of the genuine modal realist I motivate and propose a redundancy interpretation of advanced possibility claims and extend that interpretation to the cognate cases of necessity, impossibility and contingency. I then show that the problematic principles as they apply to advanced modal claims can be derived from these interpretations. I show further how the proposed interpretation enables the genuine modal realist to deal with a number of objections that centre on the alleged inadequacy of the genuine modal realist's expressive resources. I conclude that genuine modal realism emerges the stronger for having been shown capable of dealing with advanced modal claims on the basis of no conceptual and ontological resources beyond those it requires to deal with ordinary modal claims.

The Unprovability of Consistency is concerned with connections between two branches of logic: proof theory and modal logic. Modal logic is the study of the principles that govern the concepts of necessity and possibility; proof theory is, in part, the study of those that govern provability and consistency. In this book, George Boolos looks at the principles of provability from the standpoint of modal logic. In doing so, he provides two perspectives on a debate in modal logic that has persisted for at least thirty years between the followers of C. I. Lewis and W. V. O. Quine. The author employs semantic methods developed by Saul Kripke in his analysis of modal logical systems. The book will be of interest to advanced undergraduate and graduate students in logic, mathematics and philosophy, as well as to specialists in those fields.

Aristotle's Modal Logic presents a very new interpretation of Aristotle's logic by arguing that a proper understanding of the system depends on an appreciation of its connection to the metaphysics. Richard Patterson develops three striking theses in the book. First, there is a fundamental connection between Aristotle's logic of possibility and necessity, and his metaphysics, and that this connection extends far beyond the widely recognised tie to scientific demonstration and relates to the more basic distinction between the essential and accidental properties of a subject. Second, Aristotle's views on modal logic depend in very significant ways on his metaphysics without entailing any sacrifice in rigour. Third, once one has grasped the nature of the relationship, one can understand better certain genuine difficulties in the system of logic and appreciate its strengths in terms of the purposes for which it was created.

In this article, the author studies some central concepts in Avicenna's and sī's modal logics as presented in Avicenna's Al-Ish r t wa'l Tan īh t ( Pointers and Reminders ) and in sī's commentary. In this work, Avicenna introduces some remarkable distinctions in order to interpret Aristotle's modal syllogistic in the Prior Analytics . The author outlines a new interpretation of absolute sentences as temporally indefinite sentences and argues on the basis of this that Avicenna seems to subscribe to the Principle of Plenitude. He also shows that he has no valid proof of the modal conversion rules and that he uses some rather ad hoc distinctions to show that Aristotle's modal syllogistic is correct. The author also notes some interesting differences between Avicenna's and sī's approaches to modal logic.