My essay, “Multi-modal argumentation” was published in the journal, _Philosophy of the Social Sciences,_ in 1994. This information appeared again in my book, _Coalescent argumentation_ in 1997. In the ensuing twenty years, there have been many changes in argumentation theory, and I would like to take this opportunity to examine my now middle-aged theory in light of the developments in our discipline. I will begin by relating how a once keen intended lawyer and then formal logician ended up (...) in argumentation theory. (If you do not care to read this bit of autobiography, skip to Section 2). (shrink)

One of the standard views on plural quantification is that its use commits one to the existence of abstract objects–sets. On this view claims like ‘some logicians admire only each other’ involve ineliminable quantification over subsets of a salient domain. The main motivation for this view is that plural quantification has to be given some sort of semantics, and among the two main candidates—substitutional and set-theoretic—only the latter can provide the language of plurals with the desired expressive power (given that (...) the nominalist seems committed to the assumption that there can be at most countably many names). To counter this approach I develop a modal-substitutional semantics of plural quantification (on which plural variables, roughly speaking, range over ways names could be) and argue for its nominalistic acceptability. (shrink)

Russell's objections to object-theory have been refuted by the proofs of the consistency of Meinong's system given by various writers. These proofs exploit technical distinctions that Meinong apparently uses very little if at all. Instead, Meinong introduces a theoretical postulate called the modal moment. I describe this postulate and its place in Meinong's system, and I argue that it has been much under-rated by Meinong's logician expositors.

My essay, “Multi-modal argumentation” was published in the journal, _Philosophy of the Social Sciences,_ in 1994. This information appeared again in my book, _Coalescent argumentation_ in 1997. In the ensuing twenty years, there have been many changes in argumentation theory, and I would like to take this opportunity to examine my now middle-aged theory in light of the developments in our discipline. I will begin by relating how a once keen intended lawyer and then formal logician ended up (...) in argumentation theory. (If you do not care to read this bit of autobiography, skip to Section 2). (shrink)

My essay, “Multi-modal argumentation” was published in the journal, _Philosophy of the Social Sciences,_ in 1994. This information appeared again in my book, _Coalescent argumentation_ in 1997. In the ensuing twenty years, there have been many changes in argumentation theory, and I would like to take this opportunity to examine my now middle-aged theory in light of the developments in our discipline. I will begin by relating how a once keen intended lawyer and then formal logician ended up (...) in argumentation theory. (If you do not care to read this bit of autobiography, skip to Section 2). (shrink)

In this paper, I attempt to explain one of the most controversial views attributed to the Stoic Chrysippus: that the impossible can follow from the possible. My solution finds in Chrysippus a distinction later made by the medieval logician John Buridan: that between being possible (there being a state of affairs that may occur) and being possibly-true (there being some proposition whose truth-conditions are that state of affairs). Buridan and Chrysippus have radically opposing views on the nature of propositions. (...) What their conceptions share is the conclusion that at least some propositions must be contingent beings. They argue for this while maintaining a rigorous commitment to the view that propositions are strictly bivalent. In 2. I explain the Chrysippean passage in terms of a distinction Buridan makes explicitly. In 3. I show how the distinction follows implicitly from the Stoic theory of quantification. In 4. I compare the modality with other aspects of Stoic logic. In 5. I discuss how the distinction behaves in the future tense. (shrink)

The modal logician's notion of possible world and the computer scientist's notion of state of a machine provide a point of commonality which can form the foundation of a logic of action. Extending ordinary modal logic with the calculus of binary relations leads to a very natural logic for describing the behavior of computer programs.

In lieu of an abstract, here is a brief excerpt of the content:336 BOOK REVIEWS The Nature of All Being: A Study of Wittgenstein's Modal Atomism. By RAYMOND BRADLEY. New York and Oxford: The Oxford University Press, 1992. Pp. xxi + 244. $39.95. Bradley offers as his point of departure this epigraph from Wittgenstein 's Notebooks 1914-1916, written 22 January, 1915: My whole task consists in giving the nature of the proposition. In giving the nature of all being. (And (...) here being does not stand for existence.) His aim in this contribution to the corpus of Wittgenstein scholarship is to " bring into sharp focus the crucial role which modal notions-the notions of necessity, contingency, possibility, impossibility, and so onplay in Wittgenstein's early thinking" (xiii). At this task the book is a clear success. I know of no other work on the Tractatus which is so thoroughly dedicated to the role of modal notions, nor of any which does so much to bring the early Wittgenstein into conversation with contemporary philosophers who work on modal logic and who talk about possible worlds. Bradley convincingly argues that Wittgenstein is not only the largely unrecognized Urvater of modal logic in the twentieth century, with a stature that ought to be equivalent to that of Leibniz, he also presented arguments that even now have an important role to play in contemporary debates among possible world theorists and practitioners of modal logic. Philosophers who are interested in modal logic, especially its history, or in the logical structures of the Tractatus, will find very much of interest in this book. Those who, on the other hand, are chiefly interested in Wittgenstein's views on ethics, the aesthetic, God, or the mystical, or who are interested in the relation between the Tractatus and his later work, will find little here to their taste. The book's structure perspicuously tracks Bradley's argument. Chapter One sets the reader straight on what modal atomism is, giving the history of Wittgenstein's association with Russell, and establishing that the atomism of the Tractatus is in fact modal in nature. Chapter Two argues that Wittgenstein is a possibilist, that is, that he asserts the reality of merely possible entities and that his work is consistent with the modern modal logic called SS. Chapter Three turns to the ontological side, showing how Wittgenstein's commitments to atomic objects as the " substance of the world " entails certain modal relations among objects and the complex facts which they may form. In Chapter Four logic and ontology are brought together in an account of the mirroring relation through which language depicts the world, with attention to a set of intricately argued principles that makes this relation possible. The final chapter brings Wittgenstein's thought to bear on the BOOK REVIEWS 337 whole range of contemporary problems and issues in modal logic, demonstrating by example the author's contention that the Tractatus, rightly attended to, contains important contributions, still unappre· ciated, to the field. Bradley's overall intentions could not be more clear. His strategy for each chapter is clear. Each chapter is divided into between ten and twenty short sections each of which has a specific and well-defined topic. The argumentation within each such section, with a pardonable lapse or two, is well-structured and easy to follow, even for one whose expertise in modal logic is not honed by regular practice. And yet the hook has, on reading, the feel of a very dense landscape of trees, with· out much overview of the forest. This impression is given, I believe, by the style of argumentation. While the topics are sequentially related and build to a general picture, the arguments that must be articulated and answered in each section have little relation, one to the next: a problem in this view, a contradiction or a paradox in that, a resolution, and then on to the next section, where an apparent inconsistency of some sort must be addressed, and so on. There is the feeling of dealing serially with connected topics, but not the feeling of cumulative argumentation building toward a conclusion. A conclusion, however, there is: that Wittgenstein was the modal logician par excellence of the early 20th century, who has much... (shrink)

Let us call “tense logic” the programme of explaining tense in natural languages by means of a model theory similar in structure to possible worlds semantics for modality. This programme would make the following claims.

In our everyday discourse, most of us use modal statements to express possibility, necessity, or contingency. Logicians, linguists, and philosophers of language tend to use the possible world discourse to analyse the semantics of this kind of sentences. There is a disadvantage of this method: in the usual Quinean meta-ontology it commits the users to the existence of possible worlds. Even though there are many theories on metaphysics of these possible worlds, I will focus on the fictionalist approach, which (...) aims to suspend the undesired ontological commitment. Therefore it is an anti-realist theory. The fictionalist strategy is based on the idea that according to our common-sense ontology, fictions and fictional entities do not exist, and one can treat the possible world discourse as a fiction, and the possible worlds as fictional entities. However, in the last two decades, realist theories about fictions and fictional entities have arisen, especially the abstract artefact theory. In this paper I try to revise the fictionalist approach to modality by keeping the main idea that possible worlds are fictional entities, while accepting that fictional entities are abstract artefacts. The result is a new realist theory of modal fictionalism. (shrink)

The received view says that possibility is the dual of necessity: a proposition is (metaphysically, logically, epistemically etc.) possible iff it is not the case that its negation is (metaphysically, logically, epistemically etc., respectively) necessary. This reading is usually taken for granted by modal logicians and indeed seems plausible when dealing with logical or metaphysical possibility. But what about epistemic possibility? We argue that the dual definition of epistemic possibility in terms of epistemic necessity generates tension when reasoning about (...) non-idealized agents and is a problem of concern for most hyperintensional epistemic logics that alleviate the problem of logical omniscience. The tension is particularly evident when knowledge is taken as a primitive to define other epistemic concepts, such as justification and belief, as done in the knowledge-first tradition. We propose a non-dual interpretation of epistemic possibility, employing a hyperintensionality filter similar to the one that makes the corresponding epistemic necessity operator hyperintensional. We employ the proposed semantics to model Stalnaker's belief as epistemic possibility of knowledge and provide a sound and complete axiomatization for a hyperintensional version of his bimodal logic of knowledge and belief. (shrink)

The combination of quantifiers with a semantics for epistemic operators in a modal framework is one of the major contributions of Hintikka in intensional logic. Hintikka’s starting point is his diagnosis of the failure of existential generalization and the substitution of identicals in terms of referential multiplicity. In this paper, I introduce Hintikka as a free logician. Indeed, Hintikka’s first-order epistemic logic is grounded on a logic free of ontological presuppositions with respect to singular terms. It is also (...) a logic free of presuppositions of uniqueness of reference. After having focused on the use of quantifiers and singular terms in Hintikka’s epistemic logic, I discuss some consequences from a semantico-logical perspective, but also from a philosophical one. By arguing against the so-called contingent a priori truths defended by Kripke, I conclude with a proposal in favour of Hintikka’s non-rigid interpretation of proper names. (shrink)

ABSTRACTHegel's Phenomenology of Spirit provides a fascinating picture of individual minds caught up in “recognitive” relations so as to constitute a realm—“spirit”—which, while necessarily embedded in nature, is not reducible to it. In this essay I suggest a contemporary path for developing Hegel's suggestive ideas in a way that broadly conforms to the demands of his own system, such that one moves from logic to a philosophy of mind. Hence I draw on Hegel's “subjective logic”, understood in the light of (...) modern modal logic, in an attempt to model the way minds might be thought as connected by way of shared intentional contents. Here, we should not be surprised at some of the parallels that emerge between the approaches of Hegel and the modal logician Arthur Prior, as Prior had testified to the influence of his teacher, John N. Findlay, who himself had strong Hegelian leanings. In the final section, Robert Stalnaker's version of possible-world semantics is suggested as a framework within which Hegel's recognitive account of the mind might be understood. (shrink)

ABSTRACT: The modal systems of the Stoic logician Chrysippus and the two Hellenistic logicians Philo and Diodorus Cronus have survived in a fragmentary state in several sources. From these it is clear that Chrysippus was acquainted with Philo’s and Diodorus’ modal notions, and also that he developed his own in contrast of Diodorus’ and in some way incorporated Philo’s. The goal of this paper is to reconstruct the three modal systems, including their modal definitions and (...) modal theorems, and to make clear the exact relations between them; moreover, to elucidate the philosophical reasons that may have led Chrysippus to modify his predessors’ modal concept in the way he did. It becomes apparent that Chrysippus skillfully combined Philo’s and Diodorus’ modal notions, with making only a minimal change to Diodorus’ concept of possibility; and that he thus obtained a modal system of modalities (logical and physical) which fit perfectly fit into Stoic philosophy. (shrink)

Modal logics, originally conceived in philosophy, have recently found many applications in computer science, artificial intelligence, the foundations of mathematics, linguistics and other disciplines. Celebrated for their good computational behaviour, modal logics are used as effective formalisms for talking about time, space, knowledge, beliefs, actions, obligations, provability, etc. However, the nice computational properties can drastically change if we combine some of these formalisms into a many-dimensional system, say, to reason about knowledge bases developing in time or moving objects. (...) To study the computational behaviour of many-dimensional modal logics is the main aim of this book. On the one hand, it is concerned with providing a solid mathematical foundation for this discipline, while on the other hand, it shows that many seemingly different applied many-dimensional systems (e.g., multi-agent systems, description logics with epistemic, temporal and dynamic operators, spatio-temporal logics, etc.) fit in perfectly with this theoretical framework, and so their computational behaviour can be analyzed using the developed machinery. We start with concrete examples of applied one- and many-dimensional modal logics such as temporal, epistemic, dynamic, description, spatial logics, and various combinations of these. Then we develop a mathematical theory for handling a spectrum of 'abstract' combinations of modal logics - fusions and products of modal logics, fragments of first-order modal and temporal logics - focusing on three major problems: decidability, axiomatizability, and computational complexity. Besides the standard methods of modal logic, the technical toolkit includes the method of quasimodels, mosaics, tilings, reductions to monadic second-order logic, algebraic logic techniques. Finally, we apply the developed machinery and obtained results to three case studies from the field of knowledge representation and reasoning: temporal epistemic logics for reasoning about multi-agent systems, modalized description logics for dynamic ontologies, and spatio-temporal logics. The genre of the book can be defined as a research monograph. It brings the reader to the front line of current research in the field by showing both recent achievements and directions of future investigations (in particular, multiple open problems). On the other hand, well-known results from modal and first-order logic are formulated without proofs and supplied with references to accessible sources. The intended audience of this book is logicians as well as those researchers who use logic in computer science and artificial intelligence. More specific application areas are, e.g., knowledge representation and reasoning, in particular, terminological, temporal and spatial reasoning, or reasoning about agents. And we also believe that researchers from certain other disciplines, say, temporal and spatial databases or geographical information systems, will benefit from this book as well. Key Features: Integrated approach to modern modal and temporal logics and their applications in artificial intelligence and computer science Written by internationally leading researchers in the field of pure and applied logic Combines mathematical theory of modal logic and applications in artificial intelligence and computer science Numerous open problems for further research Well illustrated with pictures and tables. (shrink)

This paper provides several examples of how modal logic can be used in studying the XML document navigation language XPath. More specifically, we derive complete axiomatizations, computational complexity and expressive power results for XPath fragments from known results for corresponding logics. A secondary aim of the paper is to introduce XPath in a way that makes it accessible to an audience of modal logicians.

Modal logic is one of philosophy’s many children. As a mature adult it has moved out of the parental home and is nowadays straying far from its parent. But the ties are still there: philosophy is important to modal logic, modal logic is important for philosophy. Or, at least, this is a thesis we try to defend in this chapter. Limitations of space have ruled out any attempt at writing a survey of all the work going on (...) in our field—a book would be needed for that. Instead, we have tried to select material that is of interest in its own right or exemplifies noteworthy features in interesting ways. Here are some themes that have guided us throughout the writing: • The back-and-forth between philosophy and modal logic. There has been a good deal of give-and-take in the past. Carnap tried to use his modal logic to throw light on old philosophical questions, thereby inspiring others to continue his work and still others to criticise it. He certainly provoked Quine, who in his turn provided—and continues to provide—a healthy challenge to modal logicians. And Kripke’s and David Lewis’s philosophies are connected, in interesting ways, with their modal logic. Analytic philosophy would have been a lot different without modal logic! • The interpretation problem. The problem of providing a certain modal logic with an intuitive interpretation should not be conflated with the problem of providing a formal system with a model-theoretic semantics. An intuitively appealing model-theoretic semantics may be an important step towards solving the interpretation problem, but only a step. One may compare this situation with that in probability theory, where definitions of concepts like ‘outcome space’ and ‘random variable’ are orthogonal to questions about “interpretations” of the concept of probability. • The value of formalisation. Modal logic sets standards of precision, which are a challenge to—and sometimes a model for—philosophy. Classical philosophical questions can be sharpened and seen from a new perspective when formulated in a framework of modal logic. On the other hand, representing old questions in a formal garb has its dangers, such as simplification and distortion. • Why modal logic rather than classical (first or higher order) logic? The idioms of modal logic—today there are many!—seem better to correspond to human ways of thinking than ordinary extensional logic. (Cf. Chomsky’s conjecture that the NP + VP pattern is wired into the human brain.) In his An Essay in Modal Logic (1951) von Wright distinguished between four kinds of modalities: alethic (modes of truth: necessity, possibility and impossibility), epistemic (modes of being known: known to be true, known to be false, undecided), deontic (modes of obligation: obligatory, permitted, forbidden) and existential (modes of existence: universality, existence, emptiness). The existential modalities are not usually counted as modalities, but the other three categories are exemplified in three sections into which this chapter is divided. Section 1 is devoted to alethic modal logic and reviews some main themes at the heart of philosophical modal logic. Sections 2 and 3 deal with topics in epistemic logic and deontic logic, respectively, and are meant to illustrate two different uses that modal logic or indeed any logic can have: it may be applied to already existing (non-logical) theory, or it can be used to develop new theory. (shrink)

This paper provides several examples of how modal logic can be used in studying the XML document navigation language XPath. More specifically, we derive complete axiomatizations, computational complexity and expressive power results for XPath fragments from known results for corresponding logics. A secondary aim of the paper is to introduce XPath in a way that makes it accessible to an audience of modal logicians.

Induction, conceived as the class of rational non-truth preserving inferences, has been a perennial problem in philosophy. Aside from the problem of justification of induction, a less debated issue is the problem of properly describing inductive inferences. The purpose of this book is to conceptually investigate this descriptive problem of induction from the standpoint of the nonmonotonic logical tradition raised inside the field of Artificial Intelligence in the last thirty years. As we try to show, an essential part of this (...) enterprise involves the analysis of a pragmatically oriented notion of probability, called by us inductive plausibility. This in turn is made from the viewpoint of paraconsistency ? which according to some is the unavoidable companion of nonmonotonicity ? and modal logic. We therefore put forward a nonmonotonic, paraconsistent and modal approach to induction and plausibility. The book shall be useful to philosophers, logicians and computer scientists interested in the problem of induction in general and in the logical analysis of induction and plausibility in particular. (shrink)

ABSTRACT George Gargov was an active pioneer in the ‘Sofia School’ of modal logicians. Starting in the 1970s, he and his colleagues expanded the scope of the subject by introducing new modal expressive power, of various innovative kinds. The aim of this paper is to show some general patterns behind such extensions, and review some very general results that we know by now, 20 years later. We concentrate on simulation invariance, decidability, and correspondence. What seems clear is that (...) ‘modal logic’ as a genre of logical systems has a much wider scope than originally conceived, and that we have not reached its limits yet. (shrink)

ABSTRACT George Gargov was an active pioneer in the ‘Sofia School’ of modal logicians. Starting in the 1970s, he and his colleagues expanded the scope of the subject by introducing new modal expressive power, of various innovative kinds. The aim of this paper is to show some general patterns behind such extensions, and review some very general results that we know by now, 20 years later. We concentrate on simulation invariance, decidability, and correspondence. What seems clear is that (...) ‘modal logic’ as a genre of logical systems has a much wider scope than originally conceived, and that we have not reached its limits yet. (shrink)

Una interpretación particularmente influyente de la teoría de la implicación lógica de Bertrand Russell y Clarence I. Lewis es la propuesta por Quine en su artículo “Reply to Professor Marcus”. Allí Quine sostiene que la lógica modal de Lewis nació en pecado: el pecado de confundir uso con mención, ya que cuando se afirma que una oración implica lógicamente a otra, estas oraciones no están siendo usadas sino mencionadas. Según la interpretación de Quine, Clarence I. Lewis persistió en el (...) error de Russell, que consistió en confundir la implicación material con la implicación lógica, y confundió la implicación estricta con la implicación lógica. Estos ‘pecados’ de Russell y Lewis pueden entenderse mejor si se tiene en cuenta que tanto Lewis como Russell sostienen que los axiomas y los teoremas en un cálculo lógico se usan de dos maneras: (a) como premisas a partir de las cuales se obtienen nuevos teoremas, y (b) como reglas de inferencia mediante las cuales se obtienen nuevos teoremas. En efecto, estos autores suscriben la teoría del doble uso de los principios lógicos que parece originarse en la obra de Peano y que ha sido casi completamente ignorada en la literatura acerca de la historia de la lógica. -/- A particularly influential interpretation of Bertrand Russell’s and Clarence I. Lewis’s theory of logical implication is the one proposed by Quine in his article "Reply to Professor Marcus". In that article Quine argues that Lewis's modal logic was born in sin: the sin of confusing use with mention, since when it is stated that a sentence logically implies another sentence, these sentences are not being used but mentioned. According to Quine's interpretation, Clarence I. Lewis persisted in Russell's error, which consisted in confusing material implication with logical implication, and confused strict implication with logical implication. These 'sins' of Russell and Lewis can be better understood if we take into account that both logicians argue that the axioms and theorems in a logical calculus are used in two ways: (a) as premises from which new theorems are obtained, and (b) as rules of inference by which new theorems are obtained. In fact, these authors subscribe to the theory of the double use of logical principles that seems to originate in Peano's work and that has been almost completely ignored in the literature about the history of logic. (shrink)

Salutary reading for all philosophers, and not only for inductive logicians, philosophers of science and law, this important book presents an elaborate theory of inductive reasoning whose substantive features are as strikingly original as the approach is rare. First, the theory is based on concrete, real, actual, and significant instances of inductive reasoning, e.g., Karl von Frisch’s work on bees; that is, though its aim is genuinely theoretical in the sense that it engages in the proper amounts of idealization, abstraction, (...) systematization, precision, and rigor, it never loses sight of the fact that "theories in the philosophy of science, like theories in science itself, have to face up to the challenge of appropriately accredited experience within their appointed domains". Second, a central subject matter being analyzed is legal reasoning, that is, the types of proofs and arguments used by juries, judges, and lawyers in the Anglo-American system of jurisprudence, where criminal charges have to be proved beyond reasonable doubt and civil cases have to be argued on the balance of probability. Thus, in effect, Cohen’s synthesis of juridical and of scientific reasoning is an admirable example of the bridging of two "cultures." Third, and most importantly, the concept of "inductive probability," in terms of which Cohen makes sense out of his subject matter, is incommensurable with that of "mathematical probability," namely the classical calculus of chance according to which probability is measurable, additive, transitive, and obeys a multiplicational principle for conjunction and a complementational one for negation. This is not to say that "inductive probability" is a hazy, mysterious, or unstructured concept, and Cohen goes to great lengths to show that it is ordinal, modal, and formally analyzable; what it does mean is that essential types of reasoning in science and in law involve probable inferences whose probabilities are nonquantitative. Hence, though Cohen does not stress the qualitative nature of inductive probability, his book will be welcome by those who conceive of philosophy as scientia qualitatum. Specific conclusions reached by Cohen are that, in general, probability may be conceived as degree of provability, that differences in proof-rules generate different special cases of probability, that inductive probability may be viewed as the special case where the proof-rules are incomplete, that inductive probability is to inductive support as deducibility is to logical truth, and that some recent epistemological scepticism may be refuted by Cohen’s inductive logic. Finally, he sees himself in the British inductivist tradition, interpreting his predecessors as groping toward a logic of inductive support and probability rather than a methodology of discovery: from Bacon he develops the notion of eliminative induction but rejects that of induction by enumeration; Mill’s canons are analyzed as special cases of Cohen’s own "method of relevant variables," except for the method of residues, which is rejected as invalid ; he takes his critique of mathematical probability as a vindication of Hume’s thesis that "probability or reasoning from conjecture may be divided into two kinds, viz., that which is founded on chance and that which arises from causes" ; Whewell’s consilience of induction is justified ; and from Keynes, he develops the notion of the "weight" of evidence but rejects his mathematicist approach.—M. A. F. (shrink)

Many logicians have tried to formalize a modal logic from the Prior Analytics, but the general view is that Aristotle has failed to offer a consistent modal logic there. This paper explains that Aristotle is not interested in modal logic as such. Modalities for him pertain to the relations of predication, without challenging the assertoric system of deductions simpliciter. Thus, demonstrations or dialectical deductions have modal predicates and yet are still deductions simpliciter. It is a matter (...) of distinguishing inferential necessity that applies to every deduction from the modal predicates in the two premises and conclusion. The modality of demonstrations can be either necessary or possible. The necessity is predicative, i.e., independent of inferential necessity. While the possible demonstration challenges the predicative necessity of the necessary demonstration, it preserves the inferential necessity of the deduction simpliciter. (shrink)

In spite of the philosophical significance of higher-order modal logic, the modal logician's main concern has been with sentential logic. In this paper we do not intend to go into philosophical details, but we only remark that higher-order modal logic has a close relationship with Montague's well-known idea of “universal grammar”, which is an ambitious attempt to build a logical theory of natural languages with exact syntax and semantics, comparable with the artificial languages of mathematical logic. (...) For this matter, the reader can consult, e.g., Montague [8], [9] and Gallin [2]. (shrink)

This monograph explores the logical systems of early logicians in the Arabic tradition from a theoretical perspective, providing a complete panorama of early Arabic logic and centering it within an expansive historical context. By thoroughly examining the writings of the first Arabic logicians, al-Fārābī, Avicenna and Averroes, the author analyzes their respective theories, discusses their relationship to the syllogistics of Aristotle and his followers, and measures their influence on later logical systems. Beginning with an introduction to the writings of the (...) most prominent Arabic logicians, the author scrutinizes these works to determine their categorical logic, as well as their modal and hypothetical logics. Where most other studies written on this subject focus on the Arabic logicians’ epistemology, metaphysics, and theology, this volume takes a unique approach by focusing on the actual technical aspects and features of their logics. The author then moves on to examine the original texts as closely as possible and employs the symbolism of modern propositional, predicate, and modal logics, rendering the arguments of each logician clearly and precisely while clarifying the theories themselves in order to determine the differences between the Arabic logicians’ systems and those of Aristotle. By providing a detailed examination of theories that are still not very well-known in Western countries, the author is able to assess the improvements that can be found in the Arabic writings, and to situate Arabic logic within the breadth of the history of logic. This unique study will appeal mainly to historians of logic, logicians, and philosophers who seek a better understanding of the Arabic tradition. It also will be of interest to modern logicians who wish to delve into the historical aspects and progression of their discipline. Furthermore, this book will serve as a valuable resource for graduate students who wish to complement their general knowledge of Arabic culture, logic, and sciences. (shrink)

BERTRAND RUSSELL ON MODALITY AND LOGICAL RELEVANCE - SECOND EDITION of 2015. Praise for the first edition of 1999: "In the twenty-nine years since Russell's death, much of the major scholarship has drawn heavily on his manuscripts and unpublished correspondence. The author shows that the published Russell is capable of new interpretations; in particular, that modal notions such as possibility have a greater place in various aspects of his logical and philosophical thought than has been previously imagined." -Ivor Grattan-Guinness, (...) Foreword to the first edition. EXCERPT FROM PUBLISHED REVIEW: "Dejnožka's book is the first full-length study of modality in Russell. It is useful for its very full survey of passages in which Russell makes use of or alludes to modal notions. Dejnožka's command of Russell's huge output is indeed impressive and his utilization of it thorough...." - Nicholas Griffin, Studia Logica. EXCERPT FROM PUBLISHED REVIEW: "Dejnožka's book raises a very important point in the history of formal logic. Until now the major studies on this topic have drawn heavily on the development of classical logic as standardized by Gottlob Frege and Bertrand Russell. Dejnožka challenges the reader to open his mind for a new interpretation of Russell's work, in particular that modal and relevance notions have a greater place in his philosophy of logic than has been stressed before.... "Dejnožka rightly observes that many of Russell's insights on modality are a result of his discussions with Hugh MacColl, who was indeed the first to seriously attempt to develop formal modal logic. This particularly applies to Russell's conception of a modal logic without modal operators....That is, classical logic can be used to simulate modal expressions. Thus, the notions of (logical) necessity and possibility are not 'fundamental notions'....On this basis, Dejnožka develops a higher level of modality, where the quantification scope extends to the predicates yielding what Russell calls 'fully general propositions'.... "The best studied translation method is known as the standard translation, and it is quite compatible with Dejnožka's suggestions.... "Dejnožka's book is full of material which stimulates [one] to rethink Russell's philosophy of logic and...it is greatly to the author's credit that he brings to light such a wealth of crucial issues in the history and philosophy of logic. - Shahid Rahman, History and Philosophy of Logic. BOOK DESCRIPTION: This book is the only exhaustive study of Russell on modality and logical relevance ever written. This is the second edition, revised over a period of over sixteen years, and over twice as long. Russell initially rejects the possibility of a modal logic in an unpublished paper written ca. 1903-05 and read to an audience in 1905. But the very next year, he adopts the theory of modality he had rejected, and he repeats that theory in published works for many years to come. That theory, together with other texts scattered among Russell's writings, implies eight modal logics which Russell himself never expressly accepted. And Russell's express acceptance of the early Wittgenstein's theory of entailment ("following from") as truth-ground containment implies a deductive relevance logic in Russell's writings which Russell himself never expressly accepted. The book ends with a look at John Maynard Keynes and Anglo-American evidence law as the origins of Russell's theory of probability as degree of logical relevance. Thus the book is written toward the reunion of modern classical logicians with modal logicians and relevantists. On the reunion of modern classical logicians with relevantists, please see also The Concept of Relevance and the Logic Diagram Tradition (2012). Thus I hope to have created two new fields of Russell studies. I may have created a new legal field as well: the history of logical relevance in evidence law: 132 pages (368-500) in chapter 10. (shrink)

Departing from modal logic, Jean-Yves Girard, as a logician interested in philosophy, presented a distinction between essentialism and existentialism in logic. Carlos Lobo reflected about the Girard’s concept to reinterpret the Husserlian Platonism in regard of the status of logical modalities. We start rescuing the notion of modal logic in the Edmund Husserl’s works, especially Formal and Transcendental Logic and First Philosophy. Developing this reflexion, we propose a new contribution to this discussion, reinterpreting the platonic influence in (...) the Husserlian notions of eidos and science, light of some readings of Lee Hardy and Johanna M. Tito. As a conclusion of this dialogue between Husserl and Girard, the method of eidetic variation is presented as a tool to review the idea of science, in a manner consistent with the phenomenological approach. (shrink)

The aim of the paper is to show that W. V. O. Quine's animadversions against modal logic did not get the same attention that is considered to be the case nowadays. The community of logicians focused solely on the technical aspects of C. I. Lewis’ systems and did not take Quine's arguments and remarks seriously—or at least seriously enough to respond. In order to assess Quine's place in the history, however, his relation to Carnap is considered since their notorious (...) break was about the status of extensionality and modal logic. Since much of the works about the history of analytic philosophy is centered on the relationship of Quine and Carnap, their break about modality deserves much more attention—it also sheds some light on why should anyone wonder about Quine's early arguments against modal logic. The paper ends with some further considerations regarding the early formation of modal logic and hitherto unconsidered problematic issues. (shrink)

What formulas are tense-logically valid depends on the structure of time, for example on whether it has a beginning. Logicians have investigated what formulas correspond to what physical hypotheses about time. Analogously, we can investigate what formulas of modal logic correspond to what metaphysical hypotheses about necessity. It is widely held that physical hypotheses about time may be contingent. If so, tense-logical validity may be contingent. In contrast, validity in modal logic is typically taken to be non-contingent, as (...) reflected by the general acceptance of the so-called “rule of necessitation.” But as has been argued by various authors in recent years, metaphysical hypotheses may likewise be contingent. If, in particular, hypotheses about the extent of possibility are contingent, we should expect modal-logical validity to be contingent too. Let “contingentism” be the view that everything that is not ruled out by logic is possible. I shall investigate what the right system of modal logic is, if contingentism is true. Given plausible assumptions, the system contains the McKinsey principle, and is thus not even contained in S5. It also contains simple and elegant iteration principles for the contingency operator: something is contingent if and only if it is contingently contingent. (shrink)

Many systems of quantified modal logic cannot be characterised by Kripke's well-known possible worlds semantic analysis. This book shows how they can be characterised by a more general 'admissible semantics', using models in which there is a restriction on which sets of worlds count as propositions. This requires a new interpretation of quantifiers that takes into account the admissibility of propositions. The author sheds new light on the celebrated Barcan Formula, whose role becomes that of legitimising the Kripkean interpretation (...) of quantification. The theory is worked out for systems with quantifiers ranging over actual objects, and over all possibilia, and for logics with existence and identity predicates and definite descriptions. The final chapter develops a new admissible 'cover semantics' for propositional and quantified relevant logic, adapting ideas from the Kripke-Joyal semantics for intuitionistic logic in topos theory. This book is for mathematical or philosophical logicians, computer scientists and linguists. (shrink)

Classical logic yields counterintuitive results for numerous propositional argument forms. The usual alternatives (modal logic, relevance logic, etc.) generate counterintuitive results of their own. The counterintuitive results create problems—especially pedagogical problems—for informal logicians who wish to use formal logic to analyze ordinary argumentation. This paper presents a system, PL– (propositional logic minus the funny business), based on the idea that paradigmatic valid argument forms arise from justificatory or explanatory discourse. PL– avoids the pedagogical difficulties without sacrificing insight into argument.

Nicholas Rescher. Schools.” An English translation of these sections of the text is given in Appendix A below. No matter how difficult or boring this material proved for the Muslim schoolmaster, it is of the greatest relevance for our interests.

What makes a valid argument valid? Generally speaking, in a valid argument, if the premisses are true, then the conclusion must necessarily also be true. But on its own, this doesn’t tell us all that much. What is truth? And what is necessity? In what follows, I consider answers to these questions proposed by the fourteenth century logician John Buridan († ca. 1358). My central claim is that Buridan’s logic is downstream from his metaphysics. Accordingly, I treat his metaphysical (...) discussions as the key to his logic. As has been often noted, Buridan’s metaphysics are radically anti-realist about universals, though I think the depth and scope of his anti-realism has at times been papered over. Buridan constructs his logic on an amazingly spartan ontology, and this accomplishment is overdue for reconsideration. To the foregoing questions: truth is a feature of propositions, and of propositions only—not of proposition-like states of affairs, or anything like that. It is a function of the reference or supposition (suppositio) of their terms, which depends on predication in a propositional context. And necessary truth is grounded in causation: a predication is necessary if it cannot be falsified by any power, natural or supernatural, without ii annihilating what its terms stand for. “Socrates is a human”, for instance, can only be falsified by annihilating Socrates, and so it is a necessary truth. -/- These considerations provide an ample theoretical basis for a thorough examination of Buridan’s modal logic. This is what the thesis culminates with. In the final chapter, I set forth some novel and surprising findings, chief of which is this: Buridan’s modal syntax and semantics are nothing like Kripke’s—and indeed are incompatible with them. This has significant implications not only for modal logic and metaphysics, but for how we think about medieval logic and philosophy more generally. -/- Throughout the thesis, I advocate a methodology of emphasising the differences, rather than the similarities, between past and present thought. Buridan is wildly unlike what we’re used to, and we should let him speak for himself. (shrink)

VIRTUALLY ALL MODAL LOGICIANS after Aristotle have been troubled by his insistence that, given a valid first figure categorical syllogism.

The reasoning abilities of large language models (LLMs) are the topic of a growing body of research in artificial intelligence and cognitive science. In this paper, we probe the extent to which a dozen LLMs are able to distinguish logically correct inferences from logically fallacious ones. We focus on inference patterns involving conditionals (e.g., 'If Ann has a queen, then Bob has a jack') and epistemic modals (e.g., 'Ann might have an ace', 'Bob must have a king'). These inference patterns (...) have been of special interest to logicians, philosophers, and linguists, since they plausibly play a central role in human reasoning. Assessing LLMs on these inference patterns is thus highly relevant to the question of how much the reasoning abilities of LLMs match those of humans. Among the LLMs we tested, all but GPT-4 often make basic mistakes with conditionals. Moreover, even GPT-4 displays logically inconsistent judgments across inference patterns involving epistemic modals. (shrink)

In recent years modal syllogistic provided by 14th century logician John Buridan has attracted increasing attention of historians of medieval logic. The widespread use of quantified modal logic with the apparatus of possible worlds semantics in current analytic philosophy has encouraged the investigation of the relation of Buridan’s theory of modality with the modern developments of symbolic modal logic. We focus on the semantics of and the inferential relations among the propositions that underlie Buridan’s theory of (...) modal syllogism. First, we review all inferences between propositions of necessity, possibility, contingency, and non-contingency, with or without quod est locution, that are valid in Buridan’s semantics, and offer a comprehensive diagrammatic representation that includes them all. We then ask the question if there is a way to model those results in first order modal logic. Three ways of formalizing Buridan’s propositions in quantified modal logic are considered. Comparison of inferences between the quantified formulas and Buridan’s propositions reveals that, when supplied with a suitable formalization, Buridan’s semantics of categorical statements and immediate inferences among them can be fully captured by the quantified modal system T. (shrink)

The main purpose of the paper is to introduce philosophers and philosophical logicians to dynamic logic, a subject which promises to be of interest also to philosophy. A new completeness result involving both after — and during — operators is announced.

Timothy Williamson is one of the most influential living philosophers working in the areas of logic and metaphysics. His work in these areas has been particularly influential in shaping debates about metaphysical modality, which is the topic of his recent provocative and closely-argued book *Modal Logic as Metaphysics* (2013). The present book comprises ten essays by metaphysicians and logicians responding to Williamson’s work on metaphysical modality. The authors include some of the most distinguished philosophers of modality in the world, (...) as well as several rising stars. Each essay is followed by a reply by Williamson. In addition, the book contains a major new essay by Williamson, ‘Modal science,’ concerning the role of modal claims in natural science. This book was originally published as a special issue of the *Canadian Journal of Philosophy.*. (shrink)

The author distinguishes three views of negation: 1) term-negation ; 2) predicate-negation ; 3) sentence-negation. Strictly, however, he thinks there is no genuine sentential negation; sentential negation is reducible to the denial of a predicate. The author claims to have shown this in the present book. If sentential negation is regarded as "external" and predicate negation as "internal" to sentences, the reduction of sentential negation to predicate negation is an "internalization" of negation. The author envisages a more general reductionism for (...) logical symbols: "the internalization of negation seems to be the first and most important step [in Englebretsen's program of viewing all sentences as logically categorical]. The internalization of other operations, including modality, should follow". (shrink)

The history of contemporary modal logic dates back to the writings of C. S. Lewis in the early part of this century. Since then, a growing body of literature has attested to professional interest in the area, and in a number of related issues in philosophical logic which have received wide attention. The recent development of powerful formal techniques for modal system building, together with an increasing interest in modal logic as a tool for philosophical analysis, have (...) created a need for an up-to-date text to introduce students to this material. Snyder's book attempts to answer this need. Assuming an understanding of elementary propositional logic, Snyder introduces a reductive technique called cancellation for detecting the theorems of a system of logic. This technique, analogous to the construction of Smullyan trees but more compact, is extended to modal and quantified contexts. Cancellation versions of the systems M and Mn of G. H. von Wright, and S4 and S5 of Lewis are developed. Snyder uses a Hintikka type of semantics, showing how the various formal systems are differentiated at the semantic level by differing conditions which must be imposed on the model systems used as interpretations for them. These model systems, and the conditions imposed upon them, are in turn described by means of a metalanguage which is essentially first-order quantificational logic. The power of this technique stems from the fact that for every object language formula of a system, there is a corresponding formula in the metalanguage which describes the conditions an interpretation must meet in order to satisfy the original formula. The conditions that the interpretations of a given system of logic must meet are reflected in this metalanguage as "antecedent assumptions," and these in turn correspond to cancellation rules for the object language. This correspondence is constructed in such a way that the metalinguistic counterpart of each theorem of the system will be a theorem of first-order logic. Snyder's primary interest is in showing how modal logic can be used, with the help of these techniques, to build a large variety of formal systems and to "tailor" such systems to meet a variety of analytical tasks. Simple modal systems are developed for the articulation of a number of modal concepts, in addition to the classic alethic ones. Examples are taken from temporal, deontic, and epistemic logic to show how specific interpretations of the meanings of the relevant operators lead to the incorporation of appropriate conditions in the formal systems used to articulate them. An entire chapter is devoted to a discussion of the paradoxes of material and strict implication, and to an attempt to articulate the notion of entailment through the development of formal systems which include a dyadic modal operator that is free of these paradoxes. The final chapter discusses such classical issues as proper names, reference, fictional entities, definite descriptions, and existence presuppositions. Appendices deal with such matters as the equivalence of the cancellation systems to more classical axiomatic-deduction systems, and the sketch of a proof of soundness and completeness for all the modal systems presented. While designed as a text, this volume should be of interest to both logicians and those working in metaphysics and language analysis, since the primary concern of the author is to develop techniques that will facilitate the usefulness of modal logic as a tool for philosophical analysis. The instructor's manual contains many suggestions derived from the author's experience in teaching this non-standard treatment of the subject.--K. T. (shrink)

This dissertation studies Grosseteste's attempt in his De libero arbitrio to develop a theory of non-temporal modality which severs possibility and changeability. It consists in close philosophical analysis of Grosseteste's views, based on new editions and translations of De libero arbitrio, De scientia Dei, and De veritate propositionis contained in the Appendices. ;After studying the theories of modality proposed by Boethius, Abelard, and twelfth century logic texts, together with the logico-linguistic framework thirteenth century writers drew from them, I examine the (...) basic assumptions and motivations of Grosseteste's discussion: the notions of sentences and dicta, time-variable truth-value, the notion of power, the problem of reconciling divine knowledge and human free will. ;I identify two main modal theories in Grosseteste, one linking possibility with changeability of truth-value and bearing similarities with as well as important differences from the theories of the twelfth century logicians; the other presented as a reaction to this theory and, in particular, to its unusual claim that true sentences about future contingents are necessary. I attempt to make sense of this by a reconstruction of the theory, making use of the notion of dense time. ;I then investigate in detail Grosseteste's theory of the eternal truths, with an eye to its relation to his theories of modality. I show that Grosseteste's sophisticated theory is not a theory of modality; the theory is seen, nonetheless, to highlight obscurities in Grosseteste's account of non-temporal modality. ;The account of non-temporal modality rests on the notion of a dictim's eternal power; these powers are reduced to God's eternal powers. I critically examine Grosseteste's defense of the idea of divine power, highlighting the difficulties he viewed the notion of the actualizability of a power and of its priority to act as posing. His views on these matters are compared with those of Scotus. ;I investigate the theory of non-temporal modality erected on this basis and, in particular, its reduction to divine powers, the adequacy of the strategy, and Grosseteste's use of the theory to solve the problem divine knowledge poses for free will. (shrink)

In this paper, the author derives the Dedekind-Peano axioms for number theory from a consistent and general metaphysical theory of abstract objects. The derivation makes no appeal to primitive mathematical notions, implicit definitions, or a principle of infinity. The theorems proved constitute an important subset of the numbered propositions found in Frege's *Grundgesetze*. The proofs of the theorems reconstruct Frege's derivations, with the exception of the claim that every number has a successor, which is derived from a modal axiom (...) that (philosophical) logicians implicitly accept. In the final section of the paper, there is a brief philosophical discussion of how the present theory relates to the work of other philosophers attempting to reconstruct Frege's conception of numbers and logical objects. (shrink)

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. (shrink)

Nicholas Rescher surveys and analyzes the different kinds of unreal possibilities and nonexistent objects, tying together all the diverse ways in which this area has been approached by philosophers. As he surveys the field and clarifies the kinds of unreality, he also makes a sustained argument against the philosophical fashion for dealing with nonexistent possible world as though they were authentic objects. The author holds that, while we may discuss possibilities, we ought not to accord them ontological status. The possibility (...) of existence of a certain sort of world is not the existence of possible world of a certain sort. While we may reasonable discuss possibilities at the generic level, such as a world where dogs have horns, this does not require a commitment to a possible world where they do. The work that theorists of logic and language want to accomplish with possible worlds and individuals can be managed with propositional manifolds, stories or scenarios, while the modalities of necessity and possibility that modal logicians want to analyze in terms of realization in possible worlds can be handled by turning instead to figuring in stories or scenarios. (shrink)

Jerzy Perzanowski s ideas were based on an original blend of logic and ontology in what he called onto/logic, where the slash is meant to suggest a quotient of ontology by logic. Perzanowski began as a logician, his early works being on modal logic, then gradually shifted his interest to logical philosophy, meaning not so much philosophy of logic as philosophy informed by logic. Perzanowski was a rare breed of analytical philosopher who thought that a philosophical theory of (...) everything was worthwhile. In this systematic spirit, he began with method. He presented his method of total analysis and synthesis quite simply: reduce the object of research to its simplest possible constituents, and then combine them in some way. Better still, combine them in every possible way, thereby producing a space of possibilities analogous to (and in certain cases identical with) the logical space. Thus, analysis and synthesis differ from a trivial disassembly and reassembly.". (shrink)