This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from leading Brazilian logicians and their Latin-American and European colleagues. All papers were selected by a careful refereeing processs and were revised and updated (...) by their authors for publication in this volume. There are three sections: Advances in Logic, Advances in Theoretical Computer Science, and Advances in Philosophical Logic. Well-known specialists present original research on several aspects of model theory, proof theory, algebraic logic, category theory, connections between logic and computer science, and topics of philosophical logic of current interest. Topics interweave proof-theoretical, semantical, foundational, and philosophical aspects with algorithmic and algebraic views, offering lively high-level research results. (shrink)

Frege's conception of the logical categories has vexed commentators for decades. In this dissertation, I argue that it revolves around two forms of internality. The first is the internality of its use in the expression of judgment to the sign. A proper understanding of that internality reveals how Frege's philosophical logic cannot be fit into the framework given by the contemporary syntax/semantics distinction. The second is the internality that obtains between the way in which Begriffsschrift signs stratify into (...) different categories, on the one hand, and the way in which the realm of Bedeutungen stratifies into logical categories, on the other hand. A proper understanding of this internality reveals how Frege's conception of the relation between language and the world is neither a form of realism nor a form of linguistic idealism. To come to understand Frege's conception of the logical categories, is to come to understand both these forms of internality by coming to understand how they hang together. (shrink)

It has been much debated whether Tractarian objects are what Russell would have called particulars or whether they include also properties and relations. This paper claims that the debate is misguided: there is no logical category such that Wittgenstein intended the reader of the Tractatus to understand his objects either as providing examples of or as not providing examples of that category. This is not to say that Wittgenstein set himself against the very idea of a logical category: (...) quite the contrary. However, where Russell presents his logical variety of particulars and the various types of universal, and Frege presents his of objects and the various types of function, Wittgenstein denies the propriety of such a priori expositions. Wittgenstein envisages a variety of logical types of entity but insists that the nature of these types is something to be discovered only through analysis. (shrink)

For more than 60 years, Jim Lambek has been a profoundly inspirational mathematician, with groundbreaking contributions to algebra, category theory, linguistics, theoretical physics, logic and proof theory. This Festschrift was put together on the occasion of his 90th birthday. The papers in it give a good picture of the multiple research areas where the impact of Jim Lambek's work can be felt. The volume includes contributions by prominent researchers and by their students, showing how Jim Lambek's ideas keep inspiring upcoming (...) generations of scholars. (shrink)

This paper aims at revealing the originality of Max Weber’s conception of the logical category of “historicity”, suggesting that in his writings on the methodology of the social sciences we can find a stimulating and forerunner contribution to the analysis of some logical and formal problems concerning the relationship between human knowledge and the chaos of reality (what we might call, ante-litteram, “science of chaos”). In particular, considering that in Weber’s conception scientific knowledge finds no facts “to grasp” (...) in the natural world, but rather a chaos of unique and infinitely divisible events, the analysis will be focused on the following aspects: (a) Weber’s separation of causal imputation from the notion of necessary (natural) law; (b) the importance attached to “probability judgments” with different degrees of certainty; (c) the proclaimed irreducibility of individual events to scientific models, laws, and (ideal)-types; (d) the effects imputed to the differentiation of the point of view of a scientific observer. (shrink)

During decades, the history of philosophy has kept Kierkegaardrsquo;s and Hegelrsquo;s thought apart, and their long-standing opposition has swept through the speculative greatness of Kierkegaardian existentialism and the existential power of Hegelian philosophy. In contrast to such unfortunate misinterpretation, this article aims at showing the deep convergence that relates interiorly the Kierkegaardian ethical stage with the most important Hegelian logic categories. Kierkegaard and Hegel conceive of the idea as the real power of subjective becoming, and the existence as the (...) actual concretion of the ideal. To both of them, the pure emeneacute;rgeia/em of freedom, which starts in the abstract and aesthetical possibility of the subjective immediacy, realizes itself as the actual concretion of finitude, assuming time and contingency by the eternal and necessary force of duty. The Kierkegaardian repetition is nothing but this powerful idea, mediating the flux of finite differences in the eternal identity of subject. However, for Kierkegaard as well as for Hegel there is an absolute contradiction, which promotes the overcoming of ethics. (shrink)

I provide a critical commentary regarding the attitude of the logician and the philosopher towards the physicist and physics. The commentary is intended to showcase how a general change in attitude towards making scientific inquiries can be beneficial for science as a whole. However, such a change can come at the cost of looking beyond the categories of the disciplines of logic, philosophy and physics. It is through self-inquiry that such a change is possible, along with the realization of (...) the essence of the middle that is otherwise excluded by choice. The logician, who generally holds a reverential attitude towards the physicist, can then actively contribute to the betterment of physics by improving the language through which the physicist expresses his experience. The philosopher, who otherwise chooses to follow the advancement of physics and gets stuck in the trap of sophistication of language, can then be of guidance to the physicist on intellectual grounds by having the physicist’s experience himself. In course of this commentary, I provide a glimpse of how a truthful conversion of verbal statements to physico-mathematical expressions unravels the hitherto unrealized connection between Heisenberg uncertainty relation and Cauchy’s definition of derivative that is used in physics. The commentary can be an essential reading if the reader is willing to look beyond the categories of logic, philosophy and physics by being ‘nobody’. (shrink)

In this paper we propose a very general de nition of combination of logics by means of the concept of sheaves of logics. We first discuss some properties of this general definition and list some problems, as well as connections to related work. As applications of our abstract setting, we show that the notion of possible-translations semantics, introduced in previous papers by the first author, can be described in categorial terms. Possible-translations semantics constitute illustrative cases, since they provide a new (...) semantical account for abstract logical systems, particularly for many-valued and paraconsistent logics. (shrink)

This study of the interpretations of Aristotle's "Categories" in the thirteenth century provides an introduction to some main themes of medieval philosophical ...

Developing some suggestions of Ramsey (1925), elementary logic is formulated with respect to an arbitrary categorial system rather than the categorial system of Logical Atomism which is retained in standard elementary logic. Among the many types of non-standard categorial systems allowed by this formalism, it is argued that elementary logic with predicates of variable degree occupies a distinguished position, both for formal reasons and because of its potential value for application of formal logic to natural language and natural science. (...) This is illustrated by use of such a logic to construct a theory of quantity which is argued to be scientifically superior to existing theories of quantity based on standard categorial systems, since it yields real-valued scales without the need for unrealistic existence assumptions. This provides empirical evidence for the hypothesis that the categorial structure of the physical world itself is non-standard in this sense. (shrink)

In reworking his Logical Investigations Husserl adopts two positions that were not actually incorporated into later editions of the Investigations but do appear in other writings: a new distinction between signitive and significative intentions, and the claim that even naming and perceiving acts are categorially formed. This paper investigates Husserl's notion of noematic sense and the pure grammatical ' categories ' intimated therein in order to shed light on these new positions. The paper argues that the development of (...) the theories of the noema and of pure grammar allows us to recognize how even merely perceived or named things have a certain categoriality belonging to them, but that this development also requires us to distinguish between an anticipatory categoriality and an articulated categoriality. (shrink)

Gyorgy Tamas works in the philosophy of logic, that difficult interdisciplin ary region wherein the notion of categories is both basic and subtle. To understand ways of thinking, to understand patterns of whatever is real, to recognize what is possible and to reject the nonsensical and the impossible is to comprehend the categories. This was a in thought and in fact, recurring motive of European thought from the earliest self-aware beginnings, and Tamas knows that history well, as his (...) critical respect demonstrates. Ancient, medieval, and modern thinkers appear in this book, set forth in their own words; and likewise we see that Tamas has built upon the historians and commentators, upon the pioneering historical investigation of the categories by Trendelenburg a century ago and by Bochenski in our days. Tamas has two principal goals here: to investigate the logic, which is to say the structure and the relations, of the philosophical categories; and to set forth the logic of thought which may then be based upon the critically established system of categories obtained by that investigation. Ancillary but of striking value is his style of historical relevance which enables the reader to engage in a discussion that is both analytically sharp and developmentally insightful. Furthermore, Tamas draws upon his contem porary colleagues with similar critical respect: Lukasiewicz, Quine, Patzig, Menne, Tavanets and others. (shrink)

This paper establishes a connection between structure sensitive categorial inference and classical modal logic. The embedding theorems for non-associative Lambek Calculus and the whole class of its weak Sahlqvist extensions demonstrate that various resource sensitive regimes can be modelled within the framework of unimodal temporal logic. On the semantic side, this requires decomposition of the ternary accessibility relation to provide its correlation with standard binary Kripke frames and models.

Language in Action demonstrates the viability of mathematical research into the foundations of categorial grammar, a topic at the border between logic and linguistics. Since its initial publication it has become the classic work in the foundations of categorial grammar. A new introduction to this paperback edition updates the open research problems and records relevant results through pointers to the literature. Van Benthem presents the categorial processing of syntax and semantics as a central component in a more general dynamic logic (...) of information flow, in tune with computational developments in artificial intelligence and cognitive science. Using the paradigm of categorial grammar, he describes the substructural logics driving the dynamics of natural language syntax and semantics. This is a general type-theoretic approach that lends itself easily to proof-theoretic and semantic studies in tandem with standard logic. The emphasis is on a broad landscape of substructural categorial logics and their proof-theoretical and semantic peculiarities. This provides a systematic theory for natural language understanding, admitting of significant mathematical results. Moreover, the theory makes possible dynamic interpretations that view natural languages as programming formalisms for various cognitive activities. (shrink)

In his article “Issues of Pragmaticism” published in 1905, in The Monist, Charles S. Peirce complains that “Logicians have been at fault in giving Vagueness the go-by, so far as not even to analyze it.” That same year, occupying himself with the consequences of “Critical commonsensism,” he affirmed, “I have worked out the logic of vagueness with something like completeness,” a statement that causes the majority of the commentators on his work, including the editors of the Collected Papers to ask (...) where this logic is to be found. The fever for finding Peirce's manuscripts is fed by the hope of some researchers of discovering the logic of vagueness, a hope that has grown since Carolyn Eisele's publication of his mathematical works. Others - and I count myself among them - believe that in reality this is a matter of something already known. That is, they interpret the affirmation ending the paragraph of reproach addressed to logicians, “The present writer has done his best to work out the Stechiology, Critic, and Methodeutik of the subject,” as a tripartite semiotic of the vague, still limited, according to Peirce's older works, to symbols, that is, to the signs of natural language examined from the perspective of logic. (shrink)

Many commentators have attempted to say, more clearly than Wittgenstein did in his Tractatus logico-philosophicus, what sort of things the ‘simple objects’ spoken of in that book are. A minority approach, but in my view the correct one, is to reject all such attempts as misplaced. The Tractarian notion of an object is categorially indeterminate: in contrast with both Frege's and Russell's practice, it is not the logician's task to give a specific categorial account of the internal structure of elementary (...) propositions or atomic facts, nor, correlatively, to give an account of the forms of simple objects. The few commentators who have hitherto maintained this view have mainly devoted themselves to establishing that this was Wittgenstein's intention, and do not much address the question why Wittgenstein held that it is not the logician's business to say what the objects are. The present paper means to fill this lacuna by placing this view in the context of the Tractatus's treatment of logic generally, and in particular by connecting it with Wittgenstein's treatment of generality and with his reaction to Russell's approach to logical form. (shrink)

Starting from a generalization of the standard axioms for a monoid we present a stepwise development of various, mutually equivalent foundational axiom systems for category theory. Our axiom sets have been formalized in the Isabelle/HOL interactive proof assistant, and this formalization utilizes a semantically correct embedding of free logic in classical higher-order logic. The modeling and formal analysis of our axiom sets has been significantly supported by series of experiments with automated reasoning tools integrated with Isabelle/HOL. We also address the (...) relation of our axiom systems to alternative proposals from the literature, including an axiom set proposed by Freyd and Scedrov for which we reveal a technical issue (when encoded in free logic where free variables range over defined and undefined objects): either all operations, e.g. morphism composition, are total or their axiom system is inconsistent. The repair for this problem is quite straightforward, however. (shrink)

This edited volume is a comprehensive presentation of views on the relations between metaphysics and logic from Aristotle through twentieth century philosophers who contributed to the return of metaphysics in the analytic tradition. The collection combines interest in logic and its history with interest in analytical metaphysics and the history of metaphysical thought. By so doing, it adds both to the historical understanding of metaphysical problems and to contemporary research in the field. Throughout the volume, essays focus on metaphysica generalis, (...) or the systematic study of the most general categories of being. Beginning with Aristotle and his Categories , the volume goes on to trace metaphyscis and logic through the late ancient and Arabic traditions, examining the views of Thomas Aquinas, Duns Scotus, and William Ockham. Moving into the early modern period, contributors engage with Leibniz's metaphysics, Kant's critique of metaphysics, the relation between logic and ontology in Hegel, and Bolzano's views. Subsequent chapters address: Charles S. Peirce's logic and metaphysics; the relevance of set-theory to metaphysics; Meinong's theory of objects; Husserl's formal ontology; early analytic philosophy; C.I. Lewis and his relation to Russell; and the relations between Frege, Carnap, and Heidegger. Surveying metaphysics through to the contemporary age, essays explore W.V. Quine's attitude towards metaphysics; Wilfrid Sellars's relation to antidescriptivism as it connects to Kripke's; the views of Putnam and Kaplan; Peter F. Strawson's and David M. Armstrong's metaphysics; Trope theory; and its relation to Popper's conception of three worlds. The volume ends with a chapter on transcendental philosophy as ontology. In each chapter, contributors approach their topics not merely in an historical and exegetical fashion, but also engage critically with the thought of the philosophers whose work they discuss, offering synthesis and original philosophical thought in the volume, in addition to very extensive and well-informed analysis and interpretation of important philosophical texts. The volume will serve as an essential reference for scholars of metaphysics and logic. (shrink)

First-order logic with dependent sorts, such as Makkai's first-order logic with dependent sorts (FOLDS), or Aczel's and Belo's dependently typed (intuitionistic) first-order logic (DFOL), may be regarded as logic enriched dependent type theories. Categories with families (cwfs) is an established semantical structure for dependent type theories, such as Martin-Löf type theory. We introduce in this article a notion of hyperdoctrine over a cwf, and show how FOLDS and DFOL fit in this semantical framework. A soundness and completeness theorem is (...) proved for DFOL. The semantics is functorial in the sense of Lawvere, and uses a dependent version of the Lindenbaum-Tarski algebra for a DFOL theory. Agreement with standard first-order semantics is established. Applications of DFOL to constructive mathematics and categorical foundations are given. A key feature is a local propositions-as-types principle. (shrink)

This edited volume is a comprehensive presentation of views on the relations between metaphysics and logic from Aristotle through twentieth century philosophers who contributed to the return of metaphysics in the analytic tradition.

IN THE FIRST EDITION TRANSCENDENTAL DEDUCTION of the categories Kant does not mention the logical functions of judgment. In the second edition the Deduction can be said to be dominated by the logical functions of judgment. A transcendental deduction supplies a method for showing that pure concepts can have applicability. My contention is that the two deductions constitute exactly the same method, and so are the exact same deduction. The difference between them, rather, is in the characterization (...) of the pure concepts that the method is supposed to be a method for. The undifferentiated categories of the A edition become the logical functions together with their schemata in the B edition. This does not mean that Kant has split the A edition notion of categories since the A edition categories are equivalent to just the schemata themselves. (shrink)

What is the function of logic in al-Kindī's corpus? What kind of relation does it have with mathematics? This article tackles these questions by examining al-Kindī's theory of categories as it was presented in his epistle On the Number of Aristotle's Books, from which we can learn about his special attitude towards Aristotle theory of categories and his interpretation, as well. Al-Kindī treats the Categories as a logical book, but in a manner different from that of (...) the classical Aristotelian tradition. He ascribes a special status to the categories Quantity and Quality, whereas the rest of the categories are thought to be no more than different combinations of these two categories with the category Substance. The discussion will pay special attention to the function of the categories of Quantity and Quality as mediators between logic and mathematics. (shrink)

In this paper, I present a detailed interpretation of Peirce’s cosmogony about the origin of the universe and its evolutionary development. This involves bringing together and making sense of Peirce’s disconnected statements on cosmology, which are scattered throughout his writings and which sometimes employ different terminologies. Furthermore, it shall involve identifying the categories of firstness, secondness, and thirdness that govern its conceptual structure, and ultimately the metaphysical structure of the universe to which it refers. Attending to the categories (...) at play here reveals the basic “logic” of Peirce’s theory from which its rational coherence is discernable. As such, it may be argued that given.. (shrink)

Thought: A Journal of Philosophy, EarlyView.

Gilbert Ryle never pursued research under Edmund Husserl. However, Ryle was indeed Husserl’s student in a broader sense, as much of his own work was deeply influenced by his studies of Husserl’s pre-World War I writings. While Ryle is the thinker whose name typically comes to mind in connection with the concern over category mistakes I argue that (1) Husserl deserves to be known for precisely this concern as well, and (2) the similarity between them is no accident. Developing this (...) reading of Ryle’s Husserlian pedigree forces a broader reevaluation of each of their roles in twentieth-century thought. (shrink)

Gilbert Ryle never pursued research under Edmund Husserl. However, Ryle was indeed Husserl’s student in a broader sense, as much of his own work was deeply influenced by his studies of Husserl’s pre-World War I writings. While Ryle is the thinker whose name typically comes to mind in connection with the concern over category mistakes I argue that (1) Husserl deserves to be known for precisely this concern as well, and (2) the similarity between them is no accident. Developing this (...) reading of Ryle’s Husserlian pedigree forces a broader reevaluation of each of their roles in twentieth-century thought. (shrink)

McTaggart's ideas on the unreality of time as expressed in "The Nature of Existence" have retained great interest for many years for scholars, academics and other philosophers. In this essay, there is a brief discussion which mentions some of the high points of this philosophical interest, and goes on to apply his ideas to modern physics and neuroscience. It does not discuss McTaggart's C and D series, but does emphasise how the use of derived versions of both his A and (...) B series can be of great virtue in discussing both the abstract physics of time, and the present and future importance of McTaggart's ideas to the subject of time. Indeed an experiment using human volunteers and dynamic systems modelling which was carried out is described, which illustrates this fact. The Many Bubble Interpretation, which also derives from McTaggart's ideas, is discussed and various examples of its use and effectiveness are referred to. The Schrodinger Cat paradox is essentially resolved in principle, the quantum Zeno effect interpretable, Kwiat's recent result referred to, and the newly discovered reverse Stickgold effect described. (shrink)