We re-examine the problem of existential import by using classical predicate logic. Our problem is: How to distribute the existential import among the quantified propositions in order for all the relations of the logical square to be valid? After defining existential import and scrutinizing the available solutions, we distinguish between three possible cases: explicit import, implicit non-import, explicit negative import and formalize the propositions accordingly. Then, we examine the 16 combinations between the 8 propositions having the first two kinds of (...) import, the third one being trivial and rule out the squares where at least one relation does not hold. This leads to the following results: (1) three squares are valid when the domain is non-empty; (2) one of them is valid even in the empty domain: the square can thus be saved in arbitrary domains and (3) the aforementioned eight propositions give rise to a cube, which contains two more (non-classical) valid squares and several hexagons. A classical solution to the problem of existential import is thus possible, without resorting to deviant systems and merely relying upon the symbolism of First-order Logic (FOL). Aristotle’s system appears then as a fragment of a broader system which can be developed by using FOL. (shrink)

This is identical with the first edition (see 21: 2716) except for the addition of a Supplement containing 5 previously published articles and the bringing of the bibliography (now 73 items) up to date. The 5 added articles present clarifications or modifications of views expressed in the first edition. (PsycINFO Database Record (c) 2009 APA, all rights reserved).

The work of which this is an English translation appeared originally in French as Precis de logique mathematique. In 1954 Dr. Albert Menne brought out a revised and somewhat enlarged edition in German. In making my translation I have used both editions. For the most part I have followed the original French edition, since I thought there was some advantage in keeping the work as short as possible. However, I have included the more extensive historical notes of Dr. Menne, his (...) bibliography, and the two sections on modal logic and the syntactical categories, which were not in the original. I have endeavored to correct the typo graphical errors that appeared in the original editions and have made a few additions to the bibliography. In making the translation I have profited more than words can tell from the ever-generous help of Fr. Bochenski while he was teaching at the University of Notre Dame during 1955-56. OTTO BIRD Notre Dame, 1959 I GENERAL PRINCIPLES § O. INTRODUCTION 0. 1. Notion and history. Mathematical logic, also called 'logistic', ·symbolic logic', the 'algebra of logic', and, more recently, simply 'formal logic', is the set of logical theories elaborated in the course of the last century with the aid of an artificial notation and a rigorously deductive method. (shrink)

This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and (...) made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

Up to now theories of semantic information have implicitly relied on logical monism, or the view that there is one true logic. The latter position has been explicitly challenged by logical pluralists. Adopting an unbiased attitude in the philosophy of information, we take a suggestion from Beall and Restall at heart and exploit logical pluralism to recognise another kind of pluralism. The latter is called informational pluralism, a thesis whose implications for a theory of semantic information we explore.

This book develops a view of logic as a theory of information-driven agency and intelligent interaction between many agents - with conversation, argumentation and games as guiding examples. It provides one uniform account of dynamic logics for acts of inference, observation, questions and communication, that can handle both update of knowledge and revision of beliefs. It then extends the dynamic style of analysis to include changing preferences and goals, temporal processes, group action and strategic interaction in games. Throughout, the book (...) develops a mathematical theory unifying all these systems, and positioning them at the interface of logic, philosophy, computer science and game theory. A series of further chapters explores repercussions of the 'dynamic stance' for these areas, as well as cognitive science. (shrink)

A general characterization of logical opposition is given in the present paper, where oppositions are defined by specific answers in an algebraic question-answer game. It is shown that opposition is essentially a semantic relation of truth values between syntactic opposites, before generalizing the theory of opposition from the initial Apuleian square to a variety of alter- native geometrical representations. In the light of this generalization, the famous problem of existential import is traced back to an ambiguous interpretation of assertoric sentences (...) in Aristotle's traditional logic. Following Abelard’s distinction between two alternative readings of the O-vertex: Non omnis and Quidam non, a logical difference is made between negation and denial by means of a more fine- grained modal analysis. A consistent treatment of assertoric oppositions is thus made possible by an underlying abstract theory of logical opposition, where the central concept is negation. A parallel is finally drawn between opposition and consequence, laying the ground for future works on an abstract operator of opposition that would characterize logical negation just as does Tarski’s operator of consequence for logical truth. (shrink)

"This book is valuable as expounding in full a theory of meaning that has its roots in the work of Frege and has been of the widest influence.

Peters and Westerståhl (Quantifiers in Language and Logic, 2006), and Westerståhl (New Perspectives on the Square of Opposition, 2011) draw a crucial distinction between the “classical” Aristotelian squares of opposition and the “modern” Duality squares of opposition. The classical square involves four opposition relations, whereas the modern one only involves three of them: the two horizontal connections are fundamentally distinct in the Aristotelian case (contrariety, CR vs. subcontrariety, SCR) but express the same Duality relation of internal negation (SNEG). Furthermore, the (...) vertical relations in the classical square are unidirectional, whereas in the modern square they are bidirectional. The present paper argues that these differences become even bigger when two more operators are added, namely the U ( ${{\equiv} {\rm A}\,{\vee} \,{\rm E} }$ , all or no) and Y ( ${\equiv{\rm I} \,{\wedge} \,{\rm O}}$ , some but not all) of Blanché (Structures Intellectuelles, 1969). In the resulting Aristotelian hexagon the two extra nodes are perfectly integrated, yielding two interlocking triangles of CR and SCR. In the duality hexagon by contrast, they do not enter into any relation with the original square, but constitute a independent pair of their own, since they are their own SNEGs. Hence, they not only stand in a relation of external NEG, but also in one of duality. This reflexive nature of the SNEG will be shown to result in defective monotonicity configurations for the pair, namely the absence of right-monotonicity (on the predicate argument). In the second half of the paper, we present an overview of those hexagonal structures which are both Aristotelian and Duality configurations, and those which are only Aristotelian. (shrink)

The central aim of this paper is to present a Boolean algebraic approach to the classical Aristotelian Relations of Opposition, namely Contradiction and (Sub)contrariety, and to provide a 3D visualisation of those relations based on the geometrical properties of Platonic and Archimedean solids. In the first part we start from the standard Generalized Quantifier analysis of expressions for comparative quantification to build the Comparative Quantifier Algebra CQA. The underlying scalar structure allows us to define the Aristotelian relations in Boolean terms (...) and to propose a 3D visualisation by transforming a cube into an octahedron. In part two, the architecture of the CQA is shown to carry over, both to the classical quantifiers of Predicate Calculus and to the modal operators—which are given a Generalized Quantifier style re-interpretation. In this way we provide an algebraic foundation for Blanché’s Aristotelian hexagon as well as a 3D alternative to his 2D star-like visualisation. In a final part, a richer scalar structure is argued to underly the realm of Modality, thus generalizing the 3D algebra with eight (2 3 ) operators to a 4D algebra with sixteen (2 4 ) operators. The visual representation of the latter structure involves a transformation of the hypercube to a rhombic dodecahedron. The resulting 3D visualisation allows a straightforward embedding, not only of the classical Blanché star of Aristotelian relations or the paracomplete and paraconsistent stars of Béziau (Log Investig 10, 218–232, 2003) but also of three additional isomorphic Aristotelian constellations. (shrink)

A general theory of logical oppositions is proposed by abstracting these from the Aristotelian background of quantified sentences. Opposition is a relation that goes beyond incompatibility (not being true together), and a question-answer semantics is devised to investigate the features of oppositions and opposites within a functional calculus. Finally, several theoretical problems about its applicability are considered.

En cet article nous montrons en premier lieu que la théorie de l'hexagone logique de Blanché n'est pas, comme il le pense, le résultat d'une réflexion philosophique, mais qu'elle relève véritablement de la logique scientifique, puisqu'elle s'insère tout naturellement dans la structure d'ensemble des liaisons uninaires de la logique trivalente des propositions. Cette démonstration nous conduit, en second lieu, à renverser le jugement défavorable que E. J. Lemmon avait porté sur la toute première ébauche de cette théorie, et ainsi à (...) déVelopper la logique modale dans le sens de Blanché plutôt que dans celui de von Wright. En troisième lieu, nous complétons l'application que fait Blanché de cette théorie de l'hexagone aux liaisons binaires de la logique bivalente des propositions, en intégrant en une seule structure tétrahexaédrique l'ensemble des seize liaisons binaires, lui-même s'étant contenté d'en grouper seulement dix en une espèce d'anneau bihexagonal. (shrink)

En cet article nous montrons en premier lieu que la théorie de l'hexagone logique de Blanché n'est pas, comme il le pense, le résultat d'une réflexion philosophique, mais qu'elle relève véritablement de la logique scientifique, puisqu'elle s'insère tout naturellement dans la structure d'ensemble des liaisons uninaires de la logique trivalente des propositions. Cette démonstration nous conduit, en second lieu, à renverser le jugement défavorable que E. J. Lemmon avait porté sur la toute première ébauche de cette théorie, et ainsi à (...) déVelopper la logique modale dans le sens de Blanché plutôt que dans celui de von Wright. En troisième lieu, nous complétons l'application que fait Blanché de cette théorie de l'hexagone aux liaisons binaires de la logique bivalente des propositions, en intégrant en une seule structure tétrahexaédrique l'ensemble des seize liaisons binaires, lui-même s'étant contenté d'en grouper seulement dix en une espèce d'anneau bihexagonal. (shrink)

If two statements are contraries if and only if they cannot both be true, but can both be false, then some corresponding A and E categorical statements are not contraries, even on the presupposition that something exists which satisfies the subject term. For some such statements are necessarily true and thus cannot be false. There is a similar problem with subcontraries.

Jan Lukasiewicz's treatise on Aristotle's Syllogistic, published in the 1950s, has been very influential in framing contemporary understanding of Aristotle's logical systems. However, Lukasiewicz's interpretation is based on a number of tendentious claims, not least, the claim that the syllogistic was intended to apply only to non-empty terms. I show that this interpretation is not true to Aristotle's text and that a more coherent and faithful interpretation admits empty terms while maintaining all the relations of the traditional square of opposition.

Proceedings of an international research and development conference, Tuscon, Arizona, October 1985. One hundred and twenty-eight papers are presented in this hefty volume.

This new edition of Introduction to Lattices and Order presents a radical reorganization and updating, though its primary aim is unchanged. The explosive development of theoretical computer science in recent years has, in particular, influenced the book's evolution: a fresh treatment of fixpoints testifies to this and Galois connections now feature prominently. An early presentation of concept analysis gives both a concrete foundation for the subsequent theory of complete lattices and a glimpse of a methodology for data analysis that is (...) of commercial value in social science. Classroom experience has led to numerous pedagogical improvements and many new exercises have been added. As before, exposure to elementary abstract algebra and the notation of set theory are the only prerequisites, making the book suitable for advanced undergraduates and beginning graduate students. It will also be a valuable resource for anyone who meets ordered structures. (shrink)

The logical hexagon (or hexagon of opposition) is a strange, yet beautiful, highly symmetrical mathematical figure, mysteriously intertwining fundamental logical and geometrical features. It was discovered more or less at the same time (i.e. around 1950), independently, by a few scholars. It is the successor of an equally strange (but mathematically less impressive) structure, the “logical square” (or “square of opposition”), of which it is a much more general and powerful “relative”. The discovery of the former did not raise interest, (...) neither among logicians, nor among philosophers of logic, whereas the latter played a very important theoretical role (both for logic and philosophy) for nearly two thousand years, before falling in disgrace in the first half of the twentieth century: it was, so to say, “sentenced to death” by the so-called analytical philosophers and logicians. Contrary to this, since 2004 a new, unexpected promising branch of mathematics (dealing with “oppositions”) has appeared, “oppositional geometry” (also called “n-opposition theory”, “NOT”), inside which the logical hexagon (as well as its predecessor, the logical square) is only one term of an infinite series of “logical bi-simplexes of dimension m”, itself just one term of the more general infinite series (of series) of the “logical poly-simplexes of dimension m”. In this paper we recall the main historical and the main theoretical elements of these neglected recent discoveries. After proposing some new results, among which the notion of “hybrid logical hexagon”, we show which strong reasons, inside oppositional geometry, make understand that the logical hexagon is in fact a very important and profound mathematical structure, destined to many future fruitful developments and probably bearer of a major epistemological paradigm change. (shrink)

Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic (both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”, “permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s “logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed a (...) “deontic hexagon” as being the geometrical representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie 73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter, Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic has appeared, that is “ n -opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m ” ( m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper, by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”, “deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra), whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional very regular solid). (shrink)

This is a thorough treatment of first-order modal logic. The book covers such issues as quantification, equality (including a treatment of Frege's morning star/evening star puzzle), the notion of existence, non-rigid constants and function symbols, predicate abstraction, the distinction between nonexistence and nondesignation, and definite descriptions, borrowing from both Fregean and Russellian paradigms.

. We start from the geometrical-logical extension of Aristotle’s square in [6,15] and [14], and study them from both syntactic and semantic points of view. Recall that Aristotle’s square under its modal form has the following four vertices: A is □α, E is , I is and O is , where α is a logical formula and □ is a modality which can be defined axiomatically within a particular logic known as S5 (classical or intuitionistic, depending on whether is involutive (...) or not) modal logic. [3] has proposed extensions which can be interpreted respectively within paraconsistent and paracomplete logical frameworks. [15] has shown that these extensions are subfigures of a tetraicosahedron whose vertices are actually obtained by closure of by the logical operations , under the assumption of classical S5 modal logic. We pursue these researches on the geometrical-logical extensions of Aristotle’s square: first we list all modal squares of opposition. We show that if the vertices of that geometrical figure are logical formulae and if the sub-alternation edges are interpreted as logical implication relations, then the underlying logic is none other than classical logic. Then we consider a higher-order extension introduced by [14], and we show that the same tetraicosahedron plays a key role when additional modal operators are introduced. Finally we discuss the relation between the logic underlying these extensions and the resulting geometrical-logical figures. (shrink)

_William of Sherwood's Introduction to Logic _ was first published in 1966. Minnesota Archive Editions uses digital technology to make long-unavailable books once again accessible, and are published unaltered from the original University of Minnesota Press editions. The _Introduction to Logic _ by William of Sherwood, of which this is the first English translation, is the oldest surviving treatise which contains a treatment of the most distinctive and interesting medieval contributions to logic and semantics. Sherwood was a master at Oxford (...) and Paris in the thirteenth century and the author of several logical treatises. Besides presenting material of interest in its own right, this volume is useful as an introduction to the study of those aspects of medieval philosophy that are most pertinent to the interests of contemporary philosophers. Professor Kretzmann has provided biographical, bibliographical, and philosophical backgrounds on Sherwood and an analytical table of contents. (shrink)

In this paper I explore informationalism, a pragmatic theory of modality that seems to solve some serious problems in the familiar possible worlds accounts of modality. I view the theory as an elaboration of Stalnaker's moderate modal realism, though it also derives from Dretske's semantic theory of information. Informationalism is presented in Section 2 after the prerequisite stage setting in Section 1. Some applications are sketched in Section 3. Finally, a mathematical model of the theory is developed in Section 4.How (...) many times have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the truth? (Arthur Conan Doyle)You've put me in an impossible situation. (Anonymous)[N]othing we imagine is absolutely impossible. (David Hume). (shrink)

In this paper evidence will be provided that Wittgenstein’s intuition about the logic of colour relations is to be taken near-literally. Starting from the Aristotelian oppositions between propositions as represented in the logical square of oppositions on the one hand and oppositions between primary and secondary colors as represented in an octahedron on the other, it will be shown algebraically how definitions for the former carry over to the realm of colour categories and describe very precisely the relations obtaining between (...) the known primary and secondary colours. Linguistic evidence for the reality of the resulting isomorphism will be provided. For example, the vertices that resist natural single-item lexicalization in logic (such as the O-corner, for which there is no natural lexicalization *nall (=not all)) are not naturally lexicalized in the realm of colour terms either. From the perspective of the architecture of cognition, the isomorphism suggests that the foundations of logical oppositions and negation may well be much more deeply rooted in the physiological structure of human cognition than is standardly assumed. (shrink)

In the last 25 years, the concept of information has played a crucial role in communication theory, so much so that the terms information theory and communication theory are sometimes used almost interchangeably. It seems to us, however, that the notion of information is also destined to render valuable services to the student of induction and probability, of learning and reinforcement, of semantic meaning and deductive inference, as~well as of scientific method in general. The present volume is an attempt to (...) illustrate some of these uses of information concepts. In 'On Semantic Information' Hintikka summarizes some of his and his associates' recent work on information and induction, and comments briefly on its philosophical suggestions. Jamison surveys from the sub jectivistic point of view some recent results in 'Bayesian Information Usage'. Rosenkrantz analyzes the information obtained by experimen tation from the Bayesian and Neyman-Pearson standpoints, and also from the standpoint of entropy and related concepts. The much-debated principle of total evidence prompts Hilpinen to examine the problem of measuring the information yield of observations in his paper 'On the Information Provided by Observations'. Pietarinen addresses himself to the more general task of evaluating the systematizing ('explanatory') power of hypotheses and theories, a task which quickly leads him to information concepts. Domotor develops a qualitative theory of information and entropy. His paper gives what is probably the first axiomatization of a general qualitative theory of information adequate to guarantee a numerical representation of the standard sort. (shrink)

This book provides the first comprehensive introduction to Dynamic Logic. Among the many approaches to formal reasoning about programs, Dynamic Logic enjoys the singular advantage of being strongly related to classical logic. Its variants constitute natural generalizations and extensions of classical formalisms. For example, Propositional Dynamic Logic (PDL) can be described as a blend of three complementary classical ingredients: propositional calculus, modal logic, and the algebra of regular events. In First-Order Dynamic Logic (DL), the propositional calculus is replaced by classical (...) first-order predicate calculus. Dynamic Logic is a system of remarkable unity that is theoretically rich as well as of practical value. It can be used for formalizing correctness specifications and proving rigorously that those specifications are met by a particular program. Other uses include determining the equivalence of programs, comparing the expressive power of various programming constructs, and synthesizing programs from specifications. This book provides the first comprehensive introduction to Dynamic Logic. It is divided into three parts. The first part reviews the appropriate fundamental concepts of logic and computability theory and can stand alone as an introduction to these topics. The second part discusses PDL and its variants, and the third part discusses DL and its variants. Examples are provided throughout, and exercises and a short historical section are included at the end of each chapter. (shrink)

A Mathematical Introduction to Logic, Second Edition, offers increased flexibility with topic coverage, allowing for choice in how to utilize the textbook in a course. The author has made this edition more accessible to better meet the needs of today's undergraduate mathematics and philosophy students. It is intended for the reader who has not studied logic previously, but who has some experience in mathematical reasoning. Material is presented on computer science issues such as computational complexity and database queries, with additional (...) coverage of introductory material such as sets. Increased flexibility of the text, allowing instructors more choice in how they use the textbook in courses. Reduced mathematical rigour to fit the needs of undergraduate students. (shrink)

The paper first introduces a cube of opposition that associates the traditional square of opposition with the dual square obtained by Piaget’s reciprocation. It is then pointed out that Blanché’s extension of the square-of-opposition structure into an conceptual hexagonal structure always relies on an abstract tripartition. Considering quadripartitions leads to organize the 16 binary connectives into a regular tetrahedron. Lastly, the cube of opposition, once interpreted in modal terms, is shown to account for a recent generalization of formal concept analysis, (...) where noticeable hexagons are also laid bare. This generalization of formal concept analysis is motivated by a parallel with bipolar possibility theory. The latter, albeit graded, is indeed based on four graded set functions that can be organized in a similar structure. (shrink)

Although the two volumes of _Logic, Language, and Meaning_ can be used independently of one another, together they provide a comprehensive overview of modern logic as it is used as a tool in the analysis of natural language. Both volumes provide exercises and their solutions. Volume 1, _Introduction to Logic_, begins with a historical overview and then offers a thorough introduction to standard propositional and first-order predicate logic. It provides both a syntactic and a semantic approach to inference and validity, (...) and discusses their relationship. Although language and meaning receive special attention, this introduction is also accessible to those with a more general interest in logic. In addition, the volume contains a survey of such topics as definite descriptions, restricted quantification, second-order logic, and many-valued logic. The pragmatic approach to non-truthconditional and conventional implicatures are also discussed. Finally, the relation between logic and formal syntax is treated, and the notions of rewrite rule, automation, grammatical complexity, and language hierarchy are explained. (shrink)

Is what could have happened but never did as real as what did happen? What did happen, but isn't happening now, happened at another time. Analogously, one can say that what could have happened happens in another possible world. Whatever their views about the reality of such things as possible worlds, philosophers need to take this analogy seriously. Adriane Rini and Max Cresswell exhibit, in an easy step-by-step manner, the logical structure of temporal and modal discourse, and show that every (...) temporal construction has an exact parallel that requires a language that can refer to worlds, and vice versa. They make precise, in a way which can be articulated and tested, the claim that the parallel is at work behind even ordinary talk about time and modality. The book gives metaphysicians a sturdy framework for the investigation of time and modality - one that does not presuppose any particular metaphysical view. (shrink)

Jean-Yves Béziau Abstract In this paper I relate the story about the new rising of the square of opposition: how I got in touch with it and started to develop new ideas and to organize world congresses on the topic with subsequent publications.