The square of opposition is a diagram related to a theory of oppositions that goes back to Aristotle. Both the diagram and the theory have been discussed throughout the history of logic. Initially, the diagram was employed to present the Aristotelian theory of quantification, but extensions and criticisms of this theory have resulted in various other diagrams. The strength of the theory is that it is at the same time fairly simple and quite rich. The theory of oppositions has (...) recently become a topic of intense interest due to the development of a general geometry of opposition (polygons and polyhedra) with many applications. A congress on the square with an interdisciplinary character has been organized on a regular basis (Montreux 2007, Corsica 2010, Beirut 2012, Vatican 2014, Rapa Nui 2016). The volume at hand is a sequel to two successful books: The Square of Opposition - A General Framework of Cognition, ed. by J.-Y. Béziau & G. Payette, as well as Around and beyond the Square of Opposition, ed. by J.-Y. Béziau & D. Jacquette, and, like those, a collection of selected peer-reviewed papers. The idea of this new volume is to maintain a good equilibrium between history, technical developments and applications. The volume is likely to attract a wide spectrum of readers, mathematicians, philosophers, linguists, psychologists and computer scientists, who may range from undergraduate students to advanced researchers. (shrink)

The Fifth International Congress of Logic, Methodology and Philosophy of Science was held at the University of Western Ontario, London, Canada, 27 August to 2 September 1975. The Congress was held under the auspices of the International Union of History and Philosophy of Science, Division of Logic, Methodology and Philosophy of Science, and was sponsored by the National Research Council of Canada and the University of Western Ontario. As those associated closely with the work of the Division over (...) the years know well, the work undertaken by its members varies greatly and spans a number of fields not always obviously related. In addition, the volume of work done by first rate scholars and scientists in the various fields of the Division has risen enormously. For these and related reasons it seemed to the editors chosen by the Divisional officers that the usual format of publishing the proceedings of the Congress be abandoned in favour of a somewhat more flexible, and hopefully acceptable, method of pre sentation. Accordingly, the work of the invited participants to the Congress has been divided into four volumes appearing in the University of Western Ontario Series in Philosophy of Science. The volumes are entitled, Logic, Foundations of Mathematics and Computability Theory, Foun dational Problems in the Special Sciences, Basic Problems in Methodol ogy and Linguistics, and Historical and Philosophical Dimensions of Logic, Methodology and Philosophy of Science. (shrink)

Ancient dialectic started as an art of refutation and evolved into a science akin to our logic, grammar and linguistics. Scholars of ancient philosophy have traditionally focused on Plato's and Aristotle's dialectic without paying much attention to the diverse conceptions and uses of dialectic presented by philosophers after the classical period. To bridge this gap, this volume aims at a comprehensive understanding of the competing Hellenistic and Imperial definitions of dialectic and their connections with those of the classical period. (...) It starts from the Megaric school of the fourth century BCE and the early Peripatetics, via Epicurus, the Stoics, the Academic sceptics and Cicero, to Sextus Empiricus and Galen in the second century CE. The philosophical foundations and various uses of dialectic are closely analysed and systematically examined together with the numerous objections that were raised against them. (shrink)

I read Stefan Collini’s What are Universities For? last week with very mixed feelings. In the past, I’ve much admired his polemical essays on the REF, “impact”, the Browne Report, etc. in the London Review of Books and elsewhere: they speak to my heart. If you don’t know those essays, you can get some of their flavour from his latest article in the Guardian yesterday. But I found the book a disappointment. Perhaps the trouble is that Collini is too decent, (...) too high-minded, has too unrealistically exalted a view about what actually happens in universities which is too coloured by attitudes appropriate to the traditional humanities. And he is optimistic to the point of fantasy if he thinks that people are so susceptible to “the romance of ideas and the power of beauty” that they will want, or can be brought to want, lots and lots of universities in order to promote the ideas (as if they would suppose that the task of “conserving, understanding, extending and handing on to subsequent generations the intellectual, scientific, and artistic heritage of mankind” was clearly ill-served when there were only forty universities in England, as opposed to a hundred and whatever). (shrink)

A common reaction to Hegel’s suggestion that we collapse Kant’s distinction between form and content is that, since such a move would also deprive us of any way of distinguishing the merely logical from the real possibility of our concepts, it is incoherent and ought to be rejected. It is true that these two distinctions are intimately related in Kant, such that if one goes, the other does as well. But it is less obvious that giving them up as Kant (...) conceives them is as incoherent a proposal as many of Hegel’s critics think. It has been the point of a recent account of Hegel’s idealism to demonstrate that his critique of Kant’s dichotomy between form and content or concept and intuition does not commit him to the view that human cognition is materially creative of its content in the manner of a God-like or intuitive intellect—does not, in other words, signify a return to a pre-Critical metaphysics. This is the interpretative stance I adopt here as well, in hope of giving defenders of Kant reasons for taking Hegel more seriously. (shrink)

Hegel's Science of Logic has received less attention than his Phenomenology of Spirit, but Hegel himself took it to be his highest philosophical achievement and the backbone of his system. The present book focuses on this most difficult of Hegel’s published works. Béatrice Longuenesse offers a close analysis of core issues, including discussions of what Hegel means by ‘dialectical logic’, the role and meaning of ‘contradiction’ in Hegel’s philosophy, and Hegel’s justification for the provocative statement that ‘what is (...) actual is rational, what is rational is actual’. She examines both Hegel's debt and his polemical reaction to Kant, and shows in great detail how his project of a ‘dialectical’ logic can be understood only in light of its relation to Kant’s ‘transcendental’ logic. This book will appeal to anyone interested in Hegel's philosophy and its influence on contemporary philosophical discussion. (shrink)

Universal Logic is not a new logic, but a general theory of logics, considered as mathematical structures. The name was introduced about ten years ago, but the subject is as old as the beginning of modern logic: Alfred Tarski and other Polish logicians such as Adolf Lindenbaum developed a general theory of logics at the end of the 1920s based on consequence operations and logical matrices. The subject was revived after the flowering of thousands of new logics (...) during the last thirty years: there was a need for a systematic theory of logics to put some order in this chaotic multiplicity. This book contains recent works on universal logic by first-class researchers from all around the world. The book is full of new and challenging ideas that will guide the future of this exciting subject. It will be of interest for people who want to better understand what logic is. Tools and concepts are provided here for those who want to study classes of already existing logics or want to design and build new ones. (shrink)

In this collection of essays Béatrice Longuenesse considers the three aspects of Kant's philosophy, his epistemology and metaphysics of nature, his moral philosophy and his aesthetic theory, under one unifying standpoint: Kant's conception of our capacity to form judgements. She argues that the elements which make up our cognitive access to the world - what Kant calls the 'human point of view' - have an equally important role to play in our moral evaluations and our aesthetic judgements. Her discussion ranges (...) over Kant's account of our representations of space and time, his conception of the logical forms of judgements, sufficient reason, causality, community, God, freedom, morality, and beauty in nature and art. Her book will appeal to all who are interested in Kant and his thought. (shrink)

The logic of ‘being informed’ gives a formal analysis of a cognitive state that does not coincide with either belief, or knowledge. To Floridi, who first proposed the formal analysis, the latter is supported by the fact that unlike knowledge or belief, being informed is a factive, but not a reflective state. This paper takes a closer look at the formal analysis itself, provides a pure and an applied semantics for the logic of being informed, (...) and tries to find out to what extent the formal analysis can contribute to an information-based epistemology. (shrink)

Universal Logic is not a new logic, but a general theory of logics, considered as mathematical structures. The name was introduced about ten years ago, but the subject is as old as the beginning of modern logic: Alfred Tarski and other Polish logicians such as Adolf Lindenbaum developed a general theory of logics at the end of the 1920s based on consequence operations and logical matrices. The subject was revived after the flowering of thousands of new logics (...) during the last thirty years: there was a need for a systematic theory of logics to put some order in this chaotic multiplicity. This bo. (shrink)

Let $ be the restriction of usual order relation to integers which are primes or squares of primes, and let ⊥ denote the coprimeness predicate. The elementary theory of $\langle\mathbb{N};\bot, , is undecidable. Now denote by $ the restriction of order to primary numbers. All arithmetical relations restricted to primary numbers are definable in the structure $\langle\mathbb{N};\bot, . Furthermore, the structures $\langle\mathbb{N};\mid, and $\langle\mathbb{N};=,+,x\rangle$ are interdefinable.

One of the open problems in the philosophy of information is whether there is an information logic (IL), different from epistemic (EL) and doxastic logic (DL), which formalises the relation “a is informed that p” (Iap) satisfactorily. In this paper, the problem is solved by arguing that the axiom schemata of the normal modal logic (NML) KTB (also known as B or Br or Brouwer’s system) are well suited to formalise the relation of “being informed”. After (...) having shown that IL can be constructed as an informational reading of KTB, four consequences of a KTB-based IL are explored: information overload; the veridicality thesis (Iap → p); the relation between IL and EL; and the Kp → Bp principle or entailment property, according to which knowledge implies belief. Although these issues are discussed later in the article, they are the motivations behind the development of IL. (shrink)

In recent years, several remarkable results have shown that certain theorems of finite combinatorics are unprovable in certain logical systems. These developments have been instrumental in stimulating research in both areas, with the interface between logic and combinatorics being especially important because of its relation to crucial issues in the foundations of mathematics which were raised by the work of Kurt Godel. Because of the diversity of the lines of research that have begun to shed light on these (...) issues, there was a need for a comprehensive overview which would tie the lines together. This volume fills that need by presenting a balanced mixture of high quality expository and research articles that were presented at the August 1985 AMS-IMS-SIAM Joint Summer Research Conference, held at Humboldt State University in Arcata, California.With an introductory survey to put the works into an appropriate context, the collection consists of papers dealing with various aspects of 'unprovable theorems and fast-growing functions'. Among the topics addressed are: ordinal notations, the dynamical systems approach to Ramsey theory, Hindman's finite sums theorem and related ultrafilters, well quasiordering theory, uncountable combinatorics, nonstandard models of set theory, and a length-of-proof analysis of Godel's incompleteness theorem. Many of the articles bring the reader to the frontiers of research in this area, and most assume familiarity with combinatorics and/or mathematical logic only at the senior undergraduate or first-year graduate level. (shrink)

The First International Workshop brings together researchers from the theoretical ends of the logic programming and artificial intelligence communities to discuss their mutual interests. Logic programming deals with the use of models of mathematical logic as a way of programming computers, where theoretical AI deals with abstract issues in modeling and representing human knowledge and beliefs. One common ground is nonmonotonic reasoning, a family of logics that includes room for the kinds of variations that can be found (...) in human reasoning. Topics covered: Stable Semantics. Default Logic. AutoEpistemic Logic. Truth Maintenance Systems. Implementation Issues. Diagnosis. Applications. Inheritance Reasoning. Logics of Belief. Inconsistency and Non-Monotonicity. (shrink)

In the first part of the paper we study arithmetical properties of Pascal triangles modulo a prime power; the main result is the generalization of Lucas' theorem. Then we investigate the structure N; Bpx, where p is a prime, α is an integer greater than one, and Bpx = Rem, px); it is shown that addition is first-order definable in this structure, and that its elementary theory is decidable.

This chapter contains sections titled: The Problem: Perspectives on Method, Or, How to Approach Being Hegel's “Vorbegriff” of Logical Method Absolute Method and the Truth of Being The Method of the Logic of Being Conclusion.

Let θ, θ′ be two multiplicatively independent Pisot numbers, and letU,U′ be two linear numeration systems whose characteristic polynomial is the minimal polynomial of θ and θ′, respectively. For everyn≥ 1, ifA⊆ ℕnisU-andU′ -recognizable thenAis definable in 〈ℕ: + 〉.

Letkandlbe two multiplicatively independent integers, and letL⊆ ℕnbe al-recognizable set which is not definable in 〈ℕ; +〉. We prove that the elementary theory of 〈ℕ; +,Vk, L〉, whereVk(x)denotes the greatest power ofkdividingx, is undecidable. This result leads to a new proof of the Cobham-Semënov theorem.

Rationals and countable ordinals are important examples of structures with decidable monadic second-order theories. A chain is an expansion of a linear order by monadic predicates. We show that if the monadic second-order theory of a countable chain C is decidable then C has a non-trivial expansion with decidable monadic second-order theory.

In the Logic of Being: Realism, Truth, and Time, the influential philosopher Paul M. Livingston explores and illuminates truth, time, and their relationship by employing methods from both Continental and analytic philosophy.

The last twenty years have seen remarkable developments in our understanding of how the ancient Greek thinkers handled the general concept of being and its several varieties. The most general examination of the meaning of the Greek verb 'esti'/'einai'/'on' both in common usage and in the philosophical literature has been presented by Charles H. Kahn, most extensively in his 1973 book The Verb 'Be' in Ancient Greek. These discussions are summarized in Kahn's contribution to this volume. By and large, (...) they show that conceptual schemes by means of which philosophers have recently approached Greek thought have not been very well suited to the way the concept of being was actually used by the ancients. For one thing, being in the sense of existence played a very small role in Greek thinking according to Kahn. Even more importantly, Kahn has argued that Frege and Russell's thesis that verbs for being, such as 'esti', are multiply ambiguous is ill suited for the purpose of appreciating the actual conceptual assumptions of the Greek thinkers. Frege and Russell claimed that a verb like 'is' or'esti' is ambiguous between the 'is' of identity, the 'is' of existence, the copulative 'is', and the generic 'is' (the 'is' of class-inclusion). At least a couple of generations of scholars have relied on this thesis and fre quently criticized sundry ancients for confusing these different senses of 'esti' with each other. (shrink)

In his major work, the Periphyseon, the ninth century Latin philosopher John Scottus Eriugena gives, with the help of what he calls "dialectic", a rational analysis of reality. According to him, dialectic is a science which pertains both to language and reality. Eriugena grounds this position in a realist ontological exegesis of the Aristotelian categories, which are conceived as categories of being. His interpretation tends to transform logical patterns, such as Porphyry's Tree or the doctrine of the categories, into (...) a structure which is both ontological and logical, and to use them as tools for the analysis of the sensible world. The combination of dialectic interpreted as a science of being, capable of expressing truths about the sensible world as well as about discourse, with an ontological interpretation of logical concepts allows Eriugena to develop his metaphysical theory, a strong realism. Eriugena not only supports a theological realism, but also, and principally, an ontological realism, the assertion of the immanent existence of forms. Eriugena claims that genera and species really subsist in the individuals: they are completely and simultaneously present in each of the entities which belong to them. (shrink)

"French philosopher Alain Badiou (b. 1937) introduces the concept of democratic materialism to refer to the belief that there are only bodies and languages, then employs a fair amount of science in what he calls a somewhat fastidious examine of it. He covers formal theory on the subject (meta-physics); the greater logics: the transcendental, the object, and relation; the four forms of change; the theory of points; what a body is; and what it is to live."--Provided by publisher.

The central purpose of this paper is to sketch the logic of being a democrat. That is, what is involved in being a democrat will be defined and delineated. I shall proceed by first examining Richard Wollheim's alleged paradox of democratic theory. Wollheim's solution to the paradox will then be shown to be unsatisfactory. Next, the concept of being a democrat will be clarified. The stage will then be set for showing that Wollheim's alleged paradox of (...) democratic theory dissolves upon discerning what a democrat qua democrat is committed to believe. In the process of clarifying what is involved in being a democrat and dissolving the alleged paradox of democratic theory, it will become evident that one popular argument often levelled against resistance to democratic law is without foundation. This is the argument that any democrat who conscientiously disobeys valid democratic law is necessarily behaving inconsistently with his democratic principles; and, therefore, if someone does conscientiously disobey valid democratic law, it proves he is not ‘really’ a democrat after all. (shrink)

Recently a view of Hegel’s “idealism” which hitherto had seemed unquestionable—the view that it is fundamentally a metaphysical doctrine—has been seriously challenged. Thus yesterday’s metaphysical Hegel, complete with his cosmic megasubject hidden behind the events of nature and history, has been joined by today’s “nonmetaphysical Hegel,” the postkantian categorial “genealogist.” According to the nonmetaphysical Hegelians, a century and a half of misunderstanding has been based on the confusion of two distinct philosophical projects: on the one hand, that originating in the (...) Aristotelian project of “ontology”—the reconstruction of the categorial structure of to on e on, “being qua being”—and on the other, that stemming from the very different Aristotelian project of “theology”—the investigation of to theion, primary being or that which is first in the order of substances. (shrink)

This book provides an accessible and thorough analysis of "The Doctrine of Being," the first part of Hegel's Science of Logic. Though it received much scholarly attention in the past, interpreters of this text have generally refrained from examining it in a sufficiently detailed manner. Through a rigorous and critical reading of Hegel's speculative arguments, Mehmet Tabak illustrates that Hegel meant his logic to be both a presuppositionless analysis and development of the basic categories of thought, on (...) the one hand, and a post-Kantian ontology on the other. However, the analysis of the text demonstrates that Hegel fails to deliver such logic. This volume promises to be an indispensable guide to those who wish to understand the first book of Science of Logic. (shrink)

In this work Ockham proposes a theory of simple predication, which he uses in explicating the truth conditions of progressively more complicated kinds of propositions. His discussion includes what he takes to be the correct semantic treatment of quantified propositions, past tense and future tense propositions, and modal propositions, all of which are receiving much attention from contemporary philosophers. He also illustrates the use of exponential analysis to deal with propositions that prove troublesome in both semantic theory and other disciplines, (...) such as metaphysics, physics, and theology. This type of analysis plays an essential role in his substantive philosophical and theological works, and in many cases then can hardly be understood without a prior acquaintance with this section of the Summa. An introduction by Alfred J. Freddoso clarifies and summarizes Ockham's theory, discusses certain philosophical problems which it engenders, and proposes new interpretations for parts of it. The essay will be helpful both to those interested in Ockham as a historical figure and to those interested in his substantive systematic contributions to logical theory. (shrink)

COMMUNICATION 1. Every philosophy presupposes a theory of communication and conversely every theory of communication implies a philosophy. ...

This book provides a new philosophical, semantical and historical analysis of and justification for the relevant logic of entailment. Its fresh and original perspective on the logic of entailment will be valuable for all who want to know more about the historical and philosophical origins of modern symbolic logic.

Described by the philosopher A.J. Ayer as a work of 'great originality and power', this book revolutionized contemporary thinking on science and knowledge. Ideas such as the now legendary doctrine of 'falsificationism' electrified the scientific community, influencing even working scientists, as well as post-war philosophy. This astonishing work ranks alongside _The Open Society and Its Enemies_ as one of Popper's most enduring books and contains insights and arguments that demand to be read to this day.

"Alexander's commentary on Book 1 concerns the definition of Aristotelian syllogistic argument; its resistance to the rival Stoic theory of inference; and the character of inductive inference and of rhetorical argument. Alexander distinguishes inseparable accidents, such as the whiteness of snow, from defining differentiae, such as its being frozen, and considers how these differences fit into the schemes of categories. He speaks of dialectic as a stochastic discipline in which success is to be judged not by victory but by (...) skill in argument. Alexander also investigates the subject of ambiguity, which had been richly developed since Aristotle by the rival Stoic school."--BOOK JACKET. (shrink)

This article investigates the connections between the logics of being wrong, introduced in Steinsvold (2011, Notre Dame J. Form. Log., 52, 245–253), and factive ignorance, presented in Kubyshkina and Petrolo (2021, Synthese, 198, 5917–5928). The first part of the paper provides a sound and complete axiomatization of the logic of factive ignorance that corrects errors in Kubyshkina and Petrolo (2021, Synthese, 198, 5917–5928) and resolves questions about the expressivity of the language. In the second half, it is shown (...) that the relationship between the two logics suggests an alternative axiomatization for the logic of factive ignorance, the adequacy of which is easily proved via straightforward translations. (shrink)