In his logical papers, Leo Esakia studied corresponding ordered topological spaces and order-preserving mappings. Similar spaces and mappings appear in many other application areas such the analysis of causality in space-time. It is known that under reasonable conditions, both the topology and the original order relation $${\preccurlyeq}$$ can be uniquely reconstructed if we know the “interior” $${\prec}$$ of the order relation. It is also known that in some cases, we can uniquely reconstruct $${\prec}$$ (and hence, topology) from $${\preccurlyeq}$$. In (...) this paper, we show that, in general, under reasonable conditions, the open order $${\prec}$$ (and hence, the corresponding topology) can be uniquely determined from its closure $${\preccurlyeq}$$. (shrink)

In _Modal Logic for Open Minds,_ Johan van Benthem provides an up-to-date introduction to the field of modal logic, outlining its major ideas and exploring the numerous ways in which various academic fields have adopted it. Van Benthem begins with the basic theories of modal logic, semantics, bisimulation, and axiomatics, and also covers more advanced topics, such as expressive power and computational complexity. The book then moves to a wide range of applications, including new developments in information flow, intelligent agency, (...) and games. Taken together, the chapters show modal logic at the crossroads of philosophy, mathematics, linguistics, computer science, and economics. Most of the chapters are followed by exercises, making this volume ideal for undergraduate and graduate students in philosophy, computer science, symbolic systems, cognitive science, and linguistics. (shrink)

In this paper we discuss a view of the Machine Learning technique called Explanation-Based Learning (EBL) or Explanation-Based Generalization (EBG) as a process for the interpretation of vague concepts in logic-based models of law.The open-textured nature of legal terms is a well-known open problem in the building of knowledge-based legal systems. EBG is a technique which creates generalizations of given examples on the basis of background domain knowledge. We relate these two topics by considering EBG''s domain knowledge as corresponding to (...) statute law rules, and EBG''s training example as corresponding to a precedent case. (shrink)

Part Two contains the text-in German and English-of the first two chapters of Hegel's Logic, which cover such categories as being, becoming, something, limit, ...

First published in 1988. Routledge is an imprint of Taylor & Francis, an informa company.

I am honoured with and touched by the invitation of delivering the opening address of this Congress. Firstly, to see paraconsistent logic flourishing and growing, as we can readily see by simply glacing over the programme of this conference, is among one of my greatest joys. Secondly, and equally important, because this congress takes place in the University of Toruń.I am honoured for having lectured here, a most congenial and stimulating place, and could not think of a better place for (...) a conference dedicated to the memory of Stanisław Jaśkowski. In particular, I am delighted for having had a correspondence with him, and although I was deprived of the pleasure of meeting him personally, I was fortunate enough for having collaborated with some of his disciples, such as L. Dubikajtis and T. Kotas. All and all, Toruń in particular and Poland in general are for me a second home, for all the kindness and care everyone has shown to me over several years, since my very first visit to this country. (shrink)

In this note we compare propositional logics for closed substitutions and propositional logics for open substitutions in constructive arithmetical theories. We provide a strong example where these logics diverge in an essential way. We prove that for Markov's Arithmetic, that is, Heyting's Arithmetic plus Markov's principle plus Extended Church's Thesis, the logic of closed and the logic of open substitutions are the same.

Explores the relationship between philosophers' and psychoanalysts' attempts to discover how man thinks and perceives himself.

An introductory textbook on metalogic. It covers naive set theory, first-order logic, sequent calculus and natural deduction, the completeness, compactness, and Löwenheim-Skolem theorems, Turing machines, and the undecidability of the halting problem and of first-order logic. The audience is undergraduate students with some background in formal logic.

Dov Gabbay is a prolific logician just by himself. But beyond that, he is quite good at making other people investigate the many further things he cares about. As a result, King's College London has become a powerful attractor in our field worldwide. Thus, it is a great pleasure to be an organizer for one of its flagship events: the Augustus de Morgan Workshop of 2005. Benedikt Loewe and I proposed the topic of 'interactive logic' for this occasion, with an (...) emphasis on social software – the logical analysis and design of social procedures – and on games, arguably the formal interactive setting par excellence. This choice reflects current research interests in our logic community at ILLC Amsterdam and beyond. In this broad area of interfaces between logic, computer science, and game theory, this paper is my own attempt at playing Dov. I am, perhaps not telling, but at least asking other people to find out for me what I myself cannot. (shrink)

I discuss a collection of problems in relevance logic. The main problems discussed are: the decidability of the positive semilattice system, decidability of the fragments of R in a restricted number of variables, and the complexity of the decision problem for the implicational fragment of R. Some related problems are discussed along the way.

First published in 1988. Routledge is an imprint of Taylor & Francis, an informa company.

A textbook for modal and other intensional logics based on the Open Logic Project. It covers normal modal logics, relational semantics, axiomatic and tableaux proof systems, intuitionistic logic, and counterfactual conditionals.

Modal logicism is the view that a metaphysical possibility is just a non-absurd way for the world to be. I argue that modal logicists should see metaphysical possibility as "open ended'': any given possibilities can be used to characterize further possibilities. I then develop a formal framework for modal languages that is a good fit for the modal logicist and show that it delivers some attractive results.

Putting the ‘empiricism’ in ‘logical empiricism’: the director’s cut Content Type Journal Article DOI 10.1007/s11016-010-9444-x Authors Greg Frost-Arnold, Department of Philosophy, Hobart and William Smith Colleges, Geneva, NY 14456, USA Journal Metascience Online ISSN 1467-9981 Print ISSN 0815-0796.

The mediatization concept rests on the increasing centrality of media in everyday spheres. Within academia, mediatization is explored in various ways, such as through the use of social media, news media, and researchers’ adoption of certain media logic(s). While many studies focus on media logic(s) as an explanatory device, it can also be seen as a contextual relationship between actors enacted for various purposes. This paper explores how academics enact media logic(s) in climate communication and for what purpose. By drawing (...) on interviews with initiators of open letters on climate change, this paper illustrates that media logic(s) is evoked as a publishing strategy and a way to demarcate academics from news media journalists. The study thus suggests a conceptual shift from debates about what media logic(s) is to what it does in specific communication contexts. (shrink)

In this paper we aim to disentangle the thesis that the future is open from theses that often get associated or even conflated with it. In particular, we argue that the open future thesis is compatible with both the unrestricted principle of bivalence and determinism with respect to the laws of nature. We also argue that whether or not the future (and indeed the past) is open has no consequences as to the existence of (past and) future ontology.

Elementary Logic explains what logic is, how it is done, and why it can be exciting. The book covers the central part of logic that all students have to learn: propositional logic. It aims to provide a crystal-clear introduction to what is often regarded as the most technically difficult area in philosophy. The book opens with an explanation of what logic is and how it is constructed. Subsequent chapters take the reader step-by-step through all aspects of elementary logic. Throughout, ideas (...) are explained simply and directly, with the chapters packed with overviews, illustrative examples, and summaries. Each chapter builds on previous explanation and example, with the final chapters presenting more advanced methods. After a discussion of meta-logic and logical systems, the book closes with an exploration of how paradoxes can exist in the world of logic. Elementary Logic's clarity and engagement make it ideal for any reader studying logic for the first time. (shrink)

This short paper about fuzzy set-based approximate reasoning first emphasizes the three main semantics for fuzzy sets: similarity, preference and uncertainty. The difference between truth-functional many-valued logics of vague or gradual propositions and non fully compositional calculi such as possibilistic logic or similarity logics is stressed. Then, potentials of fuzzy set-based reasoning methods are briefly outlined for various kinds of approximate reasoning: deductive reasoning about flexible constraints, reasoning under uncertainty and inconsistency, hypothetical reasoning, exception-tolerant plausible reasoning using generic knowledge, interpolative (...) reasoning, and abductive reasoning. Open problems are listed in the conclusion. (shrink)

... to discuss the state of epistemology, inspired by new perspectives and possibilities opened by Hintikka's recent papers on the study of epistemic logic, information, rationality, and inquiry as questioning."--Page 4 of cover.

Logical monism is the view that there is ‘One True Logic’. This is the default position, against which pluralists react. If there were not ‘One True Logic’, it is hard to see how there could be one true theory of anything. A theory is closed under a logic! But what is logical monism? In this article, I consider semantic, logical, modal, scientific, and metaphysical proposals. I argue that, on no ‘factualist’ analysis (according to which ‘there is One (...) True Logic’ expresses a factual claim, rather than an attitude like approval), does the doctrine have both metaphysical and methodological import. Metaphysically, logics abound. Methodologically, what to infer from what is not settled by the facts, even the normative ones. I conclude that the only interesting sense in which there could be One True Logic is noncognitive. The same may be true of monism about normative areas, like moral, epistemic, and prudential ones, generally. (shrink)

The end of human history is an event that has been foreseen or announced by both messianics and dialecticians. But who is the protagonist of that history that is coming—or has come—to a close? What is man? How did he come on the scene? And how has he maintained his privileged place as the master of, or first among, the animals? In The Open, contemporary Italian philosopher Giorgio Agamben considers the ways in which the “human” has been thought of as (...) either a distinct and superior type of animal, or a kind of being that is essentially different from animal altogether. In an argument that ranges from ancient Greek, Christian, and Jewish texts to twentieth-century thinkers such as Heidegger, Benjamin, and Kojève, Agamben examines the ways in which the distinction between man and animal has been manufactured by the logical presuppositions of Western thought, and he investigates the profound implications that the man/animal distinction has had for disciplines as seemingly disparate as philosophy, law, anthropology, medicine, and politics. (shrink)

Gaining information can be modelled as a narrowing of epistemic space . Intuitively, becoming informed that such-and-such is the case rules out certain scenarios or would-be possibilities. Chalmers’s account of epistemic space treats it as a space of a priori possibility and so has trouble in dealing with the information which we intuitively feel can be gained from logical inference. I propose a more inclusive notion of epistemic space, based on Priest’s notion of open worlds yet which contains only (...) those epistemic scenarios which are not obviously impossible. Whether something is obvious is not always a determinate matter and so the resulting picture is of an epistemic space with fuzzy boundaries. (shrink)

This essay explores the validity of Gregory Boyd’s open theistic account of the nature of the future. In particular, it is an investigation into whether Boyd’s logical square of opposition for future contingents provides a model of reality for free will theists that can preserve both bivalence and a classical conception of omniscience. In what follows, I argue that it can.

This paper reframes the issue of appropriation, extraction, and dispossession through AI—an assemblage of machine learning models trained on big data—in terms of enclosure and foreclosure. While enclosures are the product of a well-studied set of operations pertaining to both the constitution of the sovereign State and the primitive accumulation of capital, here, I want to recover an older form of the enclosure operation to then contrast it with foreclosure to better understand the effects of current algorithmic rationality. I argue (...) that the act of enclosing is to be understood as a set of fundamental operations that consist in producing structural distinctions between inside and outside, inclusion and exclusion—whether by drawing lines on a map, constructing border walls, or by algorithmically categorizing and (mis)recognizing people and things. Tracking the transformation of an enclosure-logic into one of foreclosure, I show how current AI perpetuates and at the same time expends forms of extraction, exclusion, and dispossession toward a totalizing horizon. For, while the outside is essential and constitutive to the logic of enclosure, foreclosure, by contrast, is characterized by the “totalizing desire” to not leave anything out of it. The totalizing desire to move beyond the logic of the enclosure forecloses that the enclosure has just gotten bigger and that the operations of exclusion have been masked and displaced, making them harder to contest. (shrink)

This paper reframes the issue of appropriation, extraction, and dispossession through AI—an assemblage of machine learning models trained on big data—in terms of enclosure and foreclosure. While enclosures are the product of a well-studied set of operations pertaining to both the constitution of the sovereign State and the primitive accumulation of capital, here, I want to recover an older form of the enclosure operation to then contrast it with foreclosure to better understand the effects of current algorithmic rationality. I argue (...) that the act of enclosing is to be understood as a set of fundamental operations that consist in producing structural distinctions between inside and outside, inclusion and exclusion—whether by drawing lines on a map, constructing border walls, or by algorithmically categorizing and (mis)recognizing people and things. Tracking the transformation of an enclosure-logic into one of foreclosure, I show how current AI perpetuates and at the same time expends forms of extraction, exclusion, and dispossession toward a totalizing horizon. For, while the outside is essential and constitutive to the logic of enclosure, foreclosure, by contrast, is characterized by the “totalizing desire” to not leave anything out of it. The totalizing desire to move beyond the logic of the enclosure forecloses that the enclosure has just gotten bigger and that the operations of exclusion have been masked and displaced, making them harder to contest. (shrink)

In this pioneering work, the author develops a fundamental formulation of logic in terms of theory of matrices and vector spaces. The discovery of matrix logic represents a landmark in the further formalization of logic. For the first time the power of direct mathematical computation is applied to the whole set of logic operations, allowing the derivation of both the classical and modal logics from the same formal base. The new formalism allows the author to enlarge the alphabet of the (...) truth-values with negative logic antivalues and to link matrix logic descriptions with the Dirac formulation of quantum theory - a result having fundamental implications and repercussions for science as a whole. As a unified language which permits a logical examination of the underlying phenomena of quantum field theory and vice versa, matrix logic opens new avenues for the study of fundamental interactions and gives rise to a revolutionary conclusion that physics as such can be viewed and studied as a logic in the fundamental sense. Finally, modelling itself on exact sciences, matrix logic does not refute the classical logic but instead incorporates it as a special deterministic limit. The book requires multidisciplinary knowledge and will be of interest to physicists, mathematicians, computer scientists and engineers. (shrink)

Informational conceptions of logic are barely novel. We find them in the work of John Corcoran, in several papers on substructural and constructive logics by Heinrich Wansing, and in the interpretation of the Routley-Meyer semantics for relevant logics in terms of Barwises and Perrys theory of situations. Allo & Mares [2] present an informational account of logical consequence that is based on the content-nonexpantion platitude, but that also relies on a double inversion of the standard direction of explanation (in- (...) formation doesnt depend on a prior notion of meaning, but is used to naturalize meaning, and informational content is not defined relative to a pre-existing logical space, but that space is constructed relative to the level of abstraction at which information is assessed). In this paper I focus directly on one of the main ideas introduced in that paper, namely the contrast between logical discrimination and deductive strength, and use this contrast to (1) illustrate a number of open problems for an informational conception of logical consequence, (2) review its connection with the dynamic turn in logic, and (3) situate it relative to the research agenda of the philosophy of information. (shrink)

With his Logic of Incarnation, James K. A. Smith has provided a compelling critique of the universalizing tendencies in some strands of postmodern philosophy of religion. A truly postmodern account of religion must take seriously the preference for particularity first evidenced in the Christian account of the incarnation of God. Moving beyond the urge to universalize, which characterizes modern thought, Smith argues that it is only by taking seriously particular differences--historical, religious, and doctrinal--that we can be authentically religious and authentically (...) postmodern. Smith remains hugely influential in both academic discourse and church movements. This book is the first organized attempt to bring both of these aspects of Smith's work into conversation with each other and with him. With articles from an internationally respected group of philosophers, theologians, pastors, and laypeople, the entire range of Smith's considerable influence is represented here. Discussing questions of embodiment, eschatology, inter-religious dialogue, dogma, and difference, this book opens all the most relevant issues in postmodern religious life to a unique and penetrating critique. (shrink)

When discussing Logical Pluralism several critics argue that such an open-minded position is untenable. The key to this conclusion is that, given a number of widely accepted assumptions, the pluralist view collapses into Logical Monism. In this paper we show that the arguments usually employed to arrive at this conclusion do not work. The main reason for this is the existence of certain substructural logics which have the same set of valid inferences as Classical Logic—although they are, in (...) a clear sense, non-identical to it. We argue that this phenomenon can be generalized, given the existence of logics which coincide with Classical Logic regarding a number of metainferential levels—although they are, again, clearly different systems. We claim this highlights the need to arrive at a more refined version of the Collapse Argument, which we discuss at the end of the paper. (shrink)

Open texture is a kind of semantic indeterminacy first systematically studied by Waismann. In this paper, extant definitions of open texture will be compared and contrasted, with a view towards the consequences of open-textured concepts in mathematics. It has been suggested that these would threaten the traditional virtues of proof, primarily the certainty bestowed by proof-possession, and this suggestion will be critically investigated using recent work on informal proof. It will be argued that informal proofs have virtues that mitigate the (...) danger posed by open texture. Moreover, it will be argued that while rigor in the guise of formalisation and axiomatisation might banish open texture from mathematical theories through implicit definition, it can do so only at the cost of restricting the tamed concepts in certain ways. (shrink)

Dense pairs of geometric topological fields have tame open core, that is, every definable open subset in the pair is already definable in the reduct. We fix a minor gap in the published version of van den Dries's seminal work on dense pairs of o-minimal groups, and show that every definable unary function in a dense pair of geometric topological fields agrees with a definable function in the reduct, off a small definable subset, that is, a definable set internal to (...) the predicate. For certain dense pairs of geometric topological fields without the independence property, whenever the underlying set of a definable group is contained in the dense-codense predicate, the group law is locally definable in the reduct as a geometric topological field. If the reduct has elimination of imaginaries, we extend this result, up to interdefinability, to all groups internal to the predicate. (shrink)

Review by: Johanna N. Y. Franklin The Bulletin of Symbolic Logic, Volume 19, Issue 1, Page 115-118, March 2013.

This book introduces the reader to relevant logic and provides the subject with a philosophical interpretation. The defining feature of relevant logic is that it forces the premises of an argument to be really used in deriving its conclusion. The logic is placed in the context of possible world semantics and situation semantics, which are then applied to provide an understanding of the various logical particles and natural language conditionals. The book ends by examining various applications of relevant logic (...) and presenting some interesting open problems. It will be of interest to a range of readers including advanced students of logic, philosophical and mathematical logicians, and computer scientists. (shrink)

Recent work in formal semantics suggests that the language system includes not only a structure building device, as standardly assumed, but also a natural deductive system which can determine when expressions have trivial truth-conditions (e.g., are logically true/false) and mark them as unacceptable. This hypothesis, called the `logicality of language', accounts for many acceptability patterns, including systematic restrictions on the distribution of quantifiers. To deal with apparent counter-examples consisting of acceptable tautologies and contradictions, the logicality of language is often paired (...) with an additional assumption according to which logical forms are radically underspecified: i.e., the language system can see functional terms but is `blind' to open class terms to the extent that different tokens of the same term are treated as if independent. This conception of logical form has profound implications: it suggests an extreme version of the modularity of language, and can only be paired with non-classical---indeed quite exotic---kinds of deductive systems. The aim of this paper is to show that we can pair the logicality of language with a different and ultimately more traditional account of logical form. This framework accounts for the basic acceptability patterns which motivated the logicality of language, can explain why some tautologies and contradictions are acceptable, and makes better predictions in key cases. As a result, we can pursue versions of the logicality of language in frameworks compatible with the view that the language system is not radically modular vis-a-vis its open class terms and employs a deductive system that is basically classical. (shrink)

Book review. Stephen Houlgate on Hegel's logic.

Many of the terms of our language, such as “jar”, are open-textured in the sense that their applicability to novel objects is not entirely determined by their past usage. Many others, such as the verbs “use” and “have”, are schematic in the sense that they have only a very general meaning although on any particular occasion of use they denote some more particular relation. The phenomena of open texture and schematicity constitute a sharp challenge to referential semantics, which assumes that (...) every non-logical term has a definite extension. A different, non-referential approach to formal semantics defines truth as relative to a context and defines contexts as built up from exclusively linguistic entities. For any given utterance of a sentence, there will be one of these contexts that pertains to it. In this framework, open texture and schematicity can be understood as consequences of the complex nature of the pertaining relation between contexts and utterances. (shrink)