"An Introduction to the History of Philosophical and Formal Logic" introduces ideas and thinkers central to the development of philosophical and formal logic. From its Aristotelian origins to the present-day arguments, logic is broken down into four main time periods: -Antiquity and the Middle Ages (Aristotle and The Stoics) -The early modern period (Leibniz, Bolzano, Boole) -High modern period (Frege, Peano & Russell and Hilbert)-Early 20th century: (Godel and Tarski) Each new time frame begins with an introductory overview highlighting themes and points of importance. Chapters discuss the significance and reception of influential works and look at historical arguments in the context of contemporary debates. To support independent study, comprehensive lists of primary and secondary reading are included at the end of chapters, along with exercises and discussion questions.By clearly presenting and explaining the changes to logic across the history of philosophy, "An Introduction to the History of Philosophical and Formal Logic" constructs an easy-to-follow narrative. This is an ideal starting point for students looking to understand the historical development of logic.

Sortal concepts are at the center of certain logical discussions and have played a significant role in solutions to particular problems in philosophy. Apart from logic and philosophy, the study of sortal concepts has found its place in specific fields of psychology, such as the theory of infant cognitive development and the theory of human perception. In this monograph, different formal logics for sortal concepts and sortal-related logical notions (such as sortal identity and first-order sortal quantification) are characterized. Most of these logics are intensional in nature and possess, in addition, a bidimensional character. That is, they simultaneously represent two different logical dimensions. In most cases, the dimensions are those of time and natural necessity, and, in other cases, those of time and epistemic necessity. Another feature of the logics in question concerns second-order quantification over sortal concepts, a logical notion that is also represented in the logics. Some of the logics adopt a constant domain interpretation, others a varying domain interpretation of such quantification. Two of the above bidimensional logics are philosophically grounded on predication sortalism, that is, on the philosophical view that predication necessarily requires sortal concepts. Another bidimensional logic constitutes a logic for complex sortal predicates. These three sorts of logics are among the important novelties of this work since logics with similar features have not been developed up to now, and they might be instrumental for the solution of philosophically significant problems regarding sortal predicates. The book assumes a modern variant of conceptualism as a philosophical background. For this reason, the approach to sortal predicates is in terms of sortal concepts. Concepts, in general, are here understood as intersubjective realizable cognitive capacities. The proper features of sortal concepts are determined by an analysis of the main features of sortal predicates. Posterior to this analysis, the sortal-related logical notions represented in the above logics are discussed. There is also a discussion on the extent to which the set-theoretic formal semantic systems of the book capture different aspects of the conceptualist approach to sortals. These different semantic frameworks are also related to realist and nominalist approaches to sortal predicates, and possible modifications to them are considered that might represent those alternative approaches.

This comprehensive text shows how various notions of logic can be viewed as notions of universal algebra providing more advanced concepts for those who have an introductory knowledge of algebraic logic, as well as those wishing to delve into more theoretical aspects.

The capacity to represent things to ourselves as possible plays a crucial role both in everyday thinking and in philosophical reasoning; this volume offers much-needed philosophical illumination of conceivability, possibility, and the relations between them.

G. E. Moore famously observed that to assert, 'I went to the pictures last Tuesday but I don't believe that I did' would be 'absurd'. Moore calls it a 'paradox' that this absurdity persists despite the fact that what I say about myself might be true. Over half a century later, such sayings continue to perplex philosophers and other students of language, logic, and cognition. Ludwig Wittgenstein was fascinated by Moore's example, and the absurdity of Moore's saying was intensively discussed in the mid-20th century. Yet the source of the absurdity has remained elusive, and its recalcitrance has led researchers in recent decades to address it with greater care. In this definitive treatment of the problem of Moorean absurdity Green and Williams survey the history and relevance of the paradox and leading approaches to resolving it, and present new essays by leading thinkers in the area. Contributors Jonathan Adler, Bradley Armour-Garb, Jay D. Atlas, Thomas Baldwin, Claudio de Almeida, Andre Gallois, Robert Gordon, Mitchell Green, Alan Hajek, Roy Sorensen, John Williams

What is truth? Michael Lynch defends a bold new answer to this question. Traditional theories of truth hold that truth has only a single uniform nature. All truths are true in the same way. More recent deflationary theories claim that truth has no nature at all; the concept of truth is of no real philosophical importance. In this concise and clearly written book, Lynch argues that we should reject both these extremes and hold that truth is a functional property. To understand truth we must understand what it does, its function in our cognitive economy. Once we understand that, we'll see that this function can be performed in more than one way. And that in turn opens the door to an appealing pluralism: beliefs about the concrete physical world needn't be true in the same way as our thoughts about matters -- like morality -- where the human stain is deepest.

This is a guide to the thought and ideas of Gottlob Frege, one of the most important but also perplexing figures in the history of analytic philosophy. Gottlob Frege is regarded as one of the founders of modern logic and analytic philosophy, indeed as the greatest innovator in logic since Aristotle. His groundbreaking work identified many of the basic conceptions and distinctions that later came to dominate analytic philosophy. The literature on him is legion and ever-growing in complexity, representing a considerable challenge to the non-expert. The details of his logic, which have come into focus in recent research, are particularly difficult to grasp, although they are crucial to the development of his grand project, the reduction of arithmetic to logic, and the associated philosophical innovations. This book offers a lucid and accessible introduction to Frege's logic, taking the reader directly to the core of his philosophy, and ultimately to some of the most pertinent issues in contemporary philosophy of language, logic, mathematics, and the mind. "Continuum's Guides for the Perplexed" are clear, concise and accessible introductions to thinkers, writers and subjects that students and readers can find especially challenging - or indeed downright bewildering. Concentrating specifically on what it is that makes the subject difficult to grasp, these books explain and explore key themes and ideas, guiding the reader towards a thorough understanding of demanding material.

Reference is a central topic in philosophy of language, and has been the main focus of discussion about how language relates to the world. R. M. Sainsbury sets out a new approach to the concept, which promises to bring to an end some long-standing debates in semantic theory. There is a single category of referring expressions, all of which deserve essentially the same kind of semantic treatment. Included in this category are both singular and plural referring expressions ('Aristotle', 'The Pleiades'), complex and non-complex referring expressions ('The President of the USA in 1970', 'Nixon'), and empty and non-empty referring expressions ('Vulcan', 'Neptune'). Referring expressions are to be described semantically by a reference condition, rather than by being associated with a referent. In arguing for these theses, Sainsbury's book promises to end the fruitless oscillation between Millian and descriptivist views. Millian views insist that every name has a referent, and find it hard to give a good account of names which appear not to have referents, or at least are not known to do so, like ones introduced through error ('Vulcan'), ones where it is disputed whether they have a bearer ('Patanjali') and ones used in fiction. Descriptivist theories require that each name be associated with some body of information. These theories fly in the face of the fact names are useful precisely because there is often no overlap of information among speakers and hearers. The alternative position for which the book argues is firmly non-descriptivist, though it also does not require a referent. A much broader view can be taken of which expressions are referring expressions: not just names and pronouns used demonstratively, but also some complex expressions and some anaphoric uses of pronouns. Sainsbury's approach brings reference into line with truth: no one would think that a semantic theory should associate a sentence with a truth value, but it is commonly held that a semantic theory should associate a sentence with a truth condition, a condition which an arbitrary state of the world would have to satisfy in order to make the sentence true. The right analogy is that a semantic theory should associate a referring expression with a reference condition, a condition which an arbitrary object would have to satisfy in order to be the expression's referent. Lucid and accessible, and written with a minimum of technicality, Sainsbury's book also includes a useful historical survey. It will be of interest to those working in logic, mind, and metaphysics as well as essential reading for philosophers of language.

Frege's Theorem collects eleven essays by Richard G Heck, Jr, one of the world's leading authorities on Frege's philosophy. The Theorem is the central contribution of Gottlob Frege's formal work on arithmetic. It tells us that the axioms of arithmetic can be derived, purely logically, from a single principle: the number of these things is the same as the number of those things just in case these can be matched up one-to-one with those. But that principle seems so utterly fundamental to thought about number that it might almost count as a definition of number. If so, Frege's Theorem shows that arithmetic follows, purely logically, from a near definition. As Crispin Wright was the first to make clear, that means that Frege's logicism, long thought dead, might yet be viable.
Heck probes the philosophical significance of the Theorem, using it to launch and then guide a wide-ranging exploration of historical, philosophical, and technical issues in the philosophy of mathematics and logic, and of their connections with metaphysics, epistemology, the philosophy of language and mind, and even developmental psychology. The book begins with an overview that introduces the Theorem and the issues surrounding it, and explores how the essays that follow contribute to our understanding of those issues. There are also new postscripts to five of the essays, which discuss changes of mind, respond to published criticisms, and advance the discussion yet further.

This book is a tribute to Professor Ewa Orlowska, a Polish logician who was celebrating the 60th year of her scientific career in 2017. It offers a collection of contributed papers by different authors and covers the most important areas of her research. Prof. Orlowska made significant contributions to many fields of logic, such as proof theory, algebraic methods in logic and knowledge representation, and her work has been published in 3 monographs and over 100 articles in internationally acclaimed journals and conference proceedings. The book also includes Prof. Orlowska's autobiography, bibliography and a trialogue between her and the editors of the volume, as well as contributors' biographical notes, and is suitable for scholars and students of logic who are interested in understanding more about Prof. Orlowska's work.

Gary Kemp presents a penetrating investigation of key issues in the philosophy of language, by means of a comparative study of two great figures of late twentieth-century philosophy. So far as language and meaning are concerned, Willard Van Orman Quine and Donald Davidson are usually regarded as birds of a feather. The two disagreed in print on various matters over the years, but fundamentally they seem to be in agreement; most strikingly, Davidson's thought experiment of Radical Interpretation looks to be a more sophisticated, technically polished version of Quinean Radical Translation. Yet Quine's most basic and general philosophical commitment is to his methodological naturalism, which is ultimately incompatible with Davidson's main commitments. In particular, it is impossible to endorse, from Quine's perspective, the roles played by the concepts of truth and reference in Davidson's philosophy of language: Davidson's employment of the concept of truth is from Quine's point of view needlessly adventurous, and his use of the concept of reference cannot be divorced from unscientific 'intuition'. From Davidson's point of view, Quine's position looks needlessly scientistic, and seems blind to the genuine problems of language and meaning. Gary Kemp offers a powerful argument for Quine's position, and in favour of methodological naturalism and its corollary, naturalized epistemology. It is possible to give a consistent and explanatory account of language and meaning without problematic uses of the concepts truth and reference, which in turn makes a strident naturalism much more plausible.

This book is a consideration of Hegel's view on logic and basic logical concepts such as truth, form, validity, and contradiction, and aims to assess this view's relevance for contemporary philosophical logic. The literature on Hegel's logic is fairly rich. The attention to contemporary philosophical logic places the present research closer to those works interested in the link between Hegel's thought and analytical philosophy (Stekeler-Weithofer 1992 and 2019, Berto 2005, Rockmore 2005, Redding 2007, Nuzzo 2010 (ed.), Koch 2014, Brandom 2014, 1-15, Pippin 2016, Moyar 2017, Quante & Mooren 2018 among others). In this context, one particularity of this book consists in focusing on something that has been generally underrated in the literature: the idea that, for Hegel as well as for Aristotle and many other authors (including Frege), logic is the study of the forms of truth, i.e. the forms that our thought can (or ought to) assume in searching for truth. In this light, Hegel's thinking about logic is a fundamental reference point for anyone interested in a philosophical foundation of logic.

Contents: Introduction; I. ONTOLOGY; 1. Existence (1987); 2. Nonexistence (1998); 3. Mythical Objects (2002); II. NECESSITY; 4. Modal Logic Kalish-and-Montague Style (1994); 5. Impossible Worlds (1984); 6. An Empire of Thin Air (1988); 7. The Logic of What Might Have Been (1989); III. IDENTITY; 8. The fact that x=y (1987); 9. This Side of Paradox (1993); 10. Identity Facts (2003); 11. Personal Identity: What's the Problem? (1995); IV. PHILOSOPHY OF MATHEMATICS; 12. Wholes, Parts, and Numbers (1997); 13. The Limits of Human Mathematics (2001); V. THEORY OF MEANING AND REFERENCE; 14. On Content (1992); 15. On Designating (1997); 16. A Problem in the Frege-Church Theory of Sense and Denotation (1993); 17. The Very Possibility of Language (2001); 18. Tense and Intension (2003); 19. Pronouns as Variables (2005)

This book offers insight into the nature of meaningful discourse. It presents an argument of great intellectual scope written by an author with more than four decades of experience. Readers will gain a deeper understanding into three theories of the logos: analytic, dialectical, and oceanic. The author first introduces and contrasts these three theories. He then assesses them with respect to their basic parameters: necessity, truth, negation, infinity, as well as their use in mathematics. Analytic Aristotelian logic has traditionally claimed uniqueness, most recently in its Fregean and post-Fregean variants. Dialectical logic was first proposed by Hegel. The account presented here cuts through the dense, often incomprehensible Hegelian text. Oceanic logic was never identified as such, but the author gives numerous examples of its use from the history of philosophy. The final chapter addresses the plurality of the three theories and of how we should deal with it. The author first worked in analytic logic in the 1970s and 1980s, first researched dialectical logic in the 1990s, and discovered oceanic logic in the 2000s. This book represents the culmination of reflections that have lasted an entire scholarly career.

Necessary Beings is concerned with two central areas of metaphysics: modality-the theory of necessity, possibility, and other related notions; and ontology-the general study of what kinds of entities there are. Bob Hale's overarching purpose is to develop and defend two quite general theses about what is required for the existence of entities of various kinds: that questions about what kinds of things there are cannot be properly understood or adequately answered without recourse to considerations about possibility and necessity, and that, conversely, questions about the nature and basis of necessity and possibility cannot be satisfactorily tackled without drawing on what might be called the methodology of ontology. Taken together, these two theses claim that ontology and modality are mutually dependent upon one another, neither more fundamental than the other. Hale defends a broadly Fregean approach to metaphysics, according to which ontological distinctions among different kinds of things (objects, properties, and relations) are to be drawn on the basis of prior distinctions between different logical types of expression. The claim that facts about what kinds of things exist depend upon facts about what is possible makes little sense unless one accepts that at least some modal facts are fundamental, and not reducible to facts of some other, non-modal, sort. He argues that facts about what is absolutely necessary or possible have this character, and that they have their source or basis, not in meanings or concepts nor in facts about alternative 'worlds', but in the natures or essences of things.

Now in a new edition --the classic presentation of the theory of computable functions in the context of the foundations of mathematics. Part I motivates the study of computability with discussions and readings about the crisis in the foundations of mathematics in the early 20th century, while presenting the basic ideas of whole number, function, proof, and real number. Part II starts with readings from Turing and Post leading to the formal theory of recursive functions. Part III presents sufficient formal logic to give a full development of G del's incompleteness theorems. Part IV considers the significance of the technical work with a discussion of Church's Thesis and readings on the foundations of mathematics. This new edition contains the timeline "Computability and Undecidability" as well as the essay "On mathematics."

Many systems of logic diagrams have been offered both historically and more recently. Each of them has clear limitations. An original alternative system is offered here. It is simpler, more natural, and more expressively and inferentially powerful. It can be used to analyze not only syllogisms but arguments involving relational terms and unanalyzed statement terms.

This is an open access title available under the terms of a CC BY-NC-ND 4.0 International licence. It is free to read at Oxford Scholarship Online and offered as a free PDF download from OUP and selected open access locations. We need to understand the impossible. Francesco Berto and Mark Jago start by considering what the concepts of meaning, information, knowledge, belief, fiction, conditionality, and counterfactual supposition have in common. They are all concepts which divide the world up more finely than logic does. Logically equivalent sentences may carry different meanings and information and may differ in how they're believed. Fictions can be inconsistent yet meaningful. We can suppose impossible things without collapsing into total incoherence. Yet for the leading philosophical theories of meaning, these phenomena are an unfathomable mystery. To understand these concepts, we need a metaphysical, logical, and conceptual grasp of situations that could not possibly exist: Impossible Worlds. This book discusses the metaphysics of impossible worlds and applies the concept to a range of central topics and open issues in logic, semantics, and philosophy. It considers problems in the logic of knowledge, the meaning of alternative logics, models of imagination and mental simulation, the theory of information, truth in fiction, the meaning of conditional statements, and reasoning about the impossible. In all these cases, impossible worlds have an essential role to play.

Friedrich Ueberweg (1826-71) is best remembered for both his compendious "History of Philosophy" and his "System of Logic", both of which went through several editions in the original German. It was the latter's remarkable popularity as a textbook in Germany that led Lindsay to translate it to fill a gap in the English market. As well as incorporating the most up-to-date revisions and additons to the German edition he inserted the opinions of the more important English logicians. As such this is a valuable textbook for the understanding of logic systems as taught in England and Germany before symbolic logic was a formal and distinct discipline.

This volume examines the entire logical and philosophical production of Nicolai A. Vasil'ev, studying his life and activities as a historian and man of letters. Readers will gain a comprehensive understanding of this influential Russian logician, philosopher, psychologist, and poet. The author frames Vasil'ev's work within its historical and cultural context. He takes into consideration both the situation of logic in Russia and the state of logic in Western Europe, from the end of the 19th century to the beginning of the 20th. Following this, the book considers the attempts to develop non-Aristotelian logics or ideas that present affinities with imaginary logic. It then looks at the contribution of traditional logic in elaborating non-classical ideas. This logic allows the author to deal with incomplete objects just as imaginary logic does with contradictory ones. Both logics are objects of interesting analysis by modern researchers. This volume will appeal to graduate students and scholars interested not only in Vasil'ev's work, but also in the history of non-classical logics.

The aim of this volume is to collect original contributions by the best specialists from the area of proof theory, constructivity, and computation and discuss recent trends and results in these areas. Some emphasis will be put on ordinal analysis, reductive proof theory, explicit mathematics and type-theoretic formalisms, and abstract computations. The volume is dedicated to the 60th birthday of Professor Gerhard Jager, who has been instrumental in shaping and promoting logic in Switzerland for the last 25 years. It comprises contributions from the symposium "Advances in Proof Theory", which was held in Bern in December 2013. Proof theory came into being in the twenties of the last century, when it was inaugurated by David Hilbert in order to secure the foundations of mathematics. It was substantially influenced by Goedel's famous incompleteness theorems of 1930 and Gentzen's new consistency proof for the axiom system of first order number theory in 1936. Today, proof theory is a well-established branch of mathematical and philosophical logic and one of the pillars of the foundations of mathematics. Proof theory explores constructive and computational aspects of mathematical reasoning; it is particularly suitable for dealing with various questions in computer science.