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."

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)

Strong reasoning skills are an important aspect to cultivate in life, as they directly impact decision making on a daily basis. By examining the different ways the world views logic and order, new methods and techniques can be employed to help expand on this skill further in the future. Philosophical Perceptions on Logic and Order is a pivotal scholarly resource that discusses the evolution of logical reasoning and future applications for these types of processes. Highlighting relevant topics including logic patterns, deductive logic, and inductive logic, this publication is an ideal reference source for academicians, students, and researchers that would like to expand their understanding of how society currently employs the use of logical reasoning techniques.

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.

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.

Alexius Meinong claimed to uncover a brave new world of nonexistent objects. He contended that unreal objects, such as the golden mountain and the round square, genuinely had properties (such as nonexistence itself) and therefore, deserved a place in an all-inclusive science. Meinong's notion of nonexistents was initially not well-received, largely due to the influence and criticisms of Bertrand Russell. However, it has gained considerable popularity in more recent years as academics have uncovered shortfalls in Russell's philosophy and strived to explain apparent "facts" about the beingless. Some philosophers have continued Meinong's project, further explaining nonexistent objects or formulating logic systems that incorporate them. The more recent developments beg for a re-examination of Meinongianism. This book does just that, putting the theory on trial. Part One considers if Russell truly defeated Meinongianism. It addresses Meinongian rejoinders in response to Russell's main criticisms and further defends Russell's alternative solution, his Theory of Descriptions. Part Two explores the rationale for nonexistents and their use in interpreting three types of statements: characterization, negative existential, and intentional. The book argues that, despite appearances, Meinongianism cannot plausibly account for its own paradigm claims, whereas Russell's framework, with some further elucidation, can explain these statements quite well. Part Three primarily addresses claims about fiction, exploring the short-comings of Meinongian "and" Russellian frameworks in interpreting them. The book introduces a contextualization solution and symbolic method for capturing the logical form of such claims - one with the complexity to handle cross-contextual statements, including negative existential and intentional ones. It finally considers where that leaves nonexistent objects, ultimately rejecting such so-called entities.

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.

This work addresses the primary form of energy which runs through, holds together, furthers and maintains creation or the universe. It alerts the reader or listener to the unique nature of man, which equips him with the ability to direct the flow and intensity of this energy in his material surroundings, as well as makes him responsible for whatever happens in them. In this time, when all happenings and developments are coming together at increasingly quickening rates, he must come to thoroughly understand the true nature of his abilities, so as to be able to make that final decision which must lead him quickly upwards to where he can consciously and infinitely partake in upbuilding activities in creation, or, if he decides differently, have him quickly sink to regions the activities of which must lead to his being forever unable to do so.

"You've Got to Be Kidding : How Jokes Can Help You Think" is a thoughtful and accessible analysis of the ways in which jokes illustrate how we think critically, and how the thinking process goes awry in everyday human situations Uses jokes to illustrate the various mistakes or fallacies that are typically identified and discussed in courses on critical reasoningProvides an effective way to learn critical thinking skills since jokes often describe real-life situations where it really matters whether a person thinks well or notDemonstrates how philosophy is actually very practical and clearly related to real- life human experiencesExplains how developing good reasoning habits can make a real difference in all aspects of one's life

This is a concise and readable study of five intertwined themes at the heart of Wittgenstein's thought, written by one of his most eminent interpreters. David Pears offers penetrating investigations and lucid explications of some of the most influential and yet puzzling writings of twentieth-century philosophy. He focuses on the idea of language as a picture of the world; the phenomenon of linguistic regularity; the famous "private language argument"; logical necessity; and ego and the self.

This book presents the state of the art in the fields of formal logic pioneered by Graham Priest. It includes advanced technical work on the model and proof theories of paraconsistent logic, in contributions from top scholars in the field. Graham Priest's research has had a considerable influence on the field of philosophical logic, especially with respect to the themes of dialetheism-the thesis that there exist true but inconsistent sentences-and paraconsistency-an account of deduction in which contradictory premises do not entail the truth of arbitrary sentences. Priest's work has regularly challenged researchers to reappraise many assumptions about rationality, ontology, and truth. This book collects original research by some of the most esteemed scholars working in philosophical logic, whose contributions explore and appraise Priest's work on logical approaches to problems in philosophy, linguistics, computation, and mathematics. They provide fresh analyses, critiques, and applications of Priest's work and attest to its continued relevance and topicality. The book also includes Priest's responses to the contributors, providing a further layer to the development of these themes .

This volume collects the most important articles on the metaphysics of modality by noted philosopher Alvin Plantinga. The book chronicles Plantinga's thought from the late 1960's to the present. Plantinga is here concerned with fundamental issues in metaphysics: what is the nature of abstract objects like possible worlds,properties, propositions, and such phenomena? Are there possible but non-actual objects? Can objects that do not exist exemplify properties? In this thorough and searching book, Plantinga addresses these and many other questions that continue to preoccupy philosophers in the field. This volume contains some of the best work in metaphysics from the past 30 years, and will remain a source of critical contention and keen interest among philosophers of metaphysics and philosophical logic for years to come.

Barry Taylor's book mounts an argument against one of the fundamental tenets of much contemporary philosophy, the idea that we can make sense of reality as existing objectively, independently of our capacities to come to know it. Part One sets the scene by arguings that traditional realism can be explicated as a doctrine about truth - that truth is objective, that is, public, bivalent, and epistemically independent. Part Two, the centrepiece of the book, shows how a form of Hilary Putnam's model-theoretic argument demonstrates that no such notion of truth can be founded on the idea of correspondence, as explained in model-theoretic terms (more traditional accounts of correspondence having been already disposed of in Part One). Part Three argues that non-correspondence accounts of truth - truth as superassertibility or idealized rational acceptability, formal conceptions of truth, Tarskian truth - also fail to meet the criteria for objectivity; along the way, it also dismisses the claims of the latterday views of Putnam, and of similar views articulated by John McDowell, to constitute a new, less traditional form of realism. In the Coda, Taylor bolsters some of the considerations advanced in Part Three in evaluating formal conceptions of truth, by assessing and rejecting the claims of Robert Brandom to have combined such an account of truth with a satisfactory account of semantic structure. He concludes that there is no defensible notion of truth which preserves the theses of traditional realism, nor any extant position sufficiently true to the ideals of that doctrine to inherit its title. So the only question remaining is which form of antirealism to adopt.

This book addresses the primary form of energy in Creation and the need for its right recognition and usage for the development of mankind and their surroundings. Over millennia this energy has been misdirected by man due to his ignorance of its existence and nature of working. As a consequence, only works that have led to despair, chaos and destruction have formed in great numbers and grown through the thinking and general activity of men. The right use of this energy always leads to upbuilding in Creation, because it brings together and holds forms and substances in ways that only encourage upward growth and beauty.Loss of this energy through non-recognition and lack of absorption and right use leads to a decrease in its supply and in the effects of its magnetic and upbuilding power. Disintegration or "a falling apart" results from this lack of or reduction in the flow of the currents of this primary energy and takes the form of the break-down in peace, morality and the general ability to experience true joy by mankind.In this hour of twelve, when the fruits of the wrong use of this power are returning to man in earnest, and with great force and rapidity, with the threat of destroying him and his world, it is paramount that its true nature becomes known to man so that he may respond rightly to what comes to him in this time and avert the irreversible destructive and devastating fate that lies at the end of the path upon which he has treaded now for ages.

Kit Fine has since the 1970s been one of the leading contributors to work at the intersection of logic and metaphysics. This is his eagerly-awaited first book in the area. It draws together a series of essays, three of them previously unpublished, on possibility, necessity, and tense. These puzzling aspects of the way the world is have been the focus of considerable philosophical attention in recent decades. Fine gives here the definitive exposition and defence of certain positions for which he is well known: the intelligibility of modality de re; the primitiveness of the modal; and the primacy of the actual over the possible. But the book also argues for several positions that are not so familiar: the existence of distinctive forms of natural and normative necessity, not reducible to any form of metaphysical necessity; the need to make a distinction between the worldly and the unworldly, analogous to the distinction between the tensed and the tenseless; and the viability of a non-standard form of realism about tense, which recognizes the tensed character of reality without conceding that there is any privileged standpoint from which it is to be viewed. Modality and Tense covers a wide range of topics from many different areas: the possible-worlds analysis of counterfactuals; the compatibility of special relativity with presentism; the implications of ethical naturalism; and the nature of first-personal experience. A helpful introduction orients the reader and offers a way into some of the most original work in contemporary philosophy.