Although Kurt Gödel does not figure prominently in the history of computabilty theory, he exerted a significant influence on some of the founders of the field, both through his published work and through personal interaction. In particular, Gödel’s 1931 paper on incompleteness and the methods developed therein were important for the early development of recursive function theory and the lambda calculus at the hands of Church, Kleene, and Rosser. Church and his students studied Gödel 1931, and Gödel taught a seminar (...) at Princeton in 1934. Seen in the historical context, Gödel was an important catalyst for the emergence of computability theory in the mid 1930s. (shrink)

Change in View offers an entirely original approach to the philosophical study of reasoning by identifying principles of reasoning with principles for revising one's beliefs and intentions and not with principles of logic. This crucial observation leads to a number of important and interesting consequences that impinge on psychology and artificial intelligence as well as on various branches of philosophy, from epistemology to ethics and action theory. Gilbert Harman is Professor of Philosophy at Princeton University. A Bradford Book.

In the early 1920s, the German mathematician David Hilbert (1862–1943) put forward a new proposal for the foundation of classical mathematics which has come to be known as Hilbert's Program. It calls for a formalization of all of mathematics in axiomatic form, together with a proof that this axiomatization of mathematics is consistent. The consistency proof itself was to be carried out using only what Hilbert called “finitary” methods. The special epistemological character of finitary reasoning then yields the required justification (...) of classical mathematics. Although Hilbert proposed his program in this form only in 1921, various facets of it are rooted in foundational work of his going back until around 1900, when he first pointed out the necessity of giving a direct consistency proof of analysis. Work on the program progressed significantly in the 1920s with contributions from logicians such as Paul Bernays, Wilhelm Ackermann, John von Neumann, and Jacques Herbrand. It was also a great influence on Kurt Gödel, whose work on the incompleteness theorems were motivated by Hilbert's Program. Gödel's work is generally taken to show that Hilbert's Program cannot be carried out. It has nevertheless continued to be an influential position in the philosophy of mathematics, and, starting with the work of Gerhard Gentzen in the 1930s, work on so-called Relativized Hilbert Programs have been central to the development of proof theory. (shrink)

The paper tries to spell out a connection between deductive logic and rationality, against Harman's arguments that there is no such connection, and also against the thought that any such connection would preclude rational change in logic. One might not need to connect logic to rationality if one could view logic as the science of what preserves truth by a certain kind of necessity (or by necessity plus logical form); but the paper points out a serious obstacle to any such (...) view. (shrink)

Stephen Cole Kleene was one of the greatest logicians of the twentieth century and this book is the influential textbook he wrote to teach the subject to the next generation. It was first published in 1952, some twenty years after the publication of Godel's paper on the incompleteness of arithmetic, which marked, if not the beginning of modern logic. The 1930s was a time of creativity and ferment in the subject, when the notion of computable moved from the realm of (...) philosophical speculation to the realm of science. This was accomplished by the work of Kurt Gode1, Alan Turing, and Alonzo Church, who gave three apparently different precise definitions of computable. When they all turned out to be equivalent, there was a collective realization that this was indeed the right notion. Kleene played a key role in this process. One could say that he was there at the beginning of modern logic. He showed the equivalence of lambda calculus with Turing machines and with Godel's recursion equations, and developed the modern machinery of partial recursive functions. This textbook played an invaluable part in educating the logicians of the present. It played an important role in their own logical education.". (shrink)

These two puzzles were classic if academic examples of the dilemmas Professor Ryle is concerned with.

Kant famously declares that “although all our cognition commences with experience, … it does not on that account all arise from experience”. This marks Kant’s disagreement with empiricism, and his contention that human knowledge and experience require both sensation and the use of certain a priori concepts, the Categories. However, this is only the surface of Kant’s much deeper, though neglected view about the nature of reason and judgment. Kant holds that even our a priori concepts are acquired, not from (...) sensation, but “originally,” because our mind has a fundamental capacity to judge that, upon sensory stimulation, generates the Categories through its basic logical functions of judgment. This “epigenesis” of reason and our fundamental capacity to judge that drives it is the topic of Longuenesse’s fascinating book, and the source of her title. (shrink)

In this manuscript, published here for the first time, Tarski explores the concept of logical notion. He draws on Klein's Erlanger Programm to locate the logical notions of ordinary geometry as those invariant under all transformations of space. Generalizing, he explicates the concept of logical notion of an arbitrary discipline.

We provide for the first time an exact translation into English of the Polish version of Alfred Tarski's classic 1936 paper, whose title we translate as ‘On the Concept of Following Logically’. We also provide in footnotes an exact translation of all respects in which the German version, used as the basis of the previously published and rather inexact English translation, differs from the Polish. Although the two versions are basically identical, to an extent that is even uncanny, we note (...) more than 400 differences. Several dozen of these are substantive differences due to revisions by Tarski to the Polish version which he did not incorporate in the German version. With respect to these revisions the Polish version should be regarded as more authoritative than the German. Hence scholars limited to an English translation should use ours. (shrink)

This classic undergraduate treatment examines the deductive method in its first part and explores applications of logic and methodology in constructing mathematical theories in its second part. Exercises appear throughout.

In the second volume of his Wissenschaftslehre from 1837, the Bohemian philosopher, theologian, and mathematician Bernard Bolzano introduced his concept of consequence, named derivability, together with a variety of theorems and further considerations. Derivability is an implication relation between sentences in themselves, which are not meant to be linguistic symbols but the contents of declarative sentences as well as of certain mental episodes. When Schmidt utters the sentence ‘Schnee ist weiß’, and Jones judges that snow is white, the sentence in (...) itself expressed by Schmidt is the same as the one to which Jones agrees in thought. This sentence in itself is an abstract entity: in some sense, it exists; but it is unreal insofar as it lacks a position in space and time, does not stand in causal relationships, and is independent of the existence of thinking beings and languages. (shrink)

I did something yesterday, so it was true a thousand years ago that I was going to do it. Could I help it, then? Professor Ryle shows that I could; he also shows that a dilemma like this starts with a slender base - the question whether statements in the future can be true - and opens out before one notices it into questions like 'is it worthwhile learning to swim?' In his second demonstration Professor Ryle proves that Achilles will (...) after all overtake his tortoise and will not have to pursue it through all eternity. These two puzzles were classic if academic examples of the dilemmas Professor Ryle is concerned with. Common sense tells me that I can to some extent control my life; should I then, faced with an apparently logical argument for fatalism, reject common sense? Again, because I enjoy or dislike things, are all my actions due to the seeking of pleasure and the avoidance of pain? Because my eyes and ears tell me one set of things about the world and science tels me another, am I to think my senses less reliable or accurate than science? Professor Ryle's aim is to arbitrate between these pairs of theories which compete for our allegiance, and to show that the either/or which they seem to insist on is a false dilemma. It may lead to mere muddle, or to a more subtle confusion; either way these tugs-of-war between conflicting theories may bring the ordinary man either to confused or wrong action or to intellectual apathy. (shrink)

The paper investigates the propriety of applying the form versus matter distinction to arguments and to logic in general. Its main point is that many of the currently pervasive views on form and matter with respect to logic rest on several substantive and even contentious assumptions which are nevertheless uncritically accepted. Indeed, many of the issues raised by the application of this distinction to arguments seem to be related to a questionable combination of different presuppositions and expectations; this holds in (...) particular of the vexed issue of demarcating the class of logical constants. I begin with a characterization of currently widespread views on form and matter in logic, which I refer to as 'logical hylomorphism as we know it'—LHAWKI, for short—and argue that the hylomorphism underlying LHAWKI is mereological. Next, I sketch an overview of the historical developments leading from Aristotelian, non-mereological metaphysical hylomorphism to mereological logical hylomorphism (LHAWKI). I conclude with a reassessment of the prospects for the combination of hylomorphism and logic, arguing in particular that LHAWKI is not the only and certainly not the most suitable version of logical hylomorphism. In particular, this implies that the project of demarcating the class of logical constants as a means to define the scope and nature of logic rests on highly problematic assumptions. (shrink)

Logic is usually thought to concern itself only with features that sentences and arguments possess in virtue of their logical structures or forms. The logical form of a sentence or argument is determined by its syntactic or semantic structure and by the placement of certain expressions called “logical constants.”[1] Thus, for example, the sentences Every boy loves some girl. and Some boy loves every girl. are thought to differ in logical form, even though they share a common syntactic and semantic (...) structure, because they differ in the placement of the logical constants “every” and “some”. By contrast, the sentences Every girl loves some boy. and Every boy loves some girl. are thought to have the same logical form, because “girl” and “boy” are not logical constants. Thus, in order to settle questions about logical form, and ultimately about which arguments are logically valid and which sentences logically true, we must distinguish the “logical constants” of a language from its nonlogical expressions. (shrink)

"Kant and the Capacity to Judge" will prove to be an important and influential event in Kant studies and in philosophy.

Logic is formal in the sense that all arguments of the same form as logically valid arguments are also logically valid and hence truth-preserving. However, it is not known whether all arguments that are valid in the usual model-theoretic sense are truthpreserving. Tarski claimed that it could be proved that all arguments that are valid (in the sense of validity he contemplated in his 1936 paper on logical consequence) are truthpreserving. But he did not offer the proof. The question arises (...) whether the usual modeltheoretic sense of validity and Tarski's 1936 sense are the same. I argue in this paper that they probably are not, and that the proof Tarski had in mind, although unusable to prove that model-theoretically valid arguments are truth-preserving, can be used to prove that arguments valid in Tarski's 1936 sense are truth-preserving. (shrink)

Hartry Field; XV*—The A Prioricity of Logic, Proceedings of the Aristotelian Society, Volume 96, Issue 1, 1 June 1996, Pages 359–380, https://doi.org/10.1093/ar.

Hartry Field; XV*—The A Prioricity of Logic, Proceedings of the Aristotelian Society, Volume 96, Issue 1, 1 June 1996, Pages 359–380, https://doi.org/10.1093/ar.

I examine the theory of consequentia of the medieval logician, John Buridan. Buridan advocates a strict commitment to what we now call proposition-tokens as the bearers of truth-value. The analysis of Buridan's theory shows that, within a token-based semantics, amendments to the usual notions of inference and consequence are made necessary, since pragmatic elements disrupt the semantic behaviour of propositions. In my reconstruction of Buridan's theory, I use some of the apparatus of modern two-dimensional semantics, such as two-dimensional matrices and (...) the distinction between the context of formation and the context of evaluation of utterances. (shrink)

There is a long-standing debate whether propositions, sentences, statements or utterances provide an answer to the question of what objects logical formulas stand for. Based on the traditional understanding of logic as a science of valid arguments, this question is firstly framed more exactly, making explicit that it calls not only for identifying some class of objects, but also for explaining their relationship to ordinary language utterances. It is then argued that there are strong arguments against the proposals commonly put (...) forward in the debate. The core of the problem is that an informative account of the objects formulas stand for presupposes a theory of formalization; that is, a theory that explains what formulas may adequately substitute for an inference in proofs of validity. Although such theories are still subject to research, some consequences can be drawn from an analysis of the reasons why the common accounts featuring sentences, propositions or utterances fail. Theories of formalization cannot refer to utterances qua expressions of propositions; instead they may refer to sentences and rely on additional information about linguistic structure and pragmatic context. (shrink)

Logik ist nach dem traditionellen Verständnis eine ars iudicandi, eine Kunst, die Gültigkeit von Schlüssen zu prüfen. Da mit die normalen Mittel der modernen Logik zu diesem Zweck eingesetzt werden können, müssen erst Formeln an die Stelle von Sätzen treten: umgangssprachliche Schlüsse müssen adäquat formalisiert werden. Die richtige Formel entwickelt ein theoretisches Konzept des Formalisierens und praktisch anwendbare Adäquatheitskriterien für Formalisierungen. Dabei werden zentrale Fragen der Philosophie der Logik unter dem Gesichtspunkt des Zusammenspiels von Umgangssprache und Formalismus untersucht. Die ausführliche (...) und systematische Diskussion von Formalisierungstests bietet eine wichtige Ergänzung zu den traditionellen Logiklehrbüchern. (shrink)

We provide for the first time an exact translation into English of the Polish version of Alfred Tarski's classic 1936 paper, whose title we translate as ?On the Concept of Following Logically?. We also provide in footnotes an exact translation of all respects in which the German version, used as the basis of the previously published and rather inexact English translation, differs from the Polish. Although the two versions are basically identical, to an extent that is even uncanny, we note (...) more than 400 differences. Several dozen of these are substantive differences due to revisions by Tarski to the Polish version which he did not incorporate in the German version. With respect to these revisions the Polish version should be regarded as more authoritative than the German. Hence scholars limited to an English translation should use ours. (shrink)

Truth, etc. is a wide-ranging study of ancient logic based upon the John Locke lectures given by the eminent philosopher Jonathan Barnes in Oxford. The book presupposes no knowledge of logic and no skill in ancient languages: all ancient texts are cited in English translation; and logical symbols and logical jargon are avoided so far as possible. Anyone interested in ancient philosophy, or in logic and its history, will find much to learn and enjoy here.

The essays in this volume explore the full range of Husserl's work and reveal just how systematic his philosophy is. There are treatments of his most important contributions to phenomenology, intentionality and the philosophy of mind, epistemology, the philosophy of language, ontology, and mathematics. An underlying theme of the volume is a resistance to the idea, current in much intellectual history, of a radical break between 'modern' and 'postmodern' philosophy, with Husserl as the last of the great Cartesians. Husserl is (...) seen in this volume as a philosopher constantly revising his system in order to be able to integrate philosophy with ideas emanating from science and culture. The so-called rift between analytic and 'continental' philosophy emerges as an artificial construct. (shrink)

The Bounds of Logic presents a new philosophical theory of the scope and nature of logic based on critical analysis of the principles underlying modern Tarskian logic and inspired by mathematical and linguistic development. Extracting central philosophical ideas from Tarski’s early work in semantics, Sher questions whether these are fully realized by the standard first-order system. The answer lays the foundation for a new, broader conception of logic. By generally characterizing logical terms, Sher establishes a fundamental result in semantics. Her (...) development of the notion of logicality for quantifiers and her work on branching are of great importance for linguistics. Sher outlines the boundaries of the new logic and points out some of the philosophical ramifications of the new view of logic for such issues as the logicist thesis, ontological commitment, the role of mathematics in logic, and the metaphysical underpinning of logic. She proposes a constructive definition of logical terms, reexamines and extends the notion of branching quantification, and discusses various linguistic issues and applications. (shrink)

In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing in logic. Gödel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter? Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the (...) Second Theorem, and exploring a family of related results. The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book will be accessible to philosophy students with a limited formal background. It is equally suitable for mathematics students taking a first course in mathematical logic. (shrink)

-/- A schema (plural: schemata, or schemas), also known as a scheme (plural: schemes), is a linguistic template or pattern together with a rule for using it to specify a potentially infinite multitude of phrases, sentences, or arguments, which are called instances of the schema. Schemas are used in logic to specify rules of inference, in mathematics to describe theories with infinitely many axioms, and in semantics to give adequacy conditions for definitions of truth. -/- 1. What is a Schema? (...) 2. Uses of Schemas 3. The Ontological Status of Schemas 4. Schemas in the History of Logic Bibliography. (shrink)

The current resurgence of interest in cognition and in the nature of cognitive processing has brought with it also a renewed interest in the early work of Husserl, which contains one of the most sustained attempts to come to grips with the problems of logic from a cognitive point of view. Logic, for Husserl, is a theory of science; but it is a theory which takes seriously the idea that scientific theories are constituted by the mental acts of cognitive subjects. (...) The present essay begins with an exposition of Husserl's act-based conception of what a science is, and goes on to consider his account of the role of linguistic meanings, of the ontology of scientific objects, and of evidence and truth. The essay concentrates almost exclusively on the Logical Investigations of 1900/01. This is not only because this work, which is surely Husserl's single most important masterpiece, has been overshadowed first of all by his Ideas I and then later by the Crisis. It is also because the Investigations contain, in a peculiarly clear and pregnant form, a whole panoply of ideas on logic and cognitive theory which either simply disappeared in Husserl's own later writings or became obfuscated by an admixture of that great mystery which is 'transcendental phenomenology'. (shrink)

Husserl’s philosophy, by the usual account, evolved through three stages: 1. development of an anti-psychologistic, objective foundation of logic and mathematics, rooted in Brentanian descriptive psychology; 2. development of a new discipline of "phenomenology" founded on a metaphysical position dubbed "transcendental idealism"; transformation of phenomenology from a form of methodological solipsism into a phenomenology of intersubjectivity and ultimately (in his Crisis of 1936) into an ontology of the life-world, embracing the social worlds of culture and history. We show that this (...) story of three revolutions can provide at best a preliminary orientation, and that Husserl was constantly expanding and revising his philosophical system, integrating views in phenomenology, ontology, epistemology and logic with views on the nature and tasks of philosophy and science as well as on the nature of culture and the world in ways that reveal more common elements than violent shifts of direction. We argue further that Husserl is a seminal figure in the evolution from traditional philosophy to the characteristic philosophical concerns of the late twentieth century: concerns with representation and intentionality and with problems at the borderlines of the philosophy of mind, ontology, and cognitive science. (shrink)