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The Mathematical Development Of Set Theory  From Cantor To Cohen
 The Bulletin of Symbolic Logic
, 1996
"... This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meet ..."
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This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meeting of the Association for Symbolic Logic at Haifa, in the Massachusetts Institute of Technology logic seminar, and to the Paris Logic Group. The author would like to express his thanks to the various organizers, as well as his gratitude to the Hebrew University of Jerusalem for its hospitality during the preparation of this article in the autumn of 1995.
What does it mean to say that logic is formal?
, 2000
"... Much philosophy of logic is shaped, explicitly or implicitly, by the thought that logic is distinctively formal and abstracts from material content. The distinction between formal and material does not appear to coincide with the more familiar contrasts between a priori and empirical, necessary and ..."
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Much philosophy of logic is shaped, explicitly or implicitly, by the thought that logic is distinctively formal and abstracts from material content. The distinction between formal and material does not appear to coincide with the more familiar contrasts between a priori and empirical, necessary and contingent, analytic and synthetic—indeed, it is often invoked to explain these. Nor, it turns out, can it be explained by appeal to schematic inference patterns, syntactic rules, or grammar. What does it mean, then, to say that logic is distinctively formal? Three things: logic is said to be formal (or “topicneutral”) (1) in the sense that it provides constitutive norms for thought as such, (2) in the sense that it is indifferent to the particular identities of objects, and (3) in the sense that it abstracts entirely from the semantic content of thought. Though these three notions of formality are by no means equivalent, they are frequently run together. The reason, I argue, is that modern talk of the formality of logic has its source in Kant, and these three notions come together in the context of Kant’s transcendental philosophy. Outside of this context (e.g., in Frege), they can come apart. Attending to this
Frege’s Judgement Stroke and the Conception of Logic as the Study of Inference not Consequence
"... One of the most striking differences between Frege’s Begriffsschrift (logical system) and standard contemporary systems of logic is the inclusion in the former of the judgement stroke: a symbol which marks those propositions which are being asserted, that is, which are being used to express judgemen ..."
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One of the most striking differences between Frege’s Begriffsschrift (logical system) and standard contemporary systems of logic is the inclusion in the former of the judgement stroke: a symbol which marks those propositions which are being asserted, that is, which are being used to express judgements. There has been considerable controversy regarding both the exact purpose of the judgement stroke, and whether a system of logic should include such a symbol. This paper explains the intended role of the judgement stroke in a way that renders it readily comprehensible why Frege insisted that this symbol was an essential part of his logical system. The key point here is that Frege viewed logic as the study of inference relations amongst acts of judgement, rather than—as in the typical contemporary view—of consequence relations amongst certain objects (propositions or wellformed formulae). The paper also explains why Frege’s use of the judgement stroke is not in conflict with his avowed antipsychologism, and why Wittgenstein’s criticism of the judgement stroke as “logically quite meaningless ” is unfounded. The key point here is that while the judgement stroke has no content, its use in logic and mathematics is subject to a very stringent norm of assertion. A notable feature of Frege’s logic is the presence therein of the judgement stroke—the vertical line ‘’: a symbol which marks those propositions which are being asserted. For Frege, assertion is the external act corresponding to the inner act of judgement: “we distinguish: (1) the grasp of a thought—thinking, (2) the acknowledgement of the truth of a thought—the act of judgement, (3) the manifestation of this judgement—assertion ” [Frege 1918–26, pp. 355–6]. After the twodimensional graphical nature of his symbolism, the judgement stroke is, to our eyes, the next most striking thing about Frege’s logical system(s), 1 because standard current systems of logic employ no analogue of it: that is, they give us no way of asserting a proposition—of putting it forward as being true—as opposed to presenting or displaying a proposition so that its content may be considered. The judgement stroke was equally noteworthy both for Frege himself, and for his contemporary readers. In a review of Frege’s
The Inferential Significance of Frege’s Assertion Sign
"... Frege’s celebrated distinction between judgments and their contents invites the Tractarian denigration of his assertion sign as merely indicating the holding or putting forth as true of a thought, for whatever its other merits the marking of such an event seems of little relevance to a thought’s inf ..."
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Frege’s celebrated distinction between judgments and their contents invites the Tractarian denigration of his assertion sign as merely indicating the holding or putting forth as true of a thought, for whatever its other merits the marking of such an event seems of little relevance to a thought’s inferential significance. However, in light of (a) Frege’s conception of a logically correct language serving inter alia as an organon for the acquisition or reconstruction of knowledge, and (b) his epistemic conception of inference, it is argued that the sign of assertion is a device for distinguishing from all others those thoughts lying on the path of discovery. The drawing of such a distinction is then shown to be of inferential significance by elucidating Frege’s conception of inference as involving the acquisition or reconstruction of knowledge. Frege’s view of inference rules as codifying justificatory relations among judgments is then given an interpretation as making no undue use of psychological notions, and his denial that the assertion sign can have semantic content is shown to be mistaken but not in such a way as to frustrate the aim with which it is introduced. contents: 1.
Gödel’s incompleteness theorem and the philosophy of open systems
 7, Centre de Recherches Sémiologiques, Université de Neuchâtel (Neuchâtel
, 1992
"... In recent years a number of criticisms have been raised against the formal systems of ..."
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In recent years a number of criticisms have been raised against the formal systems of
The scope of logic: deduction, abduction, analogy
"... The present form of mathematical logic originated in the twenties and early thirties from the partial merging of two different traditions, the algebra of logic and the logicist tradition (see [27], [41]). This resulted in a new form of logic in which several features of the two earlier traditions co ..."
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The present form of mathematical logic originated in the twenties and early thirties from the partial merging of two different traditions, the algebra of logic and the logicist tradition (see [27], [41]). This resulted in a new form of logic in which several features of the two earlier traditions coexist. Clearly neither the algebra of logic nor the logicist’s logic is identical to the present form of mathematical logic, yet some of their basic ideas can be distinctly recognized within it. One of such ideas is Boole’s view that logic is the study of the laws of thought. This is not to be meant in a psychologistic way. Frege himself states that the task of logic can be represented “as the investigation of the mind; [though] of the mind, not of minds” [17, p. 369]. Moreover Frege never charges Boole with being psychologistic and in a letter to Peano even distinguishes between the followers of Boole and “the psychological logicians ” [16, p. 108]. In fact for Boole the laws of thought which are the object of logic belong “to the domain of what is termed necessary truth ” [2, p. 404]. For him logic does not depend on psychology, on the contrary psychology depends on logic insofar as it is only through an investigation of logical operations that we could obtain “some probable
What Is Logic?
"... It is far from clear what is meant by logic or what should be meant by it. It is nevertheless reasonable to identify logic as the study of inferences and inferential relations. The obvious practical use of logic is in any case to help us to reason well, to draw good inferences. And the typical form ..."
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It is far from clear what is meant by logic or what should be meant by it. It is nevertheless reasonable to identify logic as the study of inferences and inferential relations. The obvious practical use of logic is in any case to help us to reason well, to draw good inferences. And the typical form the theory of any part of logic seems to be a set of rules of inference. This answer already introduces some structure into a discussion of the nature of logic, for in an inference we can distinguish the input called a premise or premises from the output known as the conclusion. The transition from a premise or a number of premises to the conclusion is governed by a rule of inference. If the inference is in accordance with the appropriate rule, it is called valid. Rules of inference are often thought of as the alpha and omega of logic. Conceiving of logic as the study of inference is nevertheless only the first approximation to the title question, in that it prompts more questions than it answers. It is not clear what counts as an inference or what a theory of such inferences might look like. What are the rules of inference based on? Where do we find them? The ultimate end
The History of Mathematical Logic (vastly abbreviated and horribly simplified)
, 1997
"... F11.95> manipulating such forms in order to arrive at new correct arguments. The other two aspects are very intimately connected with this one. 2. In order to construct valid forms of arguments one has to know what such forms can be built from, that is, determine the ultimate "building blocks". In ..."
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F11.95> manipulating such forms in order to arrive at new correct arguments. The other two aspects are very intimately connected with this one. 2. In order to construct valid forms of arguments one has to know what such forms can be built from, that is, determine the ultimate "building blocks". In particular, one has to ask the questions about the meaning of such building blocks, of various terms and categories of terms and, furthermore, of their combinations. 3. Finally, there is the question of how to represent these patterns. Although apparently of secondary importance, it is the answer to this question which can be, to a high degree, considered the beginning of modern mathematical logic. The first three sections sketch the development along the respective lines until Renessance. In section 4, we indicate the development in modern era, with particular emphasis on the last two centuries. Section 5 indicates some basic aspect