Frege's Logic Research Articles

Extensions as representative objects in Frege's logic

Matthias Schirn has argued on a number of occasions against the interpretation of Frege's ``objects of a quite special kind'' (i.e., the objects referred to by names like `the concept F') as extensions of concepts. According to Schirn, not only are these objects not extensions, but also the idea that `the concept F' refers to objects leads to some conclusions that are counter-intuitive and incompatible with Frege's thought. In this paper, I challenge Schirn's conclusion: I want to try and argue that the assumption that `the concept F' refers to the extension of F is entirely consistent with Frege's broader views on logic and language. I shall examine each of Schirn's main arguments and show that they do not support his claim.

The Primacy of Concepts and the Priority of Judgments in Frege's Logic

The Primacy of Concepts and the Priority of Judgments in Frege's Logic

Frege's Philosophy of Mathematics.

Introduction, William Demopoulos appendix, John L. Bell. Part 1 The intellectual background to Frege's logicism: Kant, Bolzano and the emergence of logicism, Alberto Coffa Frege - the last logicist, Paul Benacerraf Frege and the rigorization of analysis, William Demopoulos Frege and arbitrary functions, John P. Burgess Frege - the royal road from geometry - postscript, Mark Wilson. Part 2 The mathematical content of Begriffsschrift and Grundlageng: Reading the Begriffsschrift, Geoge Boolos Frege's Theory of number - postscript, Charles parsons The consistency of Frege's Foundation of arithmetics, George Boolos The standard of equality of numbers, George Boolos. Part 3 der arithmetik: The development of arithmetic in Frege's Grundgesetze der arithmetik - postscript, Richard G. Heck, Jr. Definition by induction in Frege's der arithmetik, Richard G. Heck, Jr. Eudoxus and Dedekind - on the ancient Greek theory of ratios and its relation to modern mathematics, Howard Stein Frege's theory of real numbers, Peter M. Simons Frege's theory of real number, Michael Dummett On a question of Frege's about right-ordered groups, - postscript, Peter M. Newmann et al On the consistency of the first-order portion of Frege's logical system, Terence Parsons Fregean extensions of first-order theories, John L. Bell Saving Frege from contradiction, George Boolos.

Frege and the rigorization of analysis

This paper has three goals: (i) to show that the foundational program begun in theBegriffsschrift, and carried forward in theGrundlagen, represented Frege's attempt to establish the autonomy of arithmetic from geometry and kinematics; the cogency and coherence of‘intuitive’ reasoning were not in question. (ii) To place Frege's logicism in the context of the nineteenth century tradition in mathematical analysis, and, in particular, to show how the modern concept of a function made it possible for Frege to pursue the goal of autonomy within the framework of the system of second-order logic of theBegriffsschrift. (iii) To address certain criticisms of Frege by Parsons and Boolos, and thereby to clarify what was and was not achieved by the development, in Part III of theBegriffsschrift, of a fragment of the theory of relations.

The Contemporary Interest of an old Doctrine

My purpose in this talk is to give an overview of the rediscovery of Frege's theorem together with certain of the issues that this rediscovery has raised concerning the evaluation of Frege's logicism—the ‘old doctrine’ of my title.The contextual definition of the cardinality operator, suggested in §63 of Grundlagen— what, after George Boolos, has come to be known as Hume's principle—assertsThe number of Fs = the number of Gs if, and only if, F ≈ G,where F ≈ G (the Fs and the Gs are in one-to-one correspondence) has its usual, second order, explicit definition. The importance of this principle for the derivation of Peano's second postulate (‘Every natural number has a successor’) was emphasized by Crispin Wright (1983, §xix) who presented an extended argument showing that, in the context of the system of second-order logic of Frege's Begriffsschrift, Peano's second postulate is derivable from Hume's principle.

The Foundations of Frege's Logic.

The Foundations of Frege's Logic.

Frege, Informative Identities, and Logicism

Frege's Grundgesetze der Arithmetik are inconsistent, as Russell first caused to be observed. My purpose here is to demonstrate that what Frege has to say about them is also inconsistent. Specifically, there is no Fregean way to reconcile the claims that arithmetical identity statements can be informative because they contain singular terms possessing co-referential but distinct senses, and that arithmetical truth is logical truth. Gregory Currie (Currie [1982]) has noticed this incompatibility and presumably regards it as a fairly obvious one, for he offers little in the way of detailed argument. Christian Thiel (Thiel [1968]) also realized that there is a problem here. Unfortunately his main supporting evidence has been dismissed by Michael Dummett. Since the incompatibility is not widely acknowledged and is therefore not so obvious to others it seems worthwhile filling in the details. The laws of logic are for Frege analytic by the definition of that term. More importantly for present concerns, they are always and everywhere true. Yet a simple arithmetical truth such as '2 + 2 =4' is conceivably false: the senses of the terms on either side of the equality-sign differ and hence the thought expressed by the whole equation is not true in virtue of the identity of the senses of its part.' Frege would have us believe that all arithmetical truths are derivable by truth-preserving rules of inference from his basic laws of logic. Evidently matters are less than transparently clear and a likely source of disquiet is the claim that the senses of '2 + 2' and of '4' are different: we shall see that it proves incompatible with the thesis that arithmetic is derivable from logic. For Frege logic has truth for its subject matter:

Truth-Functional Conditionals and Modern vs. Traditional Syllogistic

The failure of nine traditionally valid categorical syllogisms in modern logic,2 as well as the failure of implications, contrariety and sub-contrariety among A, E, I, and 0 propositions with identical subject and predicate terms, is customarily explained by references to the modern doctrine of 'existential import'. But this is a mistake. We wish to show that the differences between what should be counted as laws of logic are due, instead, to the employment of two different conditionals, one truth-functional and one not, in the two logics. By traditional logic we shall mean the standard account given in modern textbooks of the theory of the syllogism as developed from Aristotle's time down through the Port Royale Logic, with 24 valid categorical syllogisms out of 256 possible, and the theory of immediate inference as expressed in the traditional square of opposition, including conversion of E and I, contraposition of A and 0 and obversion of A, E, I, and 0 propositions. By modern logic we mean the logic of Frege, Russell, and Whitehead through the first-order predicate calculus. For convenience we shall use Quine's formulation in giving proofs later in the paper.3

Hemispheric basis for schools in mathematics

Kant based Geometry upon the intuition of space and Arithmetic upon the intuition of time. Frege's logicism accepts Kant's view on Geometry and rejects his view on Arithmetic. Logicism bases Arithmetic upon the Logics of sets. On the other hand Brouwer's Intuitionism accepts Kant's view on Arithmetic and rejects his view on Geometry.

Frege, Dedekind, and Peano on the Foundations of Arithmetic.

1. Kant's Theory of Mathematics 2. Frege's Criticisms of Kant 3. Mill's Theory of Mathematics 4. Frege's Criticisms of Mill 5. The Concept of a Statement of Number is an Assertion About a Concept 6. Frege's Platonism 7. Frege's Logicism 8. Dedekind and Set Theory 9. Dedekind's Development of Arithmetic 10. Peano's Axioms. General Comparison of Frege, Dedekind and Peano 11. Frege's Begriffsschrift 12. Frege's Grundgesetze, and Russell's Paradox

Frege on Extensions of Concepts, from 1884 to 1903

R ussell's paradox defeated Frege's attempt to demonstrate logicism-the view that the mathematics of number can be derived from the axioms of logic together with definitions in logical terms of mathematically primitive expressions. The paradox indicated that Frege's logic was inconsistent and that the notion in terms of which he tried to define the cardinal numbers was defective. The defective notion, that of the extension of a concept, has remained interesting because it is motivated by intuitions that, arguably, play an ineliminable but inadequately understood role in modern set theories.' The development of Frege's views is also historically interesting. Frege was uncertain about the crucial notion and the paradox-producing axiom from the outset. He experimented with alternatives. What can be pieced together about his reasoning suggests deep tensions in his thought on fundamental matters. This paper is purely historical. It concentrates on the period from the publication of The Foundations of Arithmetic in 1884 to the publication of The Basic Laws of Arithmetic (first volume, 1893; second, 1903). I shall trace the development during this period of Frege's notion of the extension of a concept.

Le problème de Ramsey

JACQUES BOUVERESSE: Ramsey ' s Problem One of the fundamental principles underlying aIl of Frege's logic and ontology is the idea of the existence of a constitutive and radical asymmetry between subject and predicate such that, and inspite of what the superficial grammar of sentences might suggest, they can never really exchange roles. Ramsey, however, maintained that the distinction between subject and predicate in a sentence had no significance other than a pragmatic or rhetorical one, and corresponded to a simple difference of point ofveiw, emphasis or interest. The origins of the present controversy between fol1owers of Ramsey and Frege are traced here in detail through an in-depth examination of the arguments Ramsey himselfused.

La logique intensionnelle et l'héritage de Frege en sémantique

Pascal Engel: Intensional Logic and the Fregean Heritage in Semantics. Although Frege's logic and «semantics» inspired most later developments, neither Carnap nor Church, who used fregean ideas to build their systems, were faithful to these. The fregean heritage has been distorted in two ways: intensional logic obliterates the distinction between concepts and objects, and Frege recommends an extensional · treatment of logic. This paper tries to restate the fregean doctrine of functions by using combinatory logic, and defends the «neo-fregean» point of view of a logical semantics based on Tarski's theory of truth and Davidson's programme for the semantics of natural language.

Quine's theory of logic

No philosopher of logic can afford to ignore Quine's work. Like most great philosophers, Quine has been influenced by different and even seem? ingly incompatible perspectives. From the point of view of influence Quine, I believe, is really two philosophers ; a combination of Frege on the one hand, and Dewey on the other. In this paper I want to consider the influ? ence on Quine's theory of logic of Frege's logicism, particularly his uni? versalist theory of logic, and of Dewey's pragmatism or instrumentalism. I want this paper to serve as a prelude to a much more thorough-going pragmatist or instrumentalist theory of logic and mathematics than Quine's. I shall use the terms 'pragmatist' and 'instrumentalist' interchangeably. Quine's views will be subject to criticism from this more radical perspective. An assumption behind this paper is the view that conceptions of, or better, theories of, logic are as a controversial a matter as specific issues in the philosophy of logic. With this, of course, Quine would agree.

Some Comparisons Between Frege's Logic and Navya-Nyaya Logic

Some Comparisons Between Frege's Logic and Navya-Nyaya Logic

Ineffability in Frege's logic

Ineffability in Frege's logic

Sense and Reference in Frege's Logic.

Sense and Reference in Frege's Logic.

Frege's Logic

Frege's Logic

H. R. Smart. Frege's logic. The philosophical review, vol. 54 (1945), pp. 489–505.

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