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 ofintuitive 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.     

弗协调的弗雷格(英文)

本文表明,二阶弗协调概括与弗雷格的第五公理是足道的。也表明,如果等数关系是初始符号,那么通过弗协调推理可以从第五公理可以推出休谟原则。最后表明,弗协调的休谟原则不能用作逻辑主义数学的基础。     

Frege and his groups

Frege's docent's dissertation Rechnungsmethoden, die sich auf eine Erweiterung des Grössenbegriffes gründen(1874) contains indications of a bold attempt to extend arithmetic. According to it, arithmetic means the science of magnitude, and magnitude must be understood structurally without intuitive support. The main thing is insight into the formal structure of the operation of ‘addition’. It turns out that a general ‘magnitude domain’ coincides with a (commutative) group. This is an interesting connection with simultaneous developments in abstract algebra. As his main application, Frege studies iterations of functions. He does not yet pose the question of existence proofs. Measurement of magnitudes is also connected to numbers, but the discussion is here ambiguous in a way which calls for the systematic account of numbers in Grundgesetze     

Frege and propositional unity

This paper identifies a tension in Frege’s philosophy and offers a diagnosis of its origins. Frege’s Context Principle can be used to dissolve the problem of propositional unity. However, Frege’s official response to the problem does not invoke the Context Principle, but the distinction between ‘saturated’ and ‘unsaturated’ propositional constituents. I argue that such a response involves assumptions that clash with the Context Principle. I suggest, however, that this tension is not generated by deep-seated philosophical commitments, but by Frege’s occasional attempt to take a dubious shortcut in the justification of his conception of propositional structure.     

Ramified Frege Arithmetic

?ystein Linnebo has recently shown that the existence of successors cannot be proven in predicative Frege arithmetic, using Frege’s definitions of arithmetical notions. By contrast, it is shown here that the existence of successor can be proven in ramified predicative Frege arithmetic.     

Analyticity and Justification in Frege

That there are analytic truths may challenge a principle of the homogeneity of truth. Unlike standard conceptions, in which analyticity is couched in terms of “truth in virtue of meanings”, Frege’s notions of analytic and a priori concern justification, respecting a principle of the homogeneity of truth. Where there is no justification these notions do not apply, Frege insists. Basic truths and axioms may be analytic (or a priori), though unprovable, which means there is a form of justification which is not (deductive) proof. This is also required for regarding singular factual propositions as a posteriori. A Fregean direction for explicating this wider notion of justification is suggested in terms of his notion of sense (Sinn)—modes in which what the axioms are about are given—and its general epistemological significance is sketched.     

Frege on identities

The idea underlying the Begriffsschrift account of identities was that the content of a sentence is a function of the things it is about. If so, then if an identity a=b is about the content of its contained terms and is true, then a=a and a=b have the same content. But they do not have the same content; so, Frege concluded, identities are not about the contents of their contained terms. The way Frege regarded the matter is that in an identity the terms flanking the symbol for identity do not have their ordinary contents, but instead have themselves as their contents. In ‘Uber Sinn und Bedeutung’ Frege became convinced that if an identity a=bis about the signs aand b, then it expresses no proper knowledge. So, since it is evident that many such identities do express proper knowledge, Frege concluded that identities are not about their contained signs. So he became convinced that his Begriffsschrift account was incorrect. What was the error in the argument that led Frege to that account? It was thinking that the content of a sentence is a function of the contents of its constituent signs, that is, the things it is about.     

Frege Puzzles and Mental Files

This paper proposes a novel conception of mental files, aimed at addressing Frege puzzles. Classical Frege puzzles involve ignorance and discovery of identity. These may be addressed by accounting for a more basic way for identity to figure in thought—the treatment of beliefs by the believer as being about the same thing. This manifests itself in rational inferences that presuppose the identity of what the beliefs are about. Mental files help to provide a functional characterization of a mind capable of this presupposition, but more must be said to show how it may be rational. I argue that this can be done by drawing out the way in which mental files interact with a thinker's motivational states and so come to have normative functional properties. I show how this theory works better than some other treatments of mental files.     

Frege on the psychological significance of definitions

Conclusion In contemporary work, the distinction between the proposition expressed by a sentence and its psychological significance is usually motivated by a familiar kind of counterfactual argument; and the discussion of these issues usually centers around the role of external factors in determining the meaning of our words. My primary goal in this paper has been to show that a similar, though not identical, distinction between two aspects of meaning can be developed entirely on the basis of considerations internal to language users — their cognitive limitations. To make this point, I have focused on symbols introduced through stipulative definitions. In a language containing such symbols, certain expressions and their definitional reductions will seem to differ in psychological significance for creatures with limited intellects, and so in any aspect of meaning that is supposed to correlate with psychological significance; but it seems also that there is some important aspect of meaning that they share.I have argued that a distinction in meaning like this — between sense and psychological significance — should be drawn even in the kind of languages of most concern to Frege, and that his failure to do so led to tensions in his thought. Of course, this observation only touches on the many issues involved in interpreting Frege's theory of definition more generally. I have not tried to describe here, for example, the ways in which the weak interpretation of fruitfulness might interact with the more robust interpretation mentioned earlier; I have only mentioned Frege's view on explicative definitions and the paradox of analysis, and failed even to mention either his treatment of contextual definition, or his peculiar objections to conditional definitions. I do want to emphasize, however, that the distinction drawn here is not simply a matter of Frege scholarship, but that it has some contemporary relevance as well. As we have seen, Frege's semantic goals often coincide with our own; and a number of contemporary writers are explicitly concerned, like Frege, to construct a semantic theory that is able to account for differences in meaning among logically equivalent expressions. Any such theorist should recognize a distinction like that drawn here between sense and psychological significance, and should avoid subjecting an account of one notion to constraints appropriate only for the other.     

Frege on the generality of logical laws

Frege claims that the laws of logic are characterized by their “generality,” but it is hard to see how this could identify a special feature of those laws. I argue that we must understand this talk of generality in normative terms, but that what Frege says provides a normative demarcation of the logical laws only once we connect it with his thinking about truth and science. He means to be identifying the laws of logic as those that appear in every one of the scientific systems whose construction is the ultimate aim of science, and in which all truths have a place. Though an account of logic in terms of scientific systems might seem hopelessly antiquated, I argue that it is not: A basically Fregean account of the nature of logic still looks quite promising.