Logic and Arithmetic

Bulletin of Symbolic Logic, 2004

Frege Arithmetic (FA) is the second-order theory whose sole non-logical axiom is Hume's Principle, which says that the number of F s is identical to the number of Gs if and only if the F s and the Gs can be one-to-one correlated. According to Frege's Theorem, FA and some natural definitions imply all of second-order Peano Arithmetic. This paper distinguishes two dimensions of impredicativity involved in FA-one having to do with Hume's Principle, the other, with the underlying second-order logic-and investigates how much of Frege's Theorem goes through in various partially predicative fragments of FA. Theorem 1 shows that almost everything goes through, the most important exception being the axiom that every natural number has a successor. Theorem 2 shows that the Successor Axiom cannot be proved in the theories that are predicative in either dimension.

Acta Analytica, 2016

In early analytic philosophy, one of the most central questions concerned the status of arithmetical objects. Frege argued against the popular conception that we arrive at natural numbers with a psychological process of abstraction. Instead, he wanted to show that arithmetical truths can be derived from the truths of logic, thus eliminating all psychological components. Meanwhile, Dedekind and Peano developed axiomatic systems of arithmetic. The differences between the logicist and axiomatic approaches turned out to be philosophical as well as mathematical. In this paper, I will argue that Dedekind's approach can be seen as a precursor to modern structuralism and as such, it enjoys many advantages over Frege's logicism. I also show that from a modern perspective, Frege's criticism of abstraction and psychologism is one-sided and fails against the psychological processes that modern research suggests to be at the heart of numerical cognition. The approach here is twofold. First, through historical analysis, I will try to build a clear image of what Frege's and Dedekind's views on arithmetic were. Then, I will consider those views from the perspective of modern philosophy of mathematics, and in particular, the empirical study of arithmetical cognition. I aim to show that there is nothing to suggest that the axiomatic Dedekind approach could not provide a perfectly adequate basis for philosophy of arithmetic.

Between Logic and Reality, 2011

The paper's aim is to determine and discuss in which sense, if any, Frege's and neo-Fregean logicism are responding to the epistemological challenge concerning our arithmetical knowledge. More precisely the paper analyses what the epistemological significance of Frege's logicist programme amounts to, namely, the objective justificatory connections obtaining between arithmetical and logical statements. It then contrasts this result with the self-understanding of the neo-Fregeans who allegedly follow Frege's steps, but in fact take a rather different direction.

History and Philosophy of Logic, 2024

In many accounts of the history of logic, especially from the second half of the twentieth century and partly still today, Frege's rst book, Begri sschrift (1879), is singled out as the beginning of modern logic. In the English-speaking literature, this assessment goes back to the 1950s-60s when Frege's logical writings were rediscovered, after an initial period of neglect (although thinkers like Russell, Wittgenstein, and Carnap had paid close attention to it earlier). This is also the period during which modern logic consolidated itself, with its now standard sub elds: set theory, proof theory, model theory, and recursion theory. Good illustrations of this assessment of Frege's contributions can be found in William and Martha Kneale's book, The Development of Logic (1962), and in Jean van Heijenoort's revealingly entitled collection, From Frege to Gödel: A Source Book in Mathematical Logic. 1879-1931 (1967). The assessment was grounded in writings by a number of in uential logicians, including Alonzo Church, W.V.O. Quine, and Michael Dummett. Some of these writings, especially those by Dummett, include strong claims about how utterly original Frege's logical ideas were, thus representing a radical new beginning (cf. Reck 2023). Within the last 30-40 years such claims about the originality of Frege's views have been challenged and partly refuted in a number of ways. Thus, interpreters such as Christian Thiel, Gottfried Gabriel, Hans Sluga, etc. have pointed out the roots of some of Frege's logical ideas, including aspects of his logicist project, in neo-Kantian or post-Kantian philosophers like Hermann Lotze, J. F Herbart, and Wilhelm Windelband (cf. Gabriel 2002, Gabriel and, also the literature mentioned in them). Other interpreters of Frege, including Mark Wilson, Jamie Tappenden, and I, have discussed sources for Fregean ideas in mathematics, especially in nineteenth-century geometry, Bernhard Riemann's writings, and other works to which Frege was exposed in his mathematical education (cf. Tappenden 2008. Similarly, logical and mathematical in uences on Frege in works by Hermann and Robert Grassmann, Hermann Hankel, etc. may be worth exploring further, partly to clarify what Frege was reacting against (cf. Kreiser 2001). Yet another way in which the claim that modern logic started abruptly in 1879, with Begri sschrift, has been called into question is by rediscovering and highlighting earlier contributions by other logicians, such as George Boole, members of the Boolean school, as well as Bernard Bolzano (cf. Peckhaus 1997, Rusnock and. As this shows, it CONTACT Erich H. Reck

The True and the False revisited: Frege's logic Frege is held to be largely responsible for debunking the traditional conception that all sentences are of subject-predicate form, partly because of his relational analysis of sentences, but mostly because of his analysis of quantification. The combination of quantifiers and connectives is not considered to be a subject-predicate combination. Although Frege explicitly rejects the traditional subject-predicate distinction in Begriffsschrift and does not separate notationally nonlogical predicates from non-logical subjects, his logical analysis is a sharpening of the traditional subject-predicate analysis. It is clear from Frege's ontological analysis that the combinations of quantifiers and connectives are properties, or concepts, or functions. And it is also clear from his notation, for he separates the logical properties from the non-logical subjects.

2009

In Frege’s theory of arithmetic where the definite article serves to classify something as an object, the “definite article premise” is one of the main premises along with the “context principle” and definite article premise provided a tool for Frege to define numbers as logical objects. By this tool he could distinguish linguistically concept of cardinal number and a cardinal number as an object so that they could have different terms which denote different things. I will discuss that the “definite article premise” itself is ineffectual. Frege’s logicist system is undermined for one of the premises in his argumentation is problematic. Particularly, the “definite article premise” cannot classify the concept words for numbers as objects words (proper names). Since his logicist project fails to be completed, it is not possible to secure the referent of number terms as logical objects. However the definition he has suggested can still give the Kantian intuition of individual numbers.

Confronted with Russell's Paradox, Frege wrote an appendix to volume II of his Grundgesetze der Arithmetik. In it he offered a revision to Basic Law V, and proclaimed with confidence that the major theorems for arithmetic are recoverable. This paper shows that Frege's revised system has been seriously undermined by interpretations that transcribe his system into a predicate logic that is inattentive to important details of his concept-script. By examining the revised system as a concept-script, we see how Frege imagined that minor modifications of his former proofs would recover arithmetic. † Special thanks to Kevin Klement, Ed Mares, Francesco Orilia, and anonymous referees for helpful comments on this paper.

Journal of the …, 2010

Axiomathes, 2019

Frege's famous definition of number (in)famously uses the concept of “extension”. Extensions, in the Fregean framework, are susceptible to bringing many difficulties, and, some say, even paradoxes. Therefore, neo-logicist programs want to avoid the problems and to replace the classical Fregean definition of number with Hume's Principle (where Frege does not use extensions). I argue that this move, even if it makes sense from a computational point of view, is at odds with Frege's larger philosophical project. For Frege, I claim, extensions were an important part of his philosophical program of logic-as-an-universal-language. This is why Frege places his project in line with Leibniz' philosophical project of finding a lingua characterica universalis.

One of the more distinctive features of Bob Hale and Crispin Wright's neologicism about arithmetic is their invocation of Frege's Constraint – roughly, the requirement that the core empirical applications for a class of numbers be " built directly into " their formal characterization. In particular, they maintain that, if adopted, Frege's Constraint adjudicates in favor of their preferred foundation – Hume's Principle – and against alternatives, such as the Dedekind-Peano axioms. In what follows we establish two main claims. First, we show that, if sound, Hale and Wright's arguments for Frege's Constraint at most establish a version on which the relevant application of the naturals is transitive counting – roughly, the counting procedure by which numerals are used to answer " how many "-questions. Second, we show that this version of Frege's Constraint fails to adjudicate in favor of Hume's Principle. If this is the version of Frege's Constraint that a foundation for arithmetic must respect, then Hume's Principle no more – and no less – meets the requirement than the Dedekind-Peano axioms do.