Frege's Other Program

Annals of Pure and Applied Logic, 2004

According to Frege's principle the denotation of a sentence coincides with its truthvalue. The principle is investigated within the context of abstract algebraic logic, and it is shown that taken together with the deduction theorem it characterizes intuitionistic logic in a certain strong sense.

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.

2012

Annals of Pure and Applied Logic, 2009

In this paper, we characterize the strength of the predicative Frege hierarchy, P n+1 V, introduced by John Burgess in his book [Bur05]. We show that P n+1 V and Q + con n (Q) are mutually interpretable. It follows that PV := P 1 V is mutually interpretable with Q. This fact was proved earlier by Mihai Ganea in [Gan06] using a different proof. Another consequence of the our main result is that P 2 V is mutually interpretable with Kalmar Arithmetic (a.k.a. EA, EFA, I∆0 +EXP, Q3). The fact that P 2 V interprets EA, was proved earlier by Burgess. We provide a different proof. Each of the theories P n+1 V is finitely axiomatizable. Our main result implies that the whole hierarchy taken together, P ω V, is not finitely axiomatizable. What is more: no theory that is mutually locally interpretable with P ω V is finitely axiomatizable.

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 Review of Symbolic Logic, 2018

Recent discussions on Fregean and neo-Fregean foundations for arithmetic and real analysis pay much attention to what is called either ‘Application Constraint’ ($AC$) or ‘Frege Constraint’ ($FC$), the requirement that a mathematical theory be so outlined that it immediately allows explaining for its applicability. We distinguish between two constraints, which we, respectively, denote by the latter of these two names, by showing how $AC$ generalizes Frege’s views while $FC$ comes closer to his original conceptions. Different authors diverge on the interpretation of $FC$ and on whether it applies to definitions of both natural and real numbers. Our aim is to trace the origins of $FC$ and to explore how different understandings of it can be faithful to Frege’s views about such definitions and to his foundational program. After rehearsing the essential elements of the relevant debate (§1), we appropriately distinguish $AC$ from $FC$ (§2). We discuss six rationales which may motivate the...

The primary purpose of this note is to demonstrate that predicative Frege arithmetic naturally interprets certain weak but non-trivial arithmetical theories. It will take almost as long to explain what this means and why it matters as it will to prove the results.

Haarparanta and J. Hintikka (eds.), Dordrecht: Reidel, 1986

Logicism is not dead, as is often claimed, but can be reconstructed in essentially the same form and ontological context in which it was originally proposed by Frege and Russell. We describe here not one but two alternative reconstructions of Frege's logicism both of which are consistent (relative to weak Zermelo set theory). We will also describe a reconstruction of Russell's early form of logicism, which is di¤erent from the one he later adopted in [PM]. Though Frege's and Russell's early form of logicism are not the same, they are closely related in that both can be reconstructed as second-order predicate logics with nominalized predicates as abstract singular terms.

Notre Dame Journal of Formal Logic, 1998

After a summary of earlier work it is shown that elementary or Kalmar arithmetic can be interpreted within the system of Russell's Principia Mathematica with the axiom of infinity but without the axiom of reducibility. 1 Historical introduction After discovering the inconsistency in Frege's Grundgesetze der Arithmetik, Russell proposed two changes: first, dropping the assumption that to every higher-order entity there corresponds a first-order entity; and second, restricting the assumptions on the existence of higher-order entities, so that instead of a simple hierarchy of first-order, second-order, third-order, and so on, one has a ramified hierarchy in which each order is subdivided into various types in such a way that a condition involving quantification over all entities of one type is never assumed to determine another entity of the same type, but only of a higher type. But Russell found that with these two changes he could not derive classical mathematics, so in Principia Mathematica he partially compensated for the first change by assuming the axiom of infinity and, for all mathematical purposes, wholly undid the second change by assuming his axiom of reducibility. The predicativist tradition from Weyl [21] to Feferman [2] and beyond accepts infinity but rejects reducibility and is willing to give up parts of classical mathematics. However, predicativists have been unable to derive classical arithmetic and unwilling to give it up and so have simply assumed it as axiomatic. This assumption has its defenders, as with Feferman and Hellman [3], and also its detractors, as with C. Parsons [15]. It is, therefore, of some philosophical as well as historical interest to ask how large a fragment of classical arithmetic can be developed within the Russellian system of Principia Mathematica with infinity but without reducibility. Now many subsystems of classical or Peano arithmetic have been recognized since the work of Skolem [18], Kalmar [9], Grzegorczyk [4], and other pioneers. Among these the most studied have been the subprimitive or Grzegorczyk arithmetics n. These agree in allowing definitions by primitive recursion, but only when the function F being defined recursively is bounded by some function already given; or

Erkenntnis, 2000

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. In his essay "Über Begriff und Gegenstand" (from 1892), Frege famously addresses an objection raised by Beno Kerry against his sharp ontological distinction between concept and object. Kerry claims, against Frege, that the ontological status of something as a concept or as object is not absolute, that is to say, something that seems like a concept may "behave" like an object in certain contexts, and vice-versa. Kerry uses the famous example of the expression 'the concept horse' to illustrate his point. The thing designated by this expression seems to be a concept in contexts like 'Silver falls under the concept horse'. But in contexts like 'The concept horse is easy to grasp', 'the concept horse' turns out to designate an object due to its position as grammatical subject. Since there is no reason to suppose that 'the concept horse' refers to different things in each one of these contexts, Kerry concludes that the status as concept or as object of the thing designated is relative and not absolute, contrary to Frege's view. An essential part of Frege's reply to Kerry's criticism is his well known claim that, due to the presence of the definite article in 'the concept horse' the expression has to refer (if it has a reference at all) to an object, and not to a concept. And this is so in all contexts. There is here, as Frege claims, a systematically misleading effect of ordinary language: it makes us refer to an object, while in fact we intend to refer to a concept. The expression 'the concept horse' refers, according to Frege, to an object that "represents" the intended concept in logical investigations (KS 170). 1 Hence Frege's apparently paradoxical dictum that "the concept horse is not a concept" (KS 170).