Natural Numbers and Natural Cardinals as Abstract Objects: A Partial Reconstruction of Frege" s="" grundgesetze="" in="" object="" theory"="

In this paper, the author derives the Dedekind–Peano axioms for number theory from a consistent and general metaphysical theory of abstract objects. The derivation makes no appeal to primitive mathematical notions, implicit definitions, or a principle of infinity. The theorems proved constitute an important subset of the numbered propositions found in Frege"s Grundgesetze. The proofs of the theorems reconstruct Frege"s derivations, with the exception of the claim that every number has a successor, which is derived from a modal axiom that (philosophical) logicians implicitly accept. In the final section of the paper, there is a brief philosophical discussion of how the present theory relates to the work of other philosophers attempting to reconstruct Frege"s conception of numbers and logical objects.     

Cn-Definitions of Propositional Connectives

We attempt to define the classical propositional logic by use of appropriate derivability conditions called Cn-definitions. The conditions characterize basic properties of propositional connectives.     

First-Order Frege Theory is Undecidable

The system whose only predicate is identity, whose only nonlogical vocabulary is the abstraction operator, and whose axioms are all first-order instances of Frege's Axiom V is shown to be undecidable.     

An argument against Fregean that-clause semantics

I develop a problem for the Fregean Reference Shift analysis of that-clause reference. The problem is discussed by Stephen Schiffer in his recent book The Things We Mean (2003). Either the defender of the Fregean Reference Shift analysis must count certain counterintuitive inferences as valid, or else he must reject a plausible Exportation rule. I consider several responses. I find that the best response relies on a Kaplan-inspired analysis of quantified belief reports. But I argue that this response faces some serious problems.

The ontology of meanings

In part 4 of Meaning, Expression, and Thought, Davis rejects what he calls Fregean ideational theories, according to which the meaning of an expression is an idea; and then presents his own account, which states that, e.g., the meaning of ‘Primzahl’ in German is the property of meaning prime number. Before casting doubt on the latter ontology of meanings, I come to Frege’s defence by pointing out that he was not an advocate of the position Davis named after him because Fregean senses are not lexical meanings and Fregean thoughts are not types of mental events.

Gingerbread Nuts and Pebbles: Frege and the Neo-Kantians – Two Recently Discovered Documents

We present and discuss two recently discovered pieces of correspondence by Frege: a postcard to Heinrich Rickert dated 1 July 1911 and a letter to Hinrich Knittermeyer dated 25 October 1912. The documents and their historical context shed new light on Frege's relation to the Neo-Kantians.     

AN INTUITIONISTIC CHARACTERIZATION OF CLASSICAL LOGIC

By introducing the intensional mappings and their properties, we establish a new semantical approach of characterizing intermediate logics. First prove that this new approach provides a general method of characterizing and comparing logics without changing the semantical interpretation of implication connective. Then show that it is adequate to characterize all Kripke_complete intermediate logics by showing that each of these logics is sound and complete with respect to its (unique) ‘weakest characterization property’ of intensional mappings. In particular, we show that classical logic has the weakest characterization property , which is the strongest among all possible weakest characterization properties of intermediate logics. Finally, it follows from this result that a translation is an embedding of classical logic into intuitionistic logic, iff. its semantical counterpart has the property .      

Must Differences in Cognitive Value be Transparent?

Frege’s ‘differential dubitability’ test is a test for differences in cognitive value: if one can rationally believe that p while simultaneously doubting that q, then the contents p and q amount to different ‘cognitive values’. If subject S is rational, does her simultaneous adoption of different attitudes towards p and q require that the difference between p and q (as cognitive values) be transparent to her? It is natural to think so. But I argue that, if attitude anti-individualism is true, then rational differential dubitability does not presuppose that differences in cognitive value are transparent. The significance of this argument lies in what it tells us, both about the notion of cognitive value and its relation to the differential dubitability test, but also about the prospects for a Burge-type position which aims to combine attitude anti-individualism with a (qualified) reliance on the differential dubitability test.

Ways of thinking about being: Explorations in ontology

This paper briefly explores Merleau-Ponty's notions of body subject and flesh in order to draw out some of the implications of his work for an understanding of key aspects of non-Western worldviews, notably that of Australian aboriginal people. Focusing specifically on the concept of materiality, I argue that its elaboration as flesh in Merleau-Ponty's work constitutes an important conceptual link with non-atomistic accounts of being and world, accounts characteristic of some indigenous peoples. Writing as a non-aboriginal and a relative newcomer to the study of aboriginal beliefs, I address the issue of possible contributions such understandings may make to solutions to problems we face in future both as individuals and as educators.     

Formality of logic and Frege’s Begriffsschrift

This paper challenges a standard interpretation according to which Frege’s conception of logic (early and late) is at odds with the contemporary one, because on the latter’s view logic is formal, while on Frege’s view it is not, given that logic’s subject matter is reality’s most general features. I argue that Frege – in Begriffsschrift – retained the idea that logic is formal; Frege sees logic as providing the ‘logical cement’ that ties up together the contentful concepts of specific sciences, not the most general truths. Finally, I discuss how Frege conceives of the application of Begriffsschrift, and of its status as a ‘lingua characteristica’.     

Frege,Boolos, and Logical Objects

In this paper, the authors discuss Frege's theory of logical objects (extensions, numbers, truth-values) and the recent attempts to rehabilitate it. We show that the eta relation George Boolos deployed on Frege's behalf is similar, if not identical, to the encoding mode of predication that underlies the theory of abstract objects. Whereas Boolos accepted unrestricted Comprehension for Properties and used the eta relation to assert the existence of logical objects under certain highly restricted conditions, the theory of abstract objects uses unrestricted Comprehension for Logical Objects and banishes encoding (eta) formulas from Comprehension for Properties. The relative mathematical and philosophical strengths of the two theories are discussed. Along the way, new results in the theory of abstract objects are described, involving: (a) the theory of extensions, (b) the theory of directions and shapes, and (c) the theory of truth values.     

Models,Models, and Models

Michael Dummett famously maintained that analytic philosophy was simply philosophy that followed Frege in treating the philosophy of language as the basis for all other philosophy (1978, 441). But one important insight to emerge from computer science is how difficult it is to animate the linguistic artifacts that the analysis of thought produces. Yet, modeling the effects of thought requires a new skill that goes beyond analysis: procedural literacy. Some of the most promising research in philosophy makes use of a variety of modeling techniques that go beyond basic logic and elementary probability theory. What unifies this approach is a focus on what Alan Perlis called procedural literacy. This essay argues that the future spoils in philosophical research will disproportionally go to those who are procedurally literate.     

The Middle Wittgenstein’s Critique of Frege

ABSTRACT

This article aims to analyse Wittgenstein’s 1929–1932 notes concerning Frege’s critique of what is referred to as old formalism in the philosophy of mathematics. Wittgenstein disagreed with Frege’s critique and, in his notes, outlined his own assessment of formalism. First of all, he approvingly foregrounded its mathematics-game comparison and insistence that rules precede the meanings of expressions. In this article, I recount Frege’s critique of formalism and address Wittgenstein’s assessment of it to show that his remarks are not so much a critique of Frege as rather a defence of the formalist anti-metaphysical investment.     

A repair of Frege’s theory of thoughts

Frege’s writings contain arguments for the thesis (i) that a thought expressed by a sentence S is a structured object whose composition pictures the composition of S, and for the thesis (ii) that a thought is an unstructured object. I will argue that Frege’s reasons for both (i) and (ii) are strong. Frege’s explanation of the difference in sense between logically equivalent sentences rests on assumption (i), while Frege’s claim that the same thought can be decomposed differently makes (ii) plausible. Thoughts are supposed to do work that requires that they be structured and work that requires that they be unstructured. But this cannot be! While the standard response to this problem is to reject either (i) or (ii), I propose a charitable repair in the spirit of Frege’s theory that accepts both. The key idea can be found in Frege’s Basic Laws of Arithmetic(BL, GGA). Frege argues that the thought expressed by a sentence is determined by the truth-conditions that can be derived from the semantic axioms for the sentence constituents. The fact that the same axiomatic truth-condition can be derived in different ways from different semantic axioms suggests a Fregean solution of the dilemma: A thought is a type that is instantiated by all sequences of senses (decomposed thoughts) that have the same axiomatic truth-conditions. This allows for multiple decomposability of the same thought (for different decomposed thoughts can have the same axiomatic truth-conditions) and for a notion of containment (the decomposed thought contains those senses whose semantic axioms are needed in the derivation of the truth-conditions). My proposal combines the virtues of (i) and (ii) without inheriting their vices.     

Ramsey, ‘Universals’ and atomic propositions

In ‘Universals’ (1925), Ramsey declares that we do not, and cannot, know the forms of atomic propositions. A year later, in a symposium with Braithwaite and Joseph, he announces a change of mind: atomic propositions may, after all, be discoverable by analysis. It is clear from the 1926 paper that Ramsey intends this to be a revision of the 1925 claim. Puzzlingly, however, Ramsey does not mention analysis in 1925. My task in this article is to provide a justification for that change of mind. My argument consists of two parts. I first show what relation holds between the negative 1925 assertion and the more positive 1926 assertion which allows us to read the latter claim as a retraction of the former, as Ramsey appears to have intended it. Then I shall argue that the retraction of the 1925 claim is forced upon Ramsey by reflection on the fact that three theses, all of which he held in or around the relevant period, are, rather surprisingly, revealed to be inconsistent.