On the Consistency of the Δ1 1-CA Fragment of Frege's Grundgesetze

It is well known that Frege's system in the Grundgesetze der Arithmetik is formally inconsistent. Frege's instantiation rule for the second-order universal quantifier makes his system, except for minor differences, full (i.e., with unrestricted comprehension) second-order logic, augmented by an abstraction operator that abides to Frege's basic law V. A few years ago, Richard Heck proved the consistency of the fragment of Frege's theory obtained by restricting the comprehension schema to predicative formulae. He further conjectured that the more encompassing ¹ ₁-comprehension schema would already be inconsistent. In the present paper, we show that this is not the case.     

Logic and Philosophy of Logic in Wittgenstein

This essay discusses Wittgenstein's conception of logic, early and late, and some of the types of logical system that he constructed. The essay shows that the common view according to which Wittgenstein had stopped engaging in logic as a philosophical discipline by the time of writing Philosophical Investigations is mistaken. It is argued that, on the contrary, logic continued to figure at the very heart of later Wittgenstein's philosophy; and that Wittgenstein's mature philosophy of logic contains many interesting thoughts that have gone widely unnoticed.     

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.     

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.

The logic, intentionality, and phenomenology of emotion

My concern in this paper is with the intentionality of emotions. Desires and cognitions are the traditional paradigm cases of intentional attitudes, and one very direct approach to the question of the intentionality of emotions is to treat it as sui generis—as on a par with the intentionality of desires and cognitions but in no way reducible to it. A more common approach seeks to reduce the intentionality of emotions to the intentionality of familiar intentional attitudes like desires and cognitions. In this paper, I argue for the sui generis approach.

Erkenntnistheorie der Zahldefinition und philosophische Grundlegung der Arithmetik unter Bezugnahme auf einen Vergleich von Gottlob Freges Logizismus und platonischer Philosophie (Syrian, Theon von Smyrna u.a.)

The epistomology of the definition of number and the philosophical foundation of arithmetic based on a comparison between Gottlob Frege's logicism and Platonic philosophy (Syrianus, Theo Smyrnaeus, and others). The intention of this article is to provide arithmetic with a logically and methodologically valid definition of number for construing a consistent philosophical foundation of arithmetic. The – surely astonishing – main thesis is that instead of the modern and contemporary attempts, especially in Gottlob Frege's Foundations of Arithmetic, such a definition is found in the arithmetic in Euclid's Elements. To draw this conclusion a profound reflection on the role of epistemology for the foundation of mathematics, especially for the method of definition of number, is indispensable; a reflection not to be found in the contemporary debate (the predominate ‘pragmaticformalism’ in current mathematics just shirks from trying to solve the epistemological problems raised by the debate between logicism, intuitionism, and formalism). Frege's definition of number, ‘The number of the concept F is the extension of the concept ‘numerically equal to the concept F”, which is still substantial for contemporary mathematics, does not fulfil the requirements of logical and methodological correctness because the definiens in a double way (in the concepts ‘extension of a concept’ and ‘numerically equal’) implicitly presupposes the definiendum, i.e. number itself. Number itself, on the contrary, is defined adequately by Euclid as ‘multitude composed of units’, a definition which is even, though never mentioned, an implicit presupposition of the modern concept ofset. But Frege rejects this definition and construes his own - for epistemological reasons: Frege's definition exactly fits the needs of modern epistemology, namely that for to know something like the number of a concept one must become conscious of a multitude of acts of producing units of ‘given’ representations under the condition of a 1:1 relationship to obtain between the acts of counting and the counted ‘objects’. According to this view, which has existed at least since the Renaissance stoicism and is maintained not only by Frege but also by Descartes, Kant, Husserl, Dummett, and others, there is no such thing as a number of pure units itself because the intellect or pure reason, by itself empty, must become conscious of different units of representation in order to know a multitude, a condition not fulfilled by Euclid's conception. As this is Frege's main reason to reject Euclid's definition of number (others are discussed in detail), the paper shows that the epistemological reflection in Neoplatonic mathematical philosophy, which agrees with Euclid's definition of number, provides a consistent basement for it. Therefore it is not progress in the history of science which hasled to the a poretic contemporary state of affairs but an arbitrary change of epistemology in early modern times, which is of great influence even today. This revised version was published online in August 2006 with corrections to the Cover Date.     

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.     

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.     

The use theory of meaning and semantic stipulation

According to Horwich’s use theory of meaning, the meaning of a word W is engendered by the underived acceptance of certain sentences containing W. Horwich applies this theory to provide an account of semantic stipulation: Semantic stipulation proceeds by deciding to accept sentences containing an as yet meaningless word W. Thereby one brings it about that W gets an underived acceptance property. Since a word’s meaning is constituted by its (basic) underived acceptance property, this decision endows the word with a meaning. The use-theoretic account of semantic stipulation contrasts with the standard view that semantic stipulation proceeds by assigning the meaning (reference) to W that makes a certain set of sentences express true propositions. In this paper I will argue that the use-theoretic account does not work. I take Frege to have already made the crucial point: "a definition does not assert anything but lays down something ["etwas festsetzt"]” (Frege 1899, 36). A semantic stipulation for W cannot be the decision to accept a sentence containing W or be explained in terms of such an acceptance. Semantic stipulation constitutes a problem for Horwich's use theory of meaning, especially his basic notion of acceptance.