Online research in philosophy

I present a novel interpretation of Frege’s attempt at Grundgesetze I §§29-31 to prove that every expression of his language has a unique reference. I argue that Frege’s proof is based on a contextual account of reference, similar to but more sophisticated than that enshrined in his famous Context Principle. Although Frege’s proof is incorrect, I argue that the account of reference on which it is based is of potential philosophical value, and I analyze the class of cases to which it may successfully be applied.

Frege’s Grundgesetze der Arithmetik is formally inconsistent. This system is, except for minor differences, second-order logic together with an abstraction operator governed by Frege’s Axiom V. A few years ago, Richard Heck showed that the ramified predicative second-order fragment of the Grundgesetze is consistent. In this paper, we show that the above fragment augmented with the axiom of reducibility for concepts true of only finitely many individuals is still consistent, and that elementary Peano arithmetic (and more) is interpretable in this extended system.

I argued that Frege does not have a metatheory in the following sense: the justifications he offers for his basic laws and rules of inference neither employ nor require a truth-predicate or metalinguistic variables. In ‘Does Frege Use a Truth-predicate in his "Justification" of the Laws of Logic?’, Dirk Greimann disputes this. As Greimann interprets Frege, (i) Frege's remarks commit him to giving a metatheoretic justification of the basic laws and rules of his logic, and (ii) Frege actually gives such a justification in the early sections of Grundgesetze—although the truth-predicate that Frege employs is a non-standard one: it is neither a predicate that holds of all and only true sentences nor a predicate that holds of all and only true thoughts. I argue that Greimann's interpretation is not, in the end, true to the text, and that his non-standard view of what is required of a Tarskian truth-predicate is ultimately not viable. CiteULike    Connotea    Del.icio.us    What's this?

Let me start with a well-known story. Kant held that logic and conceptual analysis alone cannot account for our knowledge of arithmetic: “however we might turn and twist our concepts, we could never, by the mere analysis of them, and without the aid of intuition, discover what is the sum [7+5]” (KrV, B16). Frege took himself to have shown that Kant was wrong about this. According to Frege’s logicist thesis, every arithmetical concept can be defined in purely logical terms, and every theorem of arithmetic can be proved using only the basic laws of logic. Hence, Kant was wrong to think that our grasp of arithmetical concepts and our knowledge of arithmetical truth depend on an extralogical source—the pure intuition of time (Frege 1884, §89, §109). Arithmetic, properly understood, is just a part of logic.

This thesis discusses some central aspects of Wittgenstein's conception of language and logic in his Tractatus Logico-Philosophicus and brings them into relation with the philosophies of Frege and Russell. The main contention is that a fruitful way of understanding the Tractatus is to see it as responding to tensions in Frege's conception of logic and Russell's theory of judgement. In the thesis the philosophy of the Tractatus is presented as developing from these two strands of criticism and thus as the culmination of the philosophy of logic and language developed in the early analytic period. Part one examines relevant features of Frege's philosophy of logic. Besides shedding light on Frege's philosophy in its own right, it aims at preparing the ground for a discussion of those aspects of the Tractatus' conception of logic which derive from Wittgenstein's critical response to Frege. Part two first presents Russell's early view on truth and judgement, before considering several variants of the multiple relation theory of judgement, devised in opposition to it. Part three discusses the development of Wittgenstein's conception of language and logic, beginning with Wittgenstein's criticism of the multiple relation theory and his early theory of sense, seen as containing the seeds of the picture theory of propositions presented in the Tractatus. I then consider the relation between Wittgenstein's pictorial conception of language and his conception of logic, arguing that Wittgenstein's understanding of sense in terms of bipolarity grounds his view of logical complexity and of the essence of logic as a whole. This view, I show, is free from the internal tensions that affect Frege's understanding of the nature of logic.

In the opening to his late essay, Der Gedanke, Frege asserts without qualification that the word "true" points the way for logic. But in a short piece from his Nachlass entitled "y Basic Logical Insights", Frege writes that the word true makes an unsuccessful attempt to point to the essence of logic, asserting instead that "what really pertains to logic lies not in the word "true" but in the assertoric force with which the sentence is uttered". Properly understanding what Frege takes to be at issue here is crucial for understanding his conception of logic and, in particular, what he takes to be its normative status vis-à-vis judgement, assertion, and inference. In this paper, I focus my attention on clarifying the latter claim and Frege's motivations for making it, exposing what I take to be a fundamental tension in Frege's conception of logic. Finally, I discuss whether Frege's deployment of the horizontal in his mature Begriffsschrift helps to resolve this tension. CiteULike    Connotea    Del.icio.us    What's this?

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.

This paper defends the view that Frege?s reduction of arithmetic to logic would, if successful, have shown that arithmetical knowledge is analytic in essentially Kant?s sense.It is argued, as against Paul Benacerraf, that Frege?s apparent acceptance of multiple reductions is compatible with this epistemological thesis.The importance of this defense is that (a) it clarifies the role of proof, definition, and analysis in Frege?s logicist works; and (b) it demonstrates that the Fregean style of reduction is a valuable tool for those who would investigate the nature of arithmetical knowledge.

Frege's account of indirect proof has been thought to be problematic. This thought seems to rest on the supposition that some notion of logical consequence ? which Frege did not have ? is indispensable for a satisfactory account of indirect proof. It is not so. Frege's account is no less workable than the account predominant today. Indeed, Frege's account may be best understood as a restatement of the latter, although from a higher order point of view. I argue that this ascent is motivated by Frege's conception of logic.

Frege's development of the theory of arithmetic in his Grundgesetze der Arithmetik has long been ignored, since the formal theory of the Grundgesetze is inconsistent. His derivations of the axioms of arithmetic from what is known as Hume's Principle do not, however, depend upon that axiom of the system--Axiom V--which is responsible for the inconsistency. On the contrary, Frege's proofs constitute a derivation of axioms for arithmetic from Hume's Principle, in (axiomatic) second-order logic. Moreover, though Frege does prove each of the now standard Dedekind-Peano axioms, his proofs are devoted primarily to the derivation of his own axioms for arithmetic, which are somewhat different (though of course equivalent). These axioms, which may be yet more intuitive than the Dedekind-Peano axioms, may be taken to be "The Basic Laws of Cardinal Number", as Frege understood them. Though the axioms of arithmetic have been known to be derivable from Hume's Principle for about ten years now, it has not been widely recognized that Frege himself showed them so to be; nor has it been known that Frege made use of any axiomatization for arithmetic whatsoever. Grundgesetze is thus a work of much greater significance than has often been thought. First, Frege's use of the inconsistent Axiom V may invalidate certain of his claims regarding the philosophical significance of his work (viz., the establishment of Logicism), but it should not be allowed to obscure his mathematical accomplishments and his contribution to our understanding of arithmetic. Second, Frege's knowledge that arithmetic is derivable from Hume's Principle raises important sorts of questions about his philosophy of arithmetic. For example, "Why did Frege not simply abandon Axiom V and take Hume's Principle as an axiom?".