Mathematical formalism is the view that numbers are “signs” and arithmetic like a game played with such signs. Frege’s colleague Thomae defended using an analogy chess, critique of this has had major influence on discussions in analytic philosophy about signs, rules, meaning, mathematics. Here I offer new interpretation as by his predecessors, paying close attention to mathematical details hist...

I mention two methodological problems with AI's study of inference that have to do with excessive faith in classical logic. The first is the familiar bias toward formalisms that incorporate deductive patterns of argument: a bias against inductive patterns of reasoning. The second is the pretension that inference can be studied with no regard for one's habits of representation. By implicating Fr...

We investigate universes axiomatized as sets with natural closure conditions over Frege structures. In the presence of a natural form of induction, we obtain a theory of proof-theoretic strength 0 .

Tarski is widely regarded as the father of formal semantics, and rightly so. But if Tarski is the father, then Frege is the grandfather. Frege’s Grundgesetze der Arithmetik contains a semantic theory for his formal language, his begriffsschrift (or ‘conceptual notation’), that is no less rigorous than the one for the calculus of classes that Tarski develops in the early parts of “The Concept of...

In this paper, we characterize the strength of the predicative Frege hierarchy, PV, introduced by John Burgess in his book [Bur05]. We show that PV and Q + con(Q) are mutually interpretable. It follows that PV := PV 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 PV is mutually i...

We investigate the complexity of proofs in Frege (F), Substitution Frege (sF) and Renaming Frege (rF) systems. Starting from a recent work of Urquhart and using Kolmogorov Complexity we give a more general framework to obtain superlogarithmic lower bounds for the number of lines in both tree-like and dag-like sF. We show the previous known lower bound, extend it to the tree-like case and, for a...

Frege’s most celebrated paper “On Sense and Reference” begins with a puzzle. The sentences “a=a” and “a=b” superficially strike the reader as being different kinds of claims. Among other things, “a=a” seems to be trivially true, even self-evident. On the other hand, “a=b” is an informative sentence. It is of a kind important to many episodes in the history of science. Deeper reflection, however...

Gottlob Frege Patricia A. Blanchette Abstract Gottlob Frege (1848-1925) made significant contributions to both pure mathematics and philosophy. His most important technical contribution, of both mathematical and philosophical significance, is the introduction of a formal system of quantified logic. His work of a more purelyphilosophical kind includes the articulation and persuasive defense of a...

In ‘What’s Puzzling Gottlob Frege?’ Michael Thau and Ben Caplan argue that, contrary to the common wisdom, Frege never abandoned his early view that, as he puts it in Begriffsschrift, a statement of identity “expresses the circumstance that two names have the same content” (Bg, §8) and thus asserts the existence of a relation between names rather than a relation between (ordinary) objects. The ...