In this paper we study the complexity of Cutting Planes (CP) refutations, and tree-like CP refutations. Tree-like CP proofs are natural and still quite powerful. In particular, the propositional pigeonhole principle (PHP) has been shown to have polynomial-sized tree-like CP proofs. Our main result shows that a family of tautologies, introduced in this paper requires exponential-sized tree-like ...

We establish some relations between the class of truth-equational logics, the class of assertional logics, other classes in the Leibniz hierarchy, and the classes in the Frege hierarchy. We argue that the class of assertional logics belongs properly in the Leibniz hierarchy. We give two new characterizations of truth-equational logics in terms of their full generalized models, and use them to o...

It is widely believed that some puzzling and provocative remarks that Frege makes in his late writings indicate he rejected independence arguments in geometry, particularly arguments for the independence of the parallels axiom. I show that this is mistaken: Frege distinguished two approaches to independence arguments and his puzzling remarks apply only to one of them. Not only did Frege not rej...

We discuss the typography of the notation used by Gottlob Frege in his Grundgesetze der Arithmetik. §1. Background to the Grundgesetze der Arithmetik. Grundgesetze der Arithmetik was to have been the pinnacle of Gottlob Frege’s life’s work — a rigorous demonstration of how the fundamental laws of classical pure mathematics of the natural and real numbers can be derived from principles which, in...

In this paper two concepts of psychologism in logic are outlined: the one which Frege and Husserl fought against and the new psychologism, or cognitivism, which underlies a cognitive turn in contemporary logic. Four issues such cognitively oriented logic should be interested in are indicated. They concern: new fields opened for logical analysis, new methods and tools needed to address these fie...

I investigate the role of subject in judgment Kant, Frege, and Husserl, situating it broader less-often-considered context their accounts presentation (Vorstellung) as well judgment. Contemporary philosophical usage “representation” tends to elide question what Kant called constitution content, because a reluctance, traced Frege’s anti-psychologism, attend subjectivity. But for anti-psychologis...

We prove lower bounds for a proof system having exponential speed-up over both polynomial calculus and constant-depth Frege systems in DeMorgan language.

Frege hoped that the theories of mathematics could be given bases of self-evident axioms. As far as possible, the aim was the reduction of mathematics to logic; the axioms grounding mathematical theories should be logical principles, and the rules of inference used to move from the axioms to the theory should be rules of logic. Frege did not believe, however, that the reduction to logic was ach...

In this paper two concepts of psychologism in logic are outlined: the one which Frege and Husserl fought against and the new psychologism, or cognitivism, which underlies a cognitive turn in contemporary logic. Four issues such cognitively oriented logic should be interested in are indicated. They concern: new fields opened for logical analysis, new methods and tools needed to address these fie...

1 Russell on the problem of the unity of the proposition . . . . . . . . . . . . . . . . . . 1 2 Frege on concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 Propositions as primitive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4 Soames on primitive mental acts of predication . . . . . . . . . . . . . . . . . . . . . . 3 5 Propo...