This article presents a simplified proof of the result that bounded depth propositional proofs of the pigeonhole principle are exponentially large. The proof uses the new techniques for proving switching lemmas developed by Razborov and Beame. A similar result is also proved for some examples based on graphs.

Recently, several authors have explored the connections between NP-complete problems for finite objects and the complexity of their analogs for infinite objects. In this paper, we will categorize infinite versions of several problems arising from finite complexity theory in terms of their recursion theoretic complexity and proof theoretic strength. These infinite analogs can behave in a variety of unexpected ways.

Unaware of Frege's 1879 Begriffsschrift, Russell's 1903 The Principles of Mathematics set out a calculus for logic whose foundation was the doctrine that any such calculus must adopt only one style of variables–entity (individual) variables. The idea was that logic is a universal and all-encompassing science, applying alike to whatever there is–propositions, universals, classes, concrete particulars. Unfortunately, Russell's early calculus has appeared archaic if not completely obscure. This paper is an attempt to recover the formal system, showing its philosophical background and its semantic completeness with respect to the tautologies of a modern sentential calculus.

We show how to interpret Heyting's arithmetic in an intuitionistic version of TT, Russell's Simple Theory of Types. We also exhibit properties of finite sets in this theory and compare them with the corresponding properties in classical TT. Finally, we prove that arithmetic can be interpreted in intuitionistic TT$_3$, the subsystem of intuitionistic TT involving only three types. The definitions of intuitionistic TT and its finite sets and natural numbers are obtained in a straightforward way from the classical definitions. This is very natural and seems to make intuitionistic TT an interesting intuitionistic set theory to study, beside intuitionistic ZF.

If what is known need not be closed under logical consequence, then a distinction arises between something's being known to be the case (by a specific agent) and its following from something known (to that subject). When each of these notions is represented by a sentence operator, we get a bimodal logic in which to explore the relations between the two notions.

For a quasi variety of algebras K, the Higman Theorem is said to be true if every recursively presented K-algebra is embeddable into a finitely presented K-algebra; the Generalized Higman Theorem is said to be true if any K-algebra which is recursively presented over its finitely generated subalgebra is embeddable into a K-algebra which is finitely presented over this subalgebra. We suggest certain general conditions on K under which (1) the Higman Theorem implies the Generalized Higman Theorem; (2) a finitely generated K-algebra A is embeddable into every existentially closed K-algebra containing a finitely generated K-algebra B if and only if the word problem for A is Q-reducible to the word problem for B. The quasi varieties of groups, torsion-free groups, and semigroups satisfy these conditions.

Field's claim that we have a notion of consistency which is neither model-theoretic nor proof-theoretic but primitive, is examined and criticized. His argument is compared to similar examinations by Kreisel and Etchemendy, and Etchemendy's distinction between interpretational and representational semantics is employed to reveal the flaw in Field's argument.