In this paper we shall meet the application of Scott domains to nominalisation and explain its problem of predication. We claim that it is not possible to nd a solution to such a problem within semantic domains without logic. Frege structures are more conclusive than a solution to domain equations and can be used as models for nominalisation. Hence we develop a type theory based on Frege struct...

The question I am interested in in this paper is "How do words obtain their meanings?" The author I shall ask for help in answering this question is Gottlob Frege. This may appear surprising, since Frege is known as not being interested in the question posed. Frege wanted to provide mathematics with solid foundations. To achieve this, he aimed at constructing from scratch logical systems that m...

We show that short bounded-depth Frege proofs of matrix identities, such as PQ = I ⊃ QP = I (over the field of two elements), imply short bounded-depth Frege proofs of the pigeonhole principle. Since the latter principle is known to require exponential-size bounded-depth Frege proofs, it follows that the propositional version of the matrix principle also requires bounded-depth Frege proofs of e...

In proof theory one distinguishes sequent proofs with cut and cut-free sequent proofs, while for proof complexity one distinguishes Frege-systems and extended Frege-systems. In this paper we show how deep inference can provide a uniform treatment for both classifications, such that we can define cut-free systems with extension, which is neither possible with Frege-systems, nor with the sequent ...

We give a general transformation which turns polynomial-size Frege proofs to subexponential-size AC0Frege proofs. This indicates that proving truly exponential lower bounds for AC0-Frege is hard, since it is a longstanding open problem to prove super-polynomial lower bounds for Frege. Our construction is optimal for proofs of formulas of unbounded depth. As a consequence of our main result, we ...

Does every Boolean tautology have a short propositional-calculus proof? Here, a propositionalcalculus (i.e. Frege) proof is any proof starting from a set of axioms and deriving new Boolean formulas using a fixed set of sound derivation rules. Establishing any super-polynomial size lower bound on Frege proofs (in terms of the size of the formula proved) is a major open problem in proof complexit...

O objetivo do presente artigo é indicar a função que distinção entre sentido e referência desempenha dentro da perspectiva logicista de Frege. Em particular, partir distinção, Frege busca introduzir os valores verdade como objetos, quais desempenham um papel central no sistema lógico das Leis básicas aritmética

There are two contributions in this paper. First, we give a direct proof of the known fact that Frege systems with substitution can be p-simulated by the calculus of structures extended with the substitution rule. This is done without referring to the p-equivalence of extended Frege systems and Frege systems with substitution. Second, we then show that the cut-free calculus of structures with s...