This introduction to the basic ideas of structural proof theory contains a thorough discussion and comparison of various types of formalization of first-order logic. Examples are given of several areas of application, namely: the metamathematics of pure first-order logic (intuitionistic as well as classical); the theory of logic programming; category theory; modal logic; linear logic; first-order arithmetic and second-order logic. In each case the aim is to illustrate the methods in relatively simple situations and then apply them elsewhere in much (...) more complex settings. There are numerous exercises throughout the text. In general, the only prerequisite is a standard course in first-order logic, making the book ideal for graduate students and beginning researchers in mathematical logic, theoretical computer science and artificial intelligence. For the new edition, many sections have been rewritten to improve clarity, new sections have been added on cut elimination, and solutions to selected exercises have been included. (shrink)

IPV is the intuitionistic theory axiomatized by Cook's equational theory PV plus PIND on NP-formulas. Two extensions of IPV were introduced by Buss and by Cook and Urquhart by adding PIND for formulas of the form A ∨ B, respectively ¬¬A, where A is NP and x is not free in B. Cook and Urquhart posed the question of whether these extensions are proper. We show that in each of the two cases the extension is proper unless the polynomial hierarchy (...) collapses. (shrink)

A notion of feasible function of finite type based on the typed lambda calculus is introduced which generalizes the familiar type 1 polynomial-time functions. An intuitionistic theory IPVω is presented for reasoning about these functions. Interpretations for IPVω are developed both in the style of Kreisel's modified realizability and Gödel's Dialectica interpretation. Applications include alternative proofs for Buss's results concerning the classical first-order system S12 and its intuitionistic counterpart IS12 as well as proofs of some of Buss's conjectures concerning IS12, (...) and a proof that IS12 cannot prove that extended Frege systems are not polynomially bounded. (shrink)

We define classes Φnof formulae of first-order arithmetic with the following properties:(i) Everyφϵ Φnis classically equivalent to a Πn-formula (n≠ 1, Φ1:= Σ1).(ii)(iii)IΠnandiΦn(i.e., Heyting arithmetic with induction schema restricted to Φn-formulae) prove the same Π2-formulae.We further generalize a result by Visser and Wehmeier. namely that prenex induction within intuitionistic arithmetic is rather weak: After closing Φnboth under existential and universal quantification (we call these classes Θn) the corresponding theoriesiΘnstill prove the same Π2-formulae. In a second part we consideriΔ0plus collection-principles. We (...) show that both the provably recursive functions and the provably total functions ofare polynomially bounded. Furthermore we show that the contrapositive of the collection-schema gives rise to instances of the law of excluded middle and hence. (shrink)

A basic result in intuitionism is Π02-conservativity. Take any proof p in classical arithmetic of some Π02-statement , with P decidable). Then we may effectively turn p in some intuitionistic proof of the same statement. In a previous paper [1], we generalized this result: any classical proof p of an arithmetical statement ∀x.∃y.P, with P of degree k, may be effectively turned into some proof of the same statement, using Excluded Middle only over degree k formulas. When k = 0, (...) we get the original conservativity result as particular case. This result was a by-product of a semantical construction. J. Avigad of Carnegie Mellon University, found a short, direct syntactical derivation of the same result, using H. Friedman's A-translation. His proof is included here with his permission. (shrink)

A number of classical theories are interpreted in analogous theories that are based on intuitionistic logic. The classical theories considered include subsystems of first- and second-order arithmetic, bounded arithmetic, and admissible set theory.

Paul Cohen’s method of forcing, together with Saul Kripke’s related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects (...) of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing model-theoretic arguments. (shrink)

Provability, Computability and Reflection.