Introduction to the constructive point of view in the foundations of mathematics, in particular intuitionism due to L.E.J. Brouwer, constructive recursive mathematics due to A.A. Markov, and Bishop’s constructive mathematics. The constructive interpretation and formalization of logic is described. For constructive (intuitionistic) arithmetic, Kleene’s realizability interpretation is given; this provides an example of the possibility of a constructive mathematical practice which diverges from classical mathematics. The crucial notion in intuitionistic analysis, choice sequence, is briefly described and some principles which are valid for choice sequences are discussed. The second half of the article deals with some aspects of proof theory, i.e., the study of formal proofs as combinatorial objects. Gentzen’s fundamental contributions are outlined: his introduction of the so-called Gentzen systems which use sequents instead of formulas and his result on first-order arithmetic showing that (suitably formalized) transfinite induction up to the ordinal "0 cannot be proved in first-order arithmetic.

This manual describes how to use the theorem prover Isabelle. For beginners, it explains how to perform simple single-step proofs in the built-in logics. These include first-order logic, a classical sequent calculus, zf set theory, Constructive Type Theory, and higher-order logic. Each of these logics is described. The manual then explains how to develop advanced tactics and tacticals and how to derive rules. Finally, it describes how to define new logics within Isabelle. Acknowledgements. Isabelle uses Dave Matthews's Standard ml compiler, Poly/ml. Philippe de Groote wrote the first version of the logic lk. Funding and equipment were provided by SERC/Alvey grant GR/E0355.7 and ESPRIT BRA grant 3245. Thanks also to Philippe Noel, Brian Monahan, Martin Coen, and Annette Schumann. Contents 1 Basic Features of Isabelle 5 1.1 Overview of Isabelle : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 1.1.1 The representation of logics : : : : : : : : : : : : : : : : : : : 6 1.1.2 The...

Abstract. Modular cut-elimination is a particular notion of ”cut-elimination in the presence of non-logical axioms ” that is preserved under the addition of suitable rules. We introduce syntactic necessary and sufficient conditions for modular cut-elimination for standard calculi, a wide class of (possibly) multipleconclusion sequent calculi with generalized quantifiers. We provide a ”universal” modular cut-elimination procedure that works uniformly for any standard calculus satisfying our conditions. The failure of these conditions generates counterexamples for modular cut-elimination and, in certain cases, for cut-elimination. 1

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. We initiate the study of proof complexity for intuitionistic propositional proof systems: It is known that the set of intuitionistic tautologies is PSPACE-complete (as opposed to coNP-complete in the classical case). We show that formulas derived from the "clique tautologies " used in classical proof complexity have only exponential-size proofs in intuitionistic sequent calculi or Frege systems (without substitution). This is in contrast to the classical case where the complexity of Frege systems is still open. Introduction The theory of classical propositional proof systems is motivated by the conjecture NP 6= co-NP. Moreover, recently there is increased interest in the impact of this theory on questions of a more algorithmic nature,for example [Bo et al.97],[Be et al. 97]. As the set of classical tautologies is co-NP complete and propositional proof systems are just non-deterministic algorithms having at least 1 accepting computation for each tautology, there should be tautologies...

1 Introduction and review of some basic analytic tools 1

We define notation system for infinitary derivations arising from cutelimination for a theory T 1 \Sigma of recursively regular ordinals by the method of local predicativity. Using these notations, we derive finitary cutelimination steps together with corresponding ordinal assignments. Introduction There is an extensive literature connecting infinitary "Schutte-style" and finitary "Gentzen-Takeuti-style" sides of proof theory. For example, in papers [Mi75, Mi75a, Mi79, Bu91, Bu97a] this was done for systems not exceeding in strength Peano Arithmetic. But most recently, there has been an interest to what one can get on the side of finitary proof theory from the methods which are used for proof-theoretical analysis of impredicative theories (see [Wei96, Bu97]). Especially we want to mention paper [Bu97], where it was shown that Takeuti's reduction steps for \Pi 1 1 \Gamma CA+ BI [Tak87, x27] can be derived from Buchholz' method of\Omega +1 -rule ([BFPS, Ch. IV--V], [BS88]). Here we ...

i Acknowledgements I would like to thank my advisor Agata Ciabattoni for introducing me to the field of systematic proof theory, for her patience and her continuous support. I really appreciate the time she spent on explanations when a proof didn’t work out as planned. I also want to thank Gernot Salzer who was of great help when I was struggling with PROLOG. I am really grateful to my parents, Patrizia and Ingmar, my brother Dominik and my family for supporting me in every possible way. I also want to thank Christoph for always being there. Lara Non-classical logics are logics different from classical, boolean logic. They encompass, amongst others, the family of intermediate, fuzzy, and substructural logics. During the past few years, these logics have gained importance, especially in many fields of computer science, artificial intelligence, and knowledge representation. By now, there are many useful and interesting non-classical logics and practitioners in various fields