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)

In 1983, Valentini presented a syntactic proof of cut elimination for a sequent calculus GLSV for the provability logic GL where we have added the subscript V for “Valentini”. The sequents in GLSV were built from sets, as opposed to multisets, thus avoiding an explicit contraction rule. From a syntactic point of view, it is more satisfying and formal to explicitly identify the applications of the contraction rule that are ‘hidden’ in these set based proofs of cut elimination. There is (...) often an underly ing assumption that the move to a proof of cut elimination for sequents built from multisets is easy. Recently, however, it has been claimed that Valentini’s arguments to eliminate cut do not terminate when applied to a multiset formulation of GLSV with an explicit rule of contraction. The claim has led to much confusion and various authors have sought new proofs of cut elimination for GL in a multiset setting. Here we refute this claim by placing Valentini’s arguments in a formal setting and proving cut elimination for sequents built from multisets. The formal setting is particularly important for sequents built from multisets, in order to accurately account for the interplay between the weakening and contraction rules. Furthermore, Valentini’s original proof relies on a novel induction parameter called “width” which is computed ‘globally’. It is diffi cult to verify the correctness of his induction argument based on “width”. In our formulation however, verification of the induction argument is straight forward. Finally, the multiset setting also introduces a new complication in the the case of contractions above cut when the cut formula is boxed. We deal with this using a new transformation based on Valentini’s original arguments. Finally, we show that the algorithm purporting to show the non termi nation of Valentini’s arguments is not a faithful representation of the original arguments, but is instead a transformation already known to be insufficient. (shrink)

This paper deals with the system of modal logicGL, in particular with a formulation of it in terms of sequents. We prove some proof theoretical properties ofGL that allow to get the cut-elimination theorem according to Gentzen's procedure, that is, by double induction on grade and rank.