Among their topics are Hilbert's t and e in

proof theory: a proof-theoretical representation of universal and existential statements, exploring and extending the landscape of conjunctive approach to verisimilitude, quantified modal justification logic with existence predicate, two days in the life of a genius, and the Wittgensteinian and the ontological (three-dimensional) reaction to the naturalistic challenge.

Krajicek (eds.) Arithmetic,

Proof Theory, and Computational Complexity.

Under the nazis there was "German science," "German mathematics and, of course, "German logic." Then there was Gentzen, the founder of modern structural

proof theory. Although his methods, rules and structures have lead to verification programs essential to computer science, and his work on natural deduction, the sequent calculus and ordinal

proof theory are still considered advanced, his life eventually ran contrary to the passions of his nation.

When we talk of Hilbert's finitism, we ordinarily mean his finitist

proof theory or metamathematics rather than what he takes to be the finitist portion of a formalized mathematical theory.

Categories and Subject Descriptors: F.1.3 [Computation by Abstract Devices]: Complexity Classes--machine independent complexity; F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems--complexity of proof procedures; F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logic--mechanical theorem proving,

proof theory; I.2.3 [Artificial Intelligence]: Deduction and Theorem Proving--deduction, resolution.

The remaining essays concern Prior's formal legacy, with articles on applications of temporal logic in computer science, particularly database theory, and articles on modal

proof theory and modal semantics.

The following areas are currently covered by the members of the Editorial Board: Automata and Temporal Logic; Automated Deduction; Automated Verification; Commonsense and Nonmonotonic Reasoning; Constraint Programming; Finite Model Theory and Complexity of Logical Theories; Functional Programming and Lambda Calculus; Concurrency Calculi and Tools; Logic and Machine Learning; Logical Aspects of Computational Complexity; Logical Aspects of Databases; Logic Programming; Logics of Uncertainty; Modal Logics, including Dynamic and Epistemic Logics; Model Checking; Program Development and Verification; Program Specification;

Proof Theory; Term Rewriting Systems; and Type Theory and Logical Frameworks.

In his

proof theory he stated counterparts of Gentzen's cut rule and Hauptsatz.

The main contribution of this article is the introduction of a

proof theory for CCP, i.e., a calculus for proving correctness of CCP programs.

(For a nice account of the development of the notion of diagrams via Leonhard Euler, John Venn, and Charles Sanders Peirce, see the second chapter.) Shin provides two theories of diagrams, Venn-I and Venn-II, in which (1) the main syntactical notion is that of (well-formed) diagrams; (2) the

proof theory consists of a set of rules that tell us how to operate with diagrams; and (3) the semantics is a generalization of the usual semantics for Venn diagrams.