M. D'Agostino - Böcker

There are several types of proof-theoretical methodologies, Hilbert style, Gentzen style, goal directed, labelled deductive system, as well as many others. One of the most popular is the analytic tableaux methodology, first proposed by Beth and Hintikka and later perfected by Smullyan and Fitting. The style is semantically based and very intuitive. It is the first style taught to students in many universities. The late-1990s have seen interest in tableaux become more widespread, and leading members of the "tableau community" here present a broad coverage of tableau systems for a variety of logics.

Labelled deduction is an approach to providing frameworks for presenting and using different logics in a uniform and natural way by enriching the language of a logic with additional information of a semantic proof-theoretical nature. Labelled deduction systems often possess attractive properties, such as modularity in the way that families of related logics are presented, parameterized proofs of metatheoretic properties, and ease of mechanizability. It is thus not surprising that labelled deduction has been applied to problems in computer science, AI, mathematical logic, cognitive science, philosophy and computational linguistics - for example, formalizing and reasoning about dynamic "state oriented" properties such as knowledge, belief, time, space, and resources.

The tableau methodology, invented in the 1950's by Beth and Hintikka and later perfected by Smullyan and Fitting, is today one of the most popular proof theoretical methodologies. Firstly because it is a very intuitive tool, and secondly because it appears to bring together the proof-theoretical and the semantical approaches to the presentation of a logical system. The increasing demand for improved tableau methods for various logics is mainly prompted by extensive applications of logic in computer science, artificial intelligence and logic programming, as well as its use as a means of conceptual analysis in mathematics, philosophy, linguistics and in the social sciences. In the last few years the renewed interest in the method of analytic tableaux has generated a plethora of new results, in classical as well as non-classical logics. On the one hand, recent advances in tableau-based theorem proving have drawn attention to tableaux as a powerful deduction method for classical first-order logic, in particular for non-clausal formulas accommodating equality.On the other hand, there is a growing need for a diversity of non-classical logics which can serve various applications, and for algorithmic presentations of these logicas in a unifying framework which can support (or suggest) a meaningful semantic interpretation. From this point of view, the methodology of analytic tableaux seems to be most suitable. Therefore, renewed research activity is being devoted to investigating tableau systems for intuitionistic, modal, temporal and many-valued logics, as well as for new families of logics, such as non-monotonic and substructural logics. The results require systematisation. This Handbook is the first to provide such a systematisation of this expanding field. It contains several chapters on the use of tableaux methods in classical logic, but also contains extensive discussions on: the uses of the methodology in intuitionistic logics modal and temporal logics substructural logics, nonmonotonic and many-valued logics the implementation of semantic tableaux a bibliography on analytic tableaux theorem proving.The result is a solid reference work to be used by students and researchers in Computer Science, Artificial Intelligence, Mathematics, Philosophy, Cognitive Sciences, Legal Studies, Linguistics, Engineering and all the areas, whether theoretical or applied, in which the algorithmic aspects of logical deduction play a role.