Handbook of philosophical logic. Vol. 9
This volume of this Handbook series is to a large extent concerned with proof-theoretic matters. It offers only chapters which do not have companions in the first edition. Their common feature is a tendency toward computer science applications. The first chapter (pp. 1--88) by \textit{N. Martí-Oliet} and \textit{J. Meseguer} is on ``Rewriting logic as a logical and semantic framework''. The authors aim is to explain the basic ideas of rewriting logic, as introduced by \textit{J. Meseguer} [Theor. Comput. Sci. 96, 73--155 (1992; Zbl 0758.68043)], and to offer it as a general semantical framework for a unified understanding of a broad class of different logics. The next chapter (pp. 89--164) discusses ``Logical frameworks'' again from a general point of view and is written by \textit{D. Basin} and \textit{S. Matthews}. Here, logics are mainly understood as being determined by syntactic calculi. The aim again is to offer a unifying metalogical framework which covers quite different systems of logic. Following this, \textit{G. Sundholm} explains in his chapter on ``Proof theory and meaning'' (pp. 165--198) the understanding that, and to what extent, the logical calculi, particularly the rules governing the behavior of the connectives, determine the meaning of these connectives, and to what extent a truth-conditional definition of the meaning of the connectives is problematic. Essentially a particular type of logical framework is presented in the fourth chapter (pp. 199--286) by \textit{D. M. Gabbay} and \textit{N. Olivetti}: ``Goal directed deductions''. This chapter's main goal is a generalized methodology for automated deduction inspired by the logic programming paradigm. The (formal) ``goals'' within the deductions may be formulas which have to be checked for satisfiability or derivability or validity, of course in certain situations. The methodology is based upon backward reasoning which comes into considerations via the splitting of goals into subgoals which imply, if satisfied, the full goals. This strategy is explained and applied for classical, intuitionistic, modal, and also substructural logics. There follows the chapter ``On negation, completeness and consistency'' (pp. 287--320), written by \textit{A. Avron}. This author takes a closer look at problems connected with the understanding of negations in different logics. And he offers syntactical discussions, centering around a general notion of consequence relation, as well as semantical ones which center around general notions of completeness (in the sense that one has ``either \(A\) or \(\neg A\)'') and consistency. And in the final chapter, ``Logic as general rationality: a survey'' (pp. 321--366), which also (using the author's own words) could have been entitled ``Logic and classical probability: recovering the lost common ground'', \textit{T. Sales} tries to convince the reader with mainly philosophical arguments that general considerations about rationality of argumentations may lay down a common basis for logic and probability.
scientific article; zbMATH DE number 2118884