Handbook of philosophical logic. Volume II: Extensions of classical logic
[For a review of Volume I (1983) see Zbl 0538.03001.] The second volume of this handbook leaves the field of classical logic and enters the very field of non-classical logics with background in philosophical topics. The leitmotif of this volume is the semantical idea of possible worlds: all its 12 chapters treat topics which are influenced by this idea. Or, to put it in other words, the whole volume is concerned with intensional logics. Point of departure is chapter 1 by \textit{R. A. Bull} and \textit{K. Segerberg} on ''Basic modal logic'' (pp. 1-88) in which the authors first present a historical survey and a catalogue of (propositional) modal logics in the style of \textit{K. Segerberg}'s book [An essay in classical modal logic (1971; Zbl 0311.02028)] and later turn to the main model theoretical constructions and results in the field with, of course, the Kripke frames at the very core. This semantic idea too is essential for chapter 2 by \textit{J. P. Burgess} on ''Basic tense logic'' (pp. 89-133), where the basic operators and systems of tense logic are introduced, and where the characterizability of the structure of time by tense logical formulas is discussed as well as the decidability of tense logics and nonclassical tense operators. An outlook on some more ''exotic'' time structures, unfortunately with only few references, closes the considerations. Chapter 3 by \textit{R. H. Thomason} immediately unites both themes, it is on ''Combinations of tense and modality'' (pp. 135-165). The topic is treated from a more philosophical perspective with lesser emphasis on pure technicalities: the essential difficulty is to understand the meaning of such combinations of tenses and modalities, e.g. of what is meant with ''historical necessity''. Some considerations on deontic resp. conditional logic combined with historical necessity close the chapter. This really philosophical intermezzo is followed by chapter 4 on ''Correspondence theory'' (pp. 167-247), written by \textit{J. van Benthem}, in which again - as in the first two chapters - a lot of deep technical results are presented. The main topic are the relations between the truth of formulas including intensional operators and properties of their corresponding (Kripke) models, and conversely the describability of classes of frames by intensional formulas. Here, not only the propositional case is discussed but the first order case as well for the intensional formulas - the frame properties often have to be formulated by second order classical logic. The intensional operators discussed are primary the modalities, but tense logical operators and conditionals as well as the case of intuitionistic logic are treated, too. The propositional systems are left with chapter 5 by \textit{J. W. Garson} on ''Quantification in modal logic'' (pp. 249-307). The main purpose of this chapter is to make sense of the difficult and seemingly chaotic terrain opened by the combination of modalities and quantifiers. The essential problem is the lack of an intuitively satisfying semantics, whereas the usual frame semantics offers a lot of - mostly independent - possibilities for (almost) free choices which result in a lot of first order modal logics that differ e.g. with respect to their semantics of (transworld) identity, terms and predicates. A review of the major systems is given together with an inquiry into their completeness problems and some incompleteness results. After having those technical results, \textit{N. B. Cocchiarella} gives in chapter 6 ''Philosophical perspectives on quantification in tense and modal logic'' (pp. 309-353) with emphasis on the philosophical background for the technical difficulties known from chapter 5 and on some proposals to overcome them. The following chapter 7 ''General intensional logic'' (pp. 355-385) by \textit{C. A. Anderson} surveys quite general possible worlds approaches to intensional logic: Montague's approach in the axiomatized form of \textit{D. Gallin} [Intensional and higher-order modal logic (1975; Zbl 0341.02014)] and the logic of sense and denotation of \textit{A. Church} [e.g. Noûs 7, 24-33 (1973) and ibid. 8, 135-156 (1974)]. The remaining chapters treat more applied topics. The first such topic is ''Conditional logic'' in chapter 8 (pp. 387-439), written by \textit{D. Nute}, i.e. the logic of counterfactuals - in a wide sense. Main theme are the subjunctive conditionals, starting point is a paper of \textit{R. C. Stalnaker} [''A theory of conditionals'' in: \textit{N. Rescher} (ed.), Studies in Logical Theory, Amer. Philos. Quart. Monogr. Ser., 2, 98-112 (1968)]. At the end some remarks on indicative, even-if, and might conditionals are given, too. Essential semantical ideas are represented by the theories of minimal, small, and maximal changes which play a central role. The next chapter 9 by \textit{C. Smoryński} on ''Modal logic and self-reference'' (pp. 441-495) presents the modal analysis of self- reference which arose mainly in the 1970s from several directions as soon as it was recognised that, modulo the background analysis by \textit{M. H. Löb} [J. Symb. Logic 20, 115-118 (1955; Zbl 0067.002)] of the representation of provability within a system, the proofs of theorems of Gödel and Löb on self-referential sentences were propositional in character, that is, they used propositional logic with an additional - modal (!) - operator. After explaining that background, a corresponding modal calculus called provability logic is presented together with key results of deJongh-Sambin concerning fixed points and of Solovay concerning completeness. At the end, Rosser sentences and the authors own unified treatment of self-referential formulae are considered. A switch from arithmetic to computer science is made with chapter 10 by \textit{D. Harel} on ''Dynamic logic'' (pp. 497-604), i.e., to a field of extensive current research that attempts to provide a precise mathematical basis for the common activities involved in reasoning about computer programs and which started essentially with a paper of \textit{V. R. Pratt} [''Semantical considerations on Floyd-Hoare logic'', Proc. 17th IEEE Symp. Found. Comput. Sci., 109-121 (1976)]. As a basic premise one has that certain classical logical systems which are ''static'' in nature can be extended with the ability to discuss ''dynamical behaviour''. The connection with modal logic essentially is made by the observation that the subsequent execution stages of a program can be considered as a structured set of possible worlds. The chapter presents an outline of a two-semester graduate course on the topic. Three of the four main parts treat (i) propositional reasoning, (ii) first order uninterpreted, and (iii) first order interpreted reasoning. Part four surveys very shortly some related resp. competitive approaches as e.g. the algorithmic logic of the Warsaw group, the logic of effective definitions of \textit{J. Tiuryn} [e.g. Lect. Notes Comput. Sci. 125, 198-245 (1981; Zbl 0468.68037)], the \(\mu\)-calculus of Scott/deBakker [cf. \textit{J. de Bakker}, Mathematical theory of program correctness (1980; Zbl 0452.68011)], and process logic [cf. e.g. \textit{D. Harel, D. Kozen} and \textit{R. Parikh}, J. Comput. Syst. Sci. 25, 144-170 (1982; Zbl 0494.03016)]. Chapter 11 on ''Deontic logic'' by \textit{L. Åqvist} (pp. 605-714) comes back to the philosophical topics. In the first half it deals with certain frequently discussed difficulties in connection with the application of formal systems of deontic logic to a natural language such as English, and the other half is devoted to the presentation and elaboration of a number of formal systems of deontic logic, monadic as well as dyadic ones. As in other chapters, deeper results often can only be stated, for proofs the reader often is referred to \textit{L. Åqvist} [An Introduction to Deontic Logic and the Theory of Normative Systems, Naples 1985]. The final chapter 12 ''The logic of questions'' by \textit{D. Harrah} (pp. 715-764) discusses the perhaps most problematic topic of the whole volume, at least a topic with much more problems already in formalization than proposals for solution. Main emphasis is on approaches by \textit{N. D. Belnap jun.} and \textit{T. B. Steel jun.} [The logic of questions and answers (1976; Zbl 0345.02015)] and \textit{L. Åqvist} [A new approach to the logical theory of interrogatives, Uppsala 1965], but other approaches are mentioned, too. This second volume is as well made as the first one. Again, relatively few and almost only inessential misprints have been observed. The overall impression is a very good one. It can be expected that this volume (and the whole handbook indeed) will become a standard reference in its broad field.
scientific article; zbMATH DE number 3912344