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**Handbook of philosophical logic. Volume III: Alternatives to classical logic.**
*(English)*
Zbl 0603.03001

Snythese Library, Vol. 166. Dordrecht etc.: D. Reidel Publishing Company, a member of the Kluwer Academic Publishers Group. XI, 520 p. Dfl. 240.00; $ 98.00; £66.50 (1986).

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This Vol. III continues the presentation and discussion of systems of non-classical logics begun in Vol. II; only the subtitle is misleading: although some of the logics studied started as rivals of classical logic, now they are not considered as ”alternatives” but as ”supplements” which under suitable conditions are preferable to classical logic.

Chapter 1 (pp. 1-70) by S. Blamey is on ”Partial logic”. Here sentences do not always have to be either true or false and terms do not always have to denote. With emphasis on the definitions of logical connectives to allow for truth value gaps the author presents an apparatus for semantics to handle such gaps together with denotation failures in a uniform way. An essential point is monotonicity which allows truth value gaps to be filled in but not to change truth values. For motivations and applications reference is made, e.g., to presuppositions, semantical paradoxes, and situation semantics. The results are usually stated without proofs but reference is given to a book of the author (Partial logic, Naples: Bibliopolis), which is to appear.

Chapter 2 (pp. 71-116) on ”Many-valued logic” by A. Urquhart is quite short with a bit of pessimistic attitude toward its topic. Łukasiewicz’s propositional and first-order logics together with Post algebras and some applicational ideas are the central themes.

Chapter 3 (pp. 117-224) by J. M. Dunn on ”Relevance logic and entailment” does not intend to survey the whole field of relevance logic, instead it concentrates on three problems: (i) the admissibility of the Ackermann’s rule, i.e. of modus ponens for material implications, (ii) the decision problem, and (iii) the problem of a suitable semantics. All this is completed by a very instructive introduction into the field, done mainly with reference to the Anderson-Belnap system R, and written in a very readable style which makes this chapter a really valuable introduction into its topic.

Chapter 4 (pp. 225-339) by D. van Dalen entitled ”Intuitionistic logic” gives a detailed account of intuitionistic proof theory and semantics as well as some discussions of issues in intuitionistic mathematics like Heyting arithmetic, the theories of equality and appartness, and of second order intuitionistic logic including choice sequences and some intuitionistic analysis. Hence, besides pure logical matters a considerable amount of mathematics is considered in this paper. As a very welcome addition to it comes Chapter 5 (pp. 341-372) by W. Felscher on ”Dialogues as a foundation for intuitionistic logic”, which provides a sound presentation of the dialogical approach to logic and is (one of) the best introduction(s) into this topic.

E. Bencivenga’s Chapter 6 (pp. 373-426) on ”Free logics” comes back to the problem of non-denoting terms which also was present in Chapter 1. Amusingly, neither of these both chapters refers to the other one. Starting from the motivating ideas the author first presents the proof theoretical side, i.e. the calculi of free logic, later he surveys the essential semantical approaches up to supervaluations, and closes with theories of descriptions in free logic.

Chapter 7 (pp. 427-469) on ”Quantum logic” is written by M. L. Dalla Chiara. After a motivating introduction, semantics (algebraic as well as Kripke-style) comes first; later, implication operators, a model interpretation, and axiomatizations are discussed. Unfortunately, only very few remarks really explain the use of quantum logic for quantum theory.

The final Chapter 8 (pp. 471-506) ”Proof theory and meaning” by G. Sundholm shortly discusses the problem of how to logically define the meaning of logical connectives and, with this, the meaning of compound sentences. Main emphasis is on Dummett’s argument against a truth- conditional view of meaning and on Martin-Löf’s type theoretical approach.

Each of the chapters give a fair amount of space for explaining the main motivations behind the systems to be discussed and in most cases also information on applications. But, often, the latter ones are not so clear as the motivations. In every case, yet, the central themes are the formal systems - in their syntactical and semantical aspects - which constitute the considered fields of logic. The presentation usually makes for good reading. The number of printing errors is not high, and all of them can easily repaired by the careful reader. For the pure mathematician it is instructive to see that the old connections of logic to philosophical matters are alive on a quite firm level and that not only pure mathematics gives rise to nice and serious developments in formal logic.

Reviewer: S. D. Latow

### MSC:

03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |

03B45 | Modal logic (including the logic of norms) |

03B50 | Many-valued logic |

03F55 | Intuitionistic mathematics |

03G12 | Quantum logic |

03A05 | Philosophical and critical aspects of logic and foundations |

03B10 | Classical first-order logic |

03B60 | Other nonclassical logic |

03F99 | Proof theory and constructive mathematics |

03G20 | Logical aspects of Łukasiewicz and Post algebras |