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Closure spaces and logic. (English) Zbl 0855.54001

Mathematics and its Applications (Dordrecht). 369. Dordrecht: Kluwer Academic Publishers. xvi, 230 p. Dfl. 190.00; $ 124.00; £84.00 (1996).
A closure space is a set \(S\) with a set \({\mathcal C}\ell\) of subsets of \(S\) such that the intersection of any subset of \({\mathcal C} \ell\) is an element of \({\mathcal C} \ell\). Although closure operations originated in topology, they turned out to be quite useful in algebra, logic, and other fields. This book presents a general theory of closure spaces and closure operations with special emphasis on results applicable to formal logic. Given a set of logical formulas, one can consider as its closure all formulas that follow from this set (and the rules of inference). Thus we arrive at a closure space. The authors mention that A. Tarski considered this approach in 1930, and they consider the book as a continuation of Tarski’s project. Numerous works of Polish logicians in the same direction are not mentioned.
The book has nine chapters, their titles give some flavor of its contents: (1) Logic and topology; (2) Basic topological properties; (3) Some theorems of Tarski; (4) Continuous functions; (5) Homeomorphisms; (6) Closed bases and closure semantics I; (7) Theory of complete lattices; (8) Closed bases and closure semantics II; (9) Truth functions.
The bibliography is extremely short, it mentions thirteen books, four papers, and two master theses. Scores (if not hundreds) of papers on closure spaces and closure operators are never mentioned. Some arguments (e.g., on truth functions) have a definite philosophical character. The authors claim that the abstract theory of derivability “and consequence is fundamentally a branch of applied topology... [T]his way of thinking has the advantage of freeing logic from the linguistic straightjacket in which it has been trapped since Frege and Russell. The concepts which are used in the development of logic need not be overtly linguistic nor do well-formed formulae need to have the syntax they are usually assumed to have. It is even the case that for many logical purposes, it is not necessary that, for instance, every pair of sentences have a disjunction.”
This is a charming little book that is well-written and reads with real pleasure. When I say “little”, I mean that for, although almost 250 pp. long, the book has no more than 28-29 lines per page and 60 or less characters per line. The font is big enough, which makes reading even easier. Of course, all this comes at a price (to be more exact, at 50 c a page). There are numerous exercises that make the book useful as a textbook.

MSC:

54A05 Topological spaces and generalizations (closure spaces, etc.)
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
03B99 General logic
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