By Norman M. Martin

This ebook examines an summary mathematical conception, putting detailed emphasis on effects appropriate to formal common sense. If a conception is mainly summary, it will possibly discover a ordinary domestic inside numerous of the extra customary branches of arithmetic. this is often the case with the speculation of closure areas. it'd be thought of a part of topology, lattice conception, common algebra or, without doubt, one of many different branches of arithmetic besides. In our improvement we've handled it, conceptually and methodologically, as a part of topology, partially simply because we first proposal ofthe easy constitution concerned (closure space), as a generalization of Frechet's notion V-space. V-spaces were utilized in a few advancements of common topology as a generalization of topological area. certainly, while within the early '50s, one in every of us begun wondering closure areas, we proposal ofit because the generalization of Frechet V house which comes from now not requiring the null set to be CLOSURE areas ANDLOGIC XlI closed(as it really is in V-spaces). This generalization has an severe virtue in reference to program to common sense, because the most vital closure concept in good judgment, deductive closure, more often than not doesn't generate a V-space, because the closure of the null set regularly involves the "logical truths" of the good judgment being examined.

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**Example text**

Is assumed only to be a closure space unless further conditions are explicitly noted. rrCl'(A) CLOSURE SPACES AND LOGIC 48 consists of all the members of S logically incompatible with A. If f is a minimal negation (Martin, p. 33), then xE""CI(A) only if f(x)ECl(A). ), then xE""Cl(A) if and only if f(x)ECl(A). ), then xE CI(A) if and only if f(x)E"" Cl(A). So, in a classical setting, ""Cl(A) consists of precisely the denials of members of Cl(A) (x and f (x) being denials of one another). If ACS, then A is complete if and only if SC(CI(A)U""CI(A».

If DCS, then D is connected if and only if there are no nonempty, separated sets A and B such that (AUB)=D. A set of sentences is connected just in case it is impossible to divide it into two "pieces" each of which is deductively inaccessible to the other. 35 If A and B are separated and D is connected, then DC(AUB) only if either DCA or DCB. Proof Suppose DC(AUB). Then ((DnA)U(DnB» = D. Suppose A and B are separated. 34, (DnA) and (DnB) are separated. Suppose D is connected. Then either (DnA) or (DnB) is empty and, hence, either (DnB)=D or (DnA)=D.

Since B is consistent, we conclude that (AU{xl) is not dense and, hence, that x~ -'Cl(A). But then xECI(A), since A is complete. 3 implies that A is equivalent to B. (right-left) Suppose each consistent set containing A is equivalent to A. And suppose (AU{xl) is consistent. Then, since (AU [xl) contains A, Cl(AU {x})= CI(A). 1, xECl(A). We conclude that xllCl(A) only if (AUIxl) is dense. That is, S\CI(A)C -,Cl(A). So SC(Cl(A)U -'Cl(A». That is, A is complete. Comment A set is complete just in case none of its consistent extensions yield any new consequences.