Closure Spaces and Logic by Norman M. Martin

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.

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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Extra resources for Closure Spaces and Logic

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.