A System of Logic Ratiocinative and Inductive, Part II (The - download pdf or read online

By John Stuart Mill

ISBN-10: 0710075065

ISBN-13: 9780710075062

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Additional info for A System of Logic Ratiocinative and Inductive, Part II (The Collected Works of John Stuart Mill - Volume 08)

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1) Proof Assume that J is logically compact. Then, trivially, v consistent implies that every finite fuzzy subset Vj such that Vj« v is consistent. In order to prove the converse implication, assume that every finite fuzzy subset Vj such that Vj« v is Vt«v, Vj consistent. Then, since v is the inductive limit of the class {VjE finite}, v is consistent. 1). 1) and let H be an inductive class of consistent fuzzy subsets. We have to prove that v = U H is consistent. Now, for every finite fuzzy set Vj such that Vj« v an element S E H exists such that Vj~ s.

A is a contradiction if m( a) = 0 for every m E M. 6. COMPACTNESS AND CONTINUITY In this section we will examine continuous abstract fuzzy logics and therefore we will examine the notion of continuity for fuzzy operators. As observed in Chapter 1, given a set S, the notion of finite subset enables us to express the continuity of the operators in the lattice peS). , for any X E P(S) J(X) = U {J(Xf ): JV. is a finite part of X}. 1) Also, this is equivalent to saying that for every X <;:; S and XES, X E J(X) ¢:> a finite subset JV exists such that x E J(Xf ).

Namely, as m is a model of v provided that m ;;2 v, the information carried on by v is that, given any formula a, "the actual truth value of a is at least v( a)". From this point of view, an initial valuation v is not a fuzzy subset since the value v( a) is not a truth degree but a constraint on the possible truth degree of a (see Chapter 5). This is in accordance with classical logic where the available information is expressed by a set T of formulas (the assumptions) arising from a partial knowledge of a world m and the information carried on by T is that "at least the formulas in Tare true in m".

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A System of Logic Ratiocinative and Inductive, Part II (The Collected Works of John Stuart Mill - Volume 08) by John Stuart Mill

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