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«As we shall see, the logical axioms are so designed that the logical consequences (in the semantic sense, cf. p. 56) of the closure of the axioms of $K$ are precisely the theorems of $K$.» Page 60 “Introduction to Mathematical Logic“ SECOND EDITION by ELLIOTT MENDELSON The same is in fourth edition.

What does it mean: “the closure of the axioms”?

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    When you say "the same is in the fourth edition", is it located elsewhere (not p.60) of the fourth edition, I searched on line, found 4th ed., but there is no statement there resembling the statement you're interested in. I suspect more is meant by "the logical consequence...of the closure of the axioms of K are precisely the theorems of K" than closure of a particular formula whose variables are bound by a universal quantifier...2011-05-20
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    See "universal closure" here http://en.wikipedia.org/wiki/Universal_quantification#universal_closure2011-05-30

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