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【正文】 deals with the nature and relations of aggregates. Historically, as logic and axiomatic systems became more and more exact, there emerged, in response to a desire for greater lucidity, a tendency to pay greater attention to the syntactic features of the languages employed rather than to concentrate exclusively on intuitive meanings. In this way, logic, the axiomatic method (such as that employed in geometry), and semiotic (the general science of signs) converged toward metalogic. Truth definition of the given language 2 The formal system N admits of different interpretations, according to findings of G246。 as standing for ordinary addition and multiplication. Relative to this interpretation, itis possible to give a truth definition of the language of N. It is necessary first to distinguish between open and closed sentences. An open sentence, such as x = 1, is one that may be either true or false depending on the value of x, but a closed sentence, such as 0 = 1 and (?x) (x = 0) or “All x39。 for example, 0 = 0 istrue, 0 + 1 = 0 is false. This specification as it stands is not syntactic, but, with some care, it is possible to give an explicit and mechanical specification of those closed atomic sentences that are true in the intuitive sense. 2. A closed sentence ～A is true if and only if A is not true. 3. A closed sentence A ∨ B is true if and only if either A or B is true. 4. A closed sentence (?ν)A(ν) is true if and only if A(ν) is true for every value of ν—., if A(0), A(1), A(1 + 1), . . . are all true. The above definition of truth is not an explicit definition。del39。del numbers of the true sentences of N.
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