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Notes on Logic: Difference between revisions
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It is easy to suppose that "individual", "particular", "complex", etc., are primitive ideas of logic. Russell, e.g., says "individual" and "matrix" are "primitive ideas". This error is presumably to be explained by the fact that, by employment of variables instead of the generality sign, it comes to seem as if logic dealt with things which have been deprived of all properties except complexity. We forget that the indefinables of symbols (''Urbilder von Zeichen'') only occur under the generality sign, never outside it. | It is easy to suppose that "individual", "particular", "complex", etc., are primitive ideas of logic. Russell, e.g., says "individual" and "matrix" are "primitive ideas". This error is presumably to be explained by the fact that, by employment of variables instead of the generality sign, it comes to seem as if logic dealt with things which have been deprived of all properties except complexity. We forget that the indefinables of symbols (''Urbilder von Zeichen'') only occur under the generality sign, never outside it. | ||
Every proposition which says something indefinable about a thing is a subject-predicate proposition; every proposition which says | Every proposition which says something indefinable about a thing is a subject-predicate proposition; every proposition which says something indefinable about two things expresses a dual relation between these things, and so on. Thus every proposition which contains only one name and one indefinable form is a subject-predicate proposition, etc. An indefinable symbol can only be a name, and therefore we can know, by the symbol of an atomic proposition, whether it is a subject-predicate proposition. | ||
A proposition cannot occur in itself. This is the fundamental truth of the theory of types. [''Cf.'' 3.332.] In a proposition convert all indefinables into variables, there then remains a class of propositions which does not include all propositions, but does include an entire type. If we change a constituent a of a proposition φ(a) into a variable, then there is a class <math>\hat{p}[( \exists x ) . \phi x = p]</math>. This class, in general, still depends upon what, by an ''arbitrary convention'', we mean by "φx". But if we change into variables all those symbols whose significance was arbitrarily determined, there is still such a class. But this is not now dependent upon any convention, but only upon the nature of the symbol "φx". It corresponds to a logical type. [''Cf.'' 3.315.] | A proposition cannot occur in itself. This is the fundamental truth of the theory of types. [''Cf.'' 3.332.] In a proposition convert all indefinables into variables, there then remains a class of propositions which does not include all propositions, but does include an entire type. If we change a constituent a of a proposition φ(a) into a variable, then there is a class <math>\hat{p}[( \exists x ) . \phi x = p]</math>. This class, in general, still depends upon what, by an ''arbitrary convention'', we mean by "φx". But if we change into variables all those symbols whose significance was arbitrarily determined, there is still such a class. But this is not now dependent upon any convention, but only upon the nature of the symbol "φx". It corresponds to a logical type. [''Cf.'' 3.315.] | ||