By Haskell B. Curry (Eds.)
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Additional info for Outlines of a Formalist Philosophy of Mathematics
Such a translation will change nothing essential. cts. Moreover, since the truth of the propositions will be determined without regard to the values given to the variables, it will be permissible to regard these variables as apparent, and the propositions will still be propositions. In this generalized sense the doctrinal function can be regarded as one of the representations of a formal system. The upshot of this is that although a formal system may be represented in various ways, yet the theorems derived according to the specifications of the primitive frame remain true without regard to changes in representation.
Such an interpretation is to be distinguished from a representation (as discussed in Chapter VI): in a representation the predicates are defined by the primitive frame; in an interpretation we associate them with notions pertaining to the subject-matter, so that the question arises as to the agreement between the truth of the propositions of the formal system and that of the associated ones relating to the subject-matter. This agreement is, then, the primary consideration in regard to acceptability.
We then have the O-expressions formalized as expression terms. The formation rules can then be taken over bodily into the system, replacing each category, etc. of expressions by the corresponding category, etc. of expression-terms. Corresponding to the 0sentences we have the formulas among the expression-terms. Next take the rules of consequence. Let the axiom-formulas correspond to those O-sentences which are direct consequences of the null class, and let the other rules of direct consequence be set up as rules of procedure.