By Haskell Brooks Curry, Robert Feys, William Craig
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Extra resources for Combinatory Logic, Volume I
Thus ‘function’ here corresponds to a ‘functive’ there; for ‘function’ there we shall use the term ‘closure of a function’, etc. This frees the term ‘functive’ for a more special purpose in 9 B1. 26 [ID FORMAL SYSTEMS in the U-language (which includes the presupposed technical terminology) before the formal system is introduced. Even the variables ‘d and ‘y’ are used in a sense known intuitively in U (therefore they may be called “U-variables”). But symbols such as ‘O’, ‘ ’ ’, ‘ = ’ are new; they are, it may be said, the elementary symbols of a certain new language-in the generalized sense mentioned above-which is embedded in the Ulanguage.
Thus in interpretations of systems containing variables (3 2C) contensive objects are ordinarily not associated with the variables, but only with the constants. Further, when such contensive objects exist, it is not always the case that distinct ones are associated with distinct obs. A valid interpretation is one such that every contensive statement which corresponds to an elementary theorem is true; in that case the system will also be said to be valid for the interpretation. A direct interpretation is one in which at the same time a contensive object is assigned to each ob (but not necessarily distinct objects to different obs).
For an intelligible account of the ideas of intuitionism see Wilder [IFM] Chapter X ; cf. also Heyting [MGL]. Hilbertism developed into the syntactical conception of formal methods, which is currently the most fashionable (see 3 2 ) . The notion of formal system expounded in this work is that developed in a series of papers by Curry. The principal papers in this series, in the order of composition (which is quite different from that of publication) are: [ALS], [OFP], [APM], [TFD] Chapter I, [LLA] Chapter I ; briefer summaries are given in [RDN], [PKR] 5 2, [SFL]; while special topics are dealt with in [MSL], [FRA], [LFS], [LMF], [LSF], [DSR], and [TEx].