By Raimo Tuomela (auth.)
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The theory T and the set aT of its observational consequences are functionally equivalent for deductive systematization with respect to the observational language Ie (A). The simple proof can be given concisely in two parts as follows: Tu {F} I- G T I- F:::> G 'F:::> G' belongs to OT {F:::> G, F} I- G (5) OTU {F} I- G A. (1) (2) (3) (4) B. ) It is important to notice that aT is not a recursive but only a recursively enumerable set of statements. ) The aim of a replacement programme (of the second kind) is to provide an effective (recursive) method for replacing the auxiliary expressions of a theory axiomatizing a science by nonauxiliary ones, so that the nonauxiliary content of the science is left unchanged.
We divide the critical remarks into three groups. The first contains more or less direct criticism of some logical features of Craig's construction, which taken as such is of course formally faultless. The second group contains various philosophical remarks directed against Craig's construction (though they are not necessarily relevant to how well Craig succeeded in his task as defined above). These remarks contain various additional criteria for an elimination programme to be philosophically interesting.
Think of (first-order) Peano arithmetic as an example of a theory with an infinite number of axioms! ) In fact, had Craig's general result (Thm. 2) rather been proved for systems containing only natural logical axioms and rules of inference (HINTIKKA (1966b)), the infinite axiom set would probably have been considerably more perspicuous. The use of distributive normal forms in the first-order case will in fact supply an infinite set of axioms that is a simple and, in a clear sense, perspicuous monotonically increasing chain.