Abstract
There is a fairly extensive philosophical and methodological literature dealing with the role of auxiliary (‘theoretical’) terms in scientific theories. In this literature, logical and foundational ideas play a surprisingly small role, despite the wealth of results concerning definability and related concepts which logicians have established. Virtually the only non-trivial result cited is Craig’s (general) elimination theorem, and the purpose in bringing it up is all too often to deny its relevance. The possible importance of Craig’s less general, but in certain respects more informative, interpolation theorem has not caught the fancy of philosophers of science, whose store of systematic logical results and techniques seems to be often rather restricted.
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References
When identities are present, an exclusive interpretation of quantifiers must be assumed in the normal forms. Then predicates can be omitted as before in the direct reduction. However, it is obvious that a modification is needed when an individual constant aµ is omitted as a part of the reduction. For when no longer explicitly mentioned, it becomes a possible substitution value of the quantifiers in a way it was not one before.
It is clear enough what we must do. Let us assume that we want to omit a from a constituent C0 (d). Then, over and above omitting all atomic sentences containing a, we must change each part of Co (d) of the form \( (Ex)C{t_1}^{\left( c \right)}(x)(Ex)C{t_2}^{\left( c \right)}(x) \cdots (Ux)\left[ {C{t_1}^{\left( c \right)}\left( x \right) \vee \cdots } \right], \) by adding to the conjunction a new (Ex) Cf 0 (x) and to the disjunction a new member Ct 0 (x). Intuitively speaking, Ct 0 (c) (x) says here of x (in relation to the individuals mentioned in it) the same thing as Co(d) said of a. It is not hard to see how Ct 0 (c)(x) can be syntactically constructed from C 0 (d)although the details of the recipe are somewhat complicated.
The latter part of this result amounts to the theorem on permutations in models which is proved in Svenonius (1959).
This meaning of ‘expansion’ is of course completely different from that of an expansion of a normal form into a deeper one.
We are indebted to Mr. David Miller for valuable suggestions concerning expositional and stylistic matters.
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© 1970 D. Reidel Publishing Company, Dordrecht-Holland
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Hintikka, J., Tuomela, R. (1970). Towards a General Theory of Auxiliary Concepts and Definability in First-Order Theories. In: Hintikka, J., Suppes, P. (eds) Information and Inference. Synthese Library, vol 28. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-3296-4_9
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