¹ C. G. Hempel “A logical appraisal of operationism”, The Scientific Monthly 79, Oct. 1954 p. 218. Hempel's notion of an “interpretative system” is much the same as, but a little wider than, Carnap's notion of “Meaning Postulates”—R. Carnap “Meaning Postulates”, Philos. Studies 3, Oct. 1952, pp. 65–73.

² F. P. Ramsey “Theories” in “Foundations of mathematics” (Edited by R. B. Braithwaite) London (Routledge), 1931, p. 230.

³ R. B. Braithwaite “Scientific Explanation” Cambridge (U.P.), 1953, pp. 76 ff.

⁴ N. Campbell “Physics: the elements” Cambridge (U.P.) 1920, p. 122.

⁵ R. Carnap “Testability and Meaning” Philos. Sci., S (1936) and 4 (1937), reprinted by the Graduate Philosophy Club, Yale University, with additions and corrections, 1950. Compare also Carnap's “Meaning Postulates” Philos. Studies, 3, (1952). Particularly pp. 70–71.

⁶ I wish to thank Prof. Braithwaite, first for his book which was actually the source which started my own thought on the subject, and secondly for his generous and helpful discussion with me.

⁷ The following argument, or an analogue, holds for a very wide range of systems. For convenience I will frame my exposition in terms of “Principia Mathematica”.

⁸ The conjunctive, disjunctive, and prenex forms are discussed in Hilbert and Ackermann 3, 6 and 8, and in most standard texts on symbolic logic.

⁹ The atomicity of these contexts is not fundamental, but relative to the matter in hand. Some predicates may “ultimately” contain a modal reference, and thus not fall within condition A (“modality” being a second-level predicate); but if this is not relevant to a particular philosophical discussion, then for the purposes of that discussion condition A is satisfied, and the interpretative system may therefore be brought to the form being described.

¹⁰ It is not necessary that each “new term” appear only once: any may appear more than once in a given sentence with different (variable or constant) arguments, or sets of arguments if the term takes more than one argument. This situation characteristically arises when a recursive definition is re-framed in terms of a set of generalised reduction sentences—see sect. 4 below. Thus the representation given, with arguments omitted, could mislead. Carnap was not considering these possibilities.

¹¹ A fuller account of the notion of a “connective” will be found in Carnap's “Formalisation of Logic”, Cambridge, Mass., (Harvard U.P.), 1947, p. 8.

¹² ∼T ₁.T ₂ and T ₂, implies ∼T ₁; and ∼.T ₁,T ₂ and T ₁ implies ∼T ₂—but this is implied by O ₁ &∼O ₁, and therefore is useless.

¹³ Primitive recursion and certain other recursive schemes may be re-expressed by generalized reduction sentences—see sections 4 and 5 below.

¹⁴ It will be remembered that if an interpretative sentence with lone T or ∼T is implied by the set but not contained explicitly in it, then the process of making the set “transparent” will lead to this sentence being added to the set: thus the interpretative sentence for ∼ T ₁ was added in the example at the end of section 2. See also next footnote. In addition to the case stated in the text, there is the degenerate case where only one of T and ∼T appears alone in a consequent, but with antecedent universal.

¹⁵ The modal status of this sentence, and others involved in reduction sentences, is a problem of some interest and difficulty which has never so far as I am aware been adequately discussed, and it can not be discussed in this article. It must be noticed, however, that if one requires necessary truth for the closure of this disjunction then it would be appropriate to say that the condition (i) was both necessary and sufficient within the general conditions stated above. But if mere factual truth is accepted, then one must reexamine the process whereby one re-frames the interpretative system so that it shall be “transparent”. Consider the system of (n+1) sentences, n of the form ‘A_i⊃α_iT_i’ together with ‘B⊃: ∼α_iT₁…α_nT_n’. Represent the variables as follows: those that appear as argument to “T_i’ only, represent by ‘t_i'; and those that appear as argument to ‘T_i’ and one or more other T-terms, represent by ‘u_i’, for all i (this representation of variables depends on the T'_i considered). Then, under the hypothesis that (u_i)(∃t_i).A_i for all i, the interpretative system is equivalent to a set of 2n sentences with ‘T_i’ and ‘∼T_i’ appearing alone. Now, if each of (u_i)(∃t_i).A_i is a necessary truth for all i one might well feel it appropriate to use them in the “reduction” of the interpretative system to “transparency”, that is to say, one would feel that the set obtained was a mere re-expression of the original. But even if they are mere contingent accidents, it would remain true that the equivalence definitions obtained by their use would be de facto adequate, and the adequacy of these “definitions” would be precisely the same as that of the “definitions” obtained on the basis of the mere de facto (and not necessary) satisfaction of condition (i).

¹⁷ Op. Cit. Footnote 3 above: see Chap. III—the example of the 3 and 4 factor theories. 18 Op. Cit. Footnote 1 above: see p. 220 note 16.

¹⁹ The first two of the following groups are together equivalent to Braithwaite's group of “idle formulae” on p. 61.

²⁰ In the limiting case of the so-called “uncertainty relations”, as they occur e.g. in the linear oscillator, the greater than or equal to sign is replaced by equality.