1. We also suppose that Carnap's sign ‘≡’ of identity of individuals is a predicator constant, and that when A and B are individual expressions, A ≡ B is to be understood merely as an abbreviation or alternate way of writing ≡ AB. This modification of S1 serves to simplify the discussion but is not otherwise essential to the conclusions we reach.
2. I follow Carnap's terminology, in spite of my own preference for a somewhat different terminology—e.g., ‘well-formed formula’ instead of ‘designator matrix.’
3. The definition of ‘L-true’ need not be repeated here. But notice should be taken of two necessary corrections to the definition as it is developed in §§ 1–2 of Carnap's book. In 2-2 the correction of Kemeny must be adopted (Journal of Symbolic Logic, 16:206 (1951)); i.e., in place of “every state-description” the restriction must be made to non-contradictory state-descriptions. Otherwise consequences will follow that are certainly not intended by Carnap, for instance that no two different atomic sentential matrices (and no two different predicator constants) can be L-equivalent. In the rules of designation 1-1 and 1–2, the way in which the English language and certain phrases of the English language are mentioned, rather than used, is inadmissible—as may be seen by the fact that it forces the tacit use, in 1–3 and 1–4, of certain rules of designation of theEnglish language, which, if stated, would have a quite different form from 1-1 and 1–2. For example, Carnap's rule of designation, “‘s’ is a symbolic translation [i.e., from English] of ‘Walter Scott’,” should be changed to a rule which mentions the man Walter Scott rather than the words ‘Walter Scott’; perhaps it should be simply “‘s’ refers to Walter Scott,” in order to justify the inference from 1–3 to 1–4. These corrections are not directly relevant to the present paper, but our discussion presupposes that suitable corrections have been made.
4. Carnap uses ‘≡’ not only between sentential matrices as a sign of material equivalence, but also between other designator matrices as a sign of identity (in place of the usual ‘≡’).
5. Because of the restriction to the single language S1 and to designator matrices containing the same free variables, we have been able to give a simplified form to Carnap's definitions of ‘L-equivalent’ and ‘intensionally isomorphic.’