¹ W. V. Quine, The problem of interpreting modal logic, Journal of symbolic logic, vol. 12 (1947), pp. 43–48. See also ibid., pp. 95–96, where Quine reviews two papers by Ruth C. Barcan. These are the second and third of the three papers by Miss Barcan referred to in footnote 6.

² R. Carnap, Meaning and necessity.

³ Though Quine perhaps ultimately rejects this notion of “individual concepts,” he nevertheless seems to feel that modal logic is forced to employ it to overcome difficulties connected with quantification. Miss Barcan, in her third paper (see footnote 6), defines two kinds of identity. The first or “weak” kind, called I_m, is Leibnitzian identity using material implication. The second or “strong” kind, called I, is Leibnitzian identity using strict implication. In his review of Miss Barcan's third paper, Quine seems to regard the “weak” identity as analogous to his own “congruence.” He says: “It should be noted that only the strong identity is … interpretable as identity in the ordinary sense of the word. The system is accordingly best understood by reconstruing the so-called individuals as ‘individual concepts.‘” Quine apparently overlooks the important fact that the two kinds of identity are materially equivalent in Miss Barcan's system S2², and even strictly equivalent in her system S4², as is established in her theorems 2.31 and 2.33*. (The quadruple bar in 2.31 should be a triple bar. A quadruple bar should be inserted between the back-to-back parentheses in the theorem 2.32*). In the same review Quine also says, “As is to be expected, only the strong kind of identity is subject to a law of substitutivity valid for all modal contexts.” But this is not true for S4², and the fact that it is apparently true for S2² can be taken as an indication of an inadequacy in S2². This defect in S2² can be overcome by adding a rule of procedure to S2² according to which □A is an axiom of S2² whenever A is. It would then follow that the two kinds of identity would be in all respects equivalent even in S2².

⁴ See the Journal of symbolic logic, vol. 12 (1947), pp. 139–141, where A. F. Smullyan reviews Quine's paper. The problem of interpreting modal logic, referred to above. See also by Smullyan, Modality and description, ibid., vol. 13 (1948), pp. 31–37. Both of these papers by Smullyan are largely in agreement with the position of the present writer. It is perhaps worth remarking that on p. 36 of Modality and description it is possible to retain S3 in preference to S1 provided that the necessity operator in S2 is applied to the formal equivalence preceding the implication symbol rather than to the whole expression. This would amount to assuming that if it is necessary that α and β have the same members, then α and β are the same class. This assumption is the same as the form of the axiom of extensionality mentioned near the end of the present paper.

⁵ In his review of Quine's paper. Actually Smullyan retains Quine's notion of congruence in the portion of the review relevant to (5), (6) and (7). He says, “On the other hand, if, more naturally we view ‘Evening Star’ and ‘Morning Star’ as abbreviations of descriptive phrases [rather than as proper names of individuals], we find that A [that is, ‘Morning Star is congruent Evening Star ·□ (Morning Star is congruent with Morning Star)‘] expresses an evidently impossible proposition. For if it is not necessary that the morning star exists then it is not necessary that the morning star is self-congruent. And if a proposition is not necessary, then necessarily it is not necessary.“

⁶ See the correction to 2.32* given in footnote 3 above. Miss Barcan's three papers on quantified modal logic are as follows: A functional calculus of first order based on strict implication, Journal of symbolic logic, vol. 11 (1946), pp. 1–16; The deduction theorem in a functional calculus of first order based on strict implication, ibid., pp. 115–118; The identity of individuals in a strict functional calculus of second order, ibid., vol. 12 (1947), pp. 12–15. The writer wishes to acknowledge his debt to Miss Barcan for some helpful discussions concerning the ideas of this paper. In particular she pointed out the importance of the equivalence of the two kinds of identity defined in her third paper.

⁷ Quine has suggested in correspondence and conversation with the writer that such a demand might be desirable.