¹ E.g., Carnap, R., Meaning and necessity (1947) p. 83 ff.Google Scholar; Church, A., On Carnap's analysis of statements of assertion and belief, Analysis 10 (1950) p. 97 ff.CrossRefGoogle Scholar, and Intensional isomorphism and identity of belief, Philosophical studies 5 (1954) p. 65 ff.CrossRefGoogle Scholar; Scheffler, I., On synonymy and indirect discourse, Philosophy of science 12 (1955) p. 39 ff.CrossRefGoogle Scholar; etc.

² Mathematical logic (1951) p. 2Google Scholar (cf. the same author's Truth by convention in Philosophical essays for A. N. Whitehead, 1936, p. 90 ff.)Google Scholar If what Quine says here is to afford, informally, a necessary and sufficient criterion of logical truth under all circumstances he needs to make certain further stipulations, such as that in most cases (which need to be further specified) the same word occurs vacuously more than once only if it can be replaced by a word having precisely the same sense and reference on each occurrence in the statement. But these stipulations are not specially relevant to the purpose of the present paper and are therefore not discussed here.

³ A formulation of the logic of sense and denotation in Structure, method, and meaning (1951) ed. Henle, P.et al. p. 3 ffGoogle Scholar.

⁴ In any case our axiom would have to be rather stronger than Russell's, and its justification correspondingly even more questionable (cf. Ramsey, F. P., The foundations of mathematics, 1931, p. 28 ff.Google Scholar), since, in order to formalise indirect discourse we should need (for familiar reasons) an order containing properties designated by functions that were not merely extensionally equivalent to all functions designating properties that belonged to other orders of the same type, but also synonymous with them. Moreover, it appears that even Russell's Axiom permits some of the semantical antinomies to reappear unless the names of some functions are omitted, (cf. Copi, I. M., The inconsistency or redundancy of Principia mathematica in Philosophy and phenomenological research 11, 1950, pp. 190–9Google Scholar and the review of this by A. Church in this Journal, vol. 16 1951, pp. 154–5). Yet such an omission would seriously impoverish any formalised language that sets out to be as rich as the everyday natural languages in which indirect discourse is ordinarily articulated.

⁵ Cf. Cohen, L. J. and Lloyd, A. C., Assertion statements, Analysis 15 (1955) p. 68CrossRefGoogle Scholar. The main argument developed in Assertion statements reinforces, though is less powerful than, the present paper's argument about the difficulty in principle of formalising indirect discourse. But in (D) of the present paper a way round this difficulty is discussed which was not mentioned in Assertion statements.

⁶ I allude in particular to certain problems about the interpretation of modal calculi which I hope to discuss elsewhere. The procedure discussed here is also indispensable for a third kind of proposal for formalising indirect discourse, unlike either Church's or Carnap's, which is sometimes found in the literature. In the analysans for, say, ‘x asserts that p’ the prefatory phrase ‘asserts that’ or ’x asserts that’ is sometimes not represented by a predicate at all — neither by an object-language function taking as one of its arguments the name of an intensional entity nor by a metalinguistic function taking as one of its arguments the name of a sentence in the object-language — but by a non-extensional operator forming statements out of statements. A paper by Łos on a system of this sort was reviewed in this Journal 14 (1949) pp. 64–5, and another such system is proposed by Prior, A. N. in Time and modality (1957) p. 130–1Google Scholar. I have been unable to consult Łos's paper and Prior's somewhat summary description of his system does not indicate explicitly how he intends to avoid the occurrence of semantical antinomies in interpreting it. But it is clear that no hierarchy principle controls the way in which Prior's operators form statements out of statements, since one category of these operators forms truth-functions of the statements they precede and even those that are prefaces to indirect discourse may precede any other such preface. Indeed, in a formalised language based on this system the analysis of (1) would be quite straightforward. The analysans is

where ‘ϒ’ and ‘δ’ are interpreted as the operators ‘the policeman testifies that’ and ‘the prisoner deposes that,’ respectively; and the formal deduction of the consequent from the antecedent of this conditional marches step by step with the informal deduction set out above. But a policy like that outlined in (D) must be pursued if ‘ΣΣϸΚϒϸΝϸ’ is not to generate a contradiction when ‘ϒ’ is interpreted as ‘I am now asserting that’ (with ‘now’ in a narrow sense).

⁷ When a set is referred to by an expression that contravenes a hierarchy principle like Russell's theory of types or Quine's stratification requirement, another mode of referring to the set may be sought that is unobjectionable on these grounds. But the referentially opaque quality of indirect discourse precludes in general any analogous method of avoiding difficulties in formalising indirect discourse according to a hierarchy principle. One may, however, observe a rough and distant analogy between the way in which (D) above seeks to avoid the semantical antinomies by specifying conditions under which a predicate letter in a syntactically well-formed formula is to be assigned a meaning and the way in which set theories in the Zermelo tradition seek to avoid the logical antinomies by setting up conditions under which a predicate is to be viewed as having an extension.

⁸ Cf. Kleene, S. C., Introduction to metamathematics (1952) p. 39Google Scholar.