¹ Work on this paper was supported by the Arts and Humanities Research Council. I am very grateful to Paul Boghossian, Paul Horwich and Steven Gross for helpful comments on an earlier version of this paper, and to Genoveva Marti for discussion when we ran a graduate seminar together on analyticity.

² The relevant publications by Quine are these. ‘Truth by Convention’, Philosophical Essays for Alfred North Whitehead, O. Lee (ed.) (New York: Longmans 1936), reprinted in Philosophy of Mathematics, 2^nd ed., P. Benacerraf & H. Putnam (eds.) (Cambridge: Cambridge University Press 1983); ‘Two Dogmas of Empiricism.’ Philosophical Review (1951), 20–43, reprinted in Quine, W., From a Logical Point of View (Cambridge, Mass: Harvard University Press 1980)Google Scholar; ‘Carnap and Logical Truth.’ Synthèse (1960), 350–74, reprinted in Quine, W., The Ways of Paradox and Other Essays (Cambridge, Mass: 1976)Google Scholar; Word and Object (Cambridge, Mass: MIT Press 1960).

³ Boghossian, Paul, ‘Analyticity Reconsidered.’ Nous (1996), 360–391CrossRefGoogle Scholar.

⁴ There are several bovine species, including ox, bison, buffalo as well as domestic cattle (bos taurus). It is not clear to me whether the French word “vache” is applied to females of bovine species other than domestic cattle, but I will suppose for the sake of argument that it is.

⁵ The Oxford English Dictionary (OED) has eight top-level entries for “cow”, six of them for nouns and two for verbs. I discuss only those corresponding to uses belonging to my idiolect, one noun and one verb. The entry for the noun has several subentries some with subentries of their own.

⁶ The OED mentions also rhinos and seals; Merriam-Webster Unabridged mentions also moose and alligators.

⁷ The following example may help to convey the difference between homonymy and polysemy. (i) An otter will sometimes take over the den of muskrats dug into a river bank. (ii) My bank does not charge interest on overdrafts of less that £100. (iii) Do not bank on Clinton's winning the nomination. The meanings of “bank” in (i) and (ii) are unrelated; so the occurrences of “bank” in (i) and (ii) are occurrences of distinct homonymous words. The meanings of “bank” in (ii) and (iii) are related; so the occurrences in (ii) and (iii) are occurrences of a single polysemous word.

⁸ For ease of exposition let us pretend that the usage of “vache” does not exhibit similar complexity. For the record, “vache” is also used for untreated cowhide (and there are slang and metaphorical uses, as for “cow”). This pretence makes no difference to the strength of the case for moderate indeterminacy.

⁹ This is not an isolated case of discrepancy between dictionaries. For other examples see Fillmore, C. and Atkins, B. ‘Describing Polysemy: The case of “Crawl”.’ in Ravin, and Leacock, (eds.) Polysemy: Theoretical and Computational Approaches. Oxford: Oxford University Press 2000Google Scholar.

¹⁰ Geeraerts, D., ‘Vagueness's puzzles, polysemy's vagaries’, Cognitive Linguistics 1993, 223–272CrossRefGoogle Scholar.

¹¹ Geeraerts, D., ‘The definitional practice of dictionaries and the Cognitive Semantic conception of polysemy’. Lexicographica 2001, 6–21Google Scholar.

¹² A. Cruse, ‘Aspects of the Micro-structure of Word Meanings.’ in Ravin and Leacock Op. Cit.

¹³ Linguists sometimes use the word “monosemy” for singularity of meaning, as in Polysemy or monosemy: interpretation of the imperative and the dative-infinitive construction in Russian by E. Fortuin.

¹⁴ These are intensional functions (algorithms), not extensional functions (sets of ordered pairs).

¹⁵ For ease of exposition I will oversimplify by writing as though (i) individuation of uses is not also afflicted by a degree of indeterminacy and (ii) the outputs are uses, rather than use-governing representations.

¹⁶ Cruse, Ibid.

¹⁷ Quine, , ‘Two Dogmas of Empiricism.’ Philosophical Review (1951), 20–43CrossRefGoogle Scholar, reprinted in Quine, W., From a Logical Point of View (Cambridge, Mass: Harvard University Press 1980)Google Scholar.

¹⁸ Putnam, H. 1962: “The Analytic and the Synthetic” reprinted in Putnam, H.Mind, Language and Reality: Philosophical Papers, volume 2. Cambridge: Cambridge University Press 1975CrossRefGoogle Scholar. Originally published in Feigl, H. and Maxwell, G. eds. Minnesota Studies in the Philosophy of Science, III. Minneapolis: University of Minnesota Press.

¹⁹ Quine, Word and Object. Ch. 2.

²⁰ This will include the belief that speakers of other languages are cognitively much like us, and so they do not parse biological entities as finely sliced temporal stages etc. See Quine, , ‘On the Reasons for Indeterminacy of Translation.’ Journal of Philosophy (1970), 178–183CrossRefGoogle Scholar; ‘Indeterminacy of Translation Again.' Journal of Philosophy (1987), 5–10.

²¹ I ignore the possibility that a person may make a statement without expressing a determinate thought.

²² Quine, ‘Two Dogmas of Empiricism.’

²³ “Even an outright equation or biconditional connection of the definiens and the definiendum is a definitional transcription of a prior logical truth of the form ‘x = x’ or ‘p ≡ p’.” Quine, ‘Carnap and Logical Truth.’

²⁴ Quine, Ibid.

²⁵ An author could explicitly revoke and replace the definition later in the text. In that case the definition is operative only up to its explicit revocation. The question of scope arising when an author uses a defined term in obvious contravention of the definition without its explicit revocation is trickier. But I would guess that such occurrences in mathematics texts are rare.

²⁶ Boghossian gives an account of basic logical knowledge substantially different from the linguistic one given in ‘Analyticity Reconsidered’ in a more recent paper, ‘Blind Reasoning’ Proceedings of the Aristotelian Society, supp. vol. 77, (2003), 225–48.

²⁷ This is taken verbatim from Boghossian, ‘Analyticity Reconsidered’ p. 376.

²⁸ This is adapted from Boghossian, ibid. p. 386. This adaptation retains a use of “means” that amounts to “refers to” or “has as its semantic value”.

²⁹ Li(x) = ∫₂^x 1/ln(u).du This is sometimes known as ‘the logarithmic integral’.

³⁰ @ = @; so ∃n [n = @]; so ∃n [n > 2 & ¬ [π(n) < Li(n)]].

³¹ It was proved by Littlewood in 1912. The least counter-example is almost certainly greater than 1.3 × 10³¹⁶. In 1914 Littlewood proved that there are infinitely many counter-examples.

³² H(‘H’) → ¬ ‘H'T‘H' [by Hi] → ¬ H(‘H’) [by Ti]. ¬ H(‘H’) → ¬ ‘H'T‘H' [by Tii] → H(‘H’) [by Hii].

³³ So do Prior's rules for ‘tonk’ in ‘The Runabout Inference Ticket’ Analysis (1960), 38–9. But one cannot generally follow those rules, whereas the rules for ‘true of’ and ‘heterological’ fail only at isolated points.

³⁴ Horwich, , ‘Stipulation, Meaning, and Apriority’ in Boghossian, P. and Peacocke, C. (eds.) New Essays on the A Priori. (Oxford: Oxford University Press 2000)Google Scholar. See also Ch. 6 of Horwich, , Reflections on Meaning. (Oxford: Oxford University Press 2005)CrossRefGoogle Scholar.

³⁵ Perhaps we can find for each basic inference form a suitable argument for it that uses only other basic inference forms. But as the totality of basic inference forms is finite, there would still be an epistemic circularity even if no individual argument were circular.

³⁶ ‘Analyticity Reconsidered’ p. 374, 386–7. Boghossian is not quite so explicit, but less than this would not meet the objection.

³⁷ Steven Gross brought this possible reply to my attention, without expressing confidence in it.

³⁸ Something akin to this is proposed by Horwich in Meaning. (Oxford: Oxford University Press 1998), ch.3 and in Reflections on Meaning, ch.2.