UPDATE (Nov 2019): I have recently published a paper on this topic, 'Quine's Poor Tom', in the European Journal of Analytic Philosophy.
I will not engage directly with these three papers, but rather aim to give a clearer objection to Quine's argument.
Here is Sayward's apt description of the argument's point:
To a first approximation, the argument purports to show that if Tom has a certain minimal level of logical acuity—a level many of us possess—then if ‘belief’ has a sense in which it is a transparent operator, then Tom, if he in that sense of the word believes anything, he in that sense of the word believes everything. (Sayward 2007, p. 54.)Quine assumes that Tom believes at least one true sentence and one false one. In fact, he assumes something much stronger: that Tom believes the true sentence 'Cicero denounced Catiline' and the false sentence 'Tully did not denounce Catiline'. (Cicero is Tully.) That these sentences are are (in a sense) contradictories, and that they are about the same object, is not essential for Quine's argument. These features of Tom were needed for earlier, separate arguments in chapter IV of Word and Object.
Here is the argument:
Where ‘p’ represents a sentence, let us write ‘#p’ (following Kronecker) as short for the description:
the number x such that ((x = 1) and p) or ((x = 0) and not p).
[In place of '#', Kronecker and Quine used a different symbol, which I can't easily reproduce here. - TH.]
We may suppose that poor Tom, whatever his limitations regarding Latin literature and local philanthropies, is enough of a logician to believe a sentence of the form ‘#p = 1’ when and only when he believes the sentence represented by ‘p’. But then we can argue from the transparency of belief that he believes everything. For, by the hypothesis already before us,
(3) Tom believes that # (Cicero denounced Catiline) = 1.
But, whenever ‘p’ represents a true sentence,
# p = #(Cicero denounced Catiline).
But then, by (3) and the transparency of belief,
Tom believes that #p = 1,
from which it follows, by the hypothesis about Tom’s logical acumen, that
(4) Tom believes that p.
But ‘p’ represented any true sentence. Repeating the argument using the falsehood ‘Tully did not denounce Catiline’ instead of the truth ‘Cicero denounced Catiline’, we establish (4) also where ‘p’ represents any falsehood. Tom ends up believing everything. (Quine 1960, pp. 148–149).First, to rehearse Quine's definition of referential transparency. (Familiar readers can skip this paragraph.) Quine defines transparency in terms of 'modes of containment ... of singular terms or sentences in singular terms or sentences'. Definite descriptions count here as singular terms. For Quine, a mode of containment M is referentially transparent iff, 'whenever an occurrence of a singular term t is purely referential in a term or sentence C(t), it is purely referential also in the containing term or sentence M(C(t)). ' (p. 144, schematic letters changed). For a singular term t to be purely referential in a term or sentence is for it to occupy a purely referential position there. Quine's 'criterion' for a position's being purely referential is that the position 'must be subject to the substitutivity of identity' (p. 142). That is, to the substitutivity of co-extensive singular terms salva veritate.
Let us begin by simply granting (3) for the sake of argument, ignoring its justification - Quine's 'by the hypothesis already before us'. (After we have identified a later fatal flaw in the argument, we will return to (3)'s justification briefly, since it seems to suffer from essentially the same flaw.)
Now, note that Quine's 'hypothesis about Tom's logical acumen' (hereafter 'the acumen hypothesis') and the steps of his argument are at different semantic levels. The hypothesis is framed in terms of belief in sentences, while in the argument, sentences appear unquoted as the contents of 'that'-clauses. Thus, the acumen hypothesis does not apply directly to 'Tom believes that #p = 1', since that sentence says nothing about Tom's belief in any sentence. Quine is, apparently, suppressing a quotational and a disquotational step here. An expanded version of this part of the argument, in which the acumen hypothesis could be applied directly, would have to run something like:
(i) Tom believes that #p = 1.
(ii) Hence Tom believes the sentence '#p = 1'. (Quotation step.)
(iii) Hence Tom believes the sentence 'p'. (Acumen hypothesis together with (ii).)
(4) Tom believes that p. (Disquotation step.)
Secondly, note that 'believes' in 'Tom believes that #p = 1' is to be taken in a transparent sense, as the piece of reasoning preceding it makes clear. (In case of any residual doubt about this: in the very next sentence after the argument as quoted, Quine summarizes it by saying 'Thus in declaring belief invariably transparent ... we would let in too much.')
Putting these things together, we can see the invalidity of Quine's argument: when (i) is taken in a transparent sense, it does not imply (ii).
To see this, consider that Delia Graff Fara believes (in the transparent sense) that Quine wrote Word and Object. We hereby introduce a new name for Quine, 'G6'. Now, since G6 is Quine - since 'G6' and 'Quine' are co-extensive - we may infer that Delia Graff Fara believes (in the transparent sense) that G6 wrote Word and Object. Plainly, we cannot infer from this that Professor Fara, who knows nothing of my convention (at the time of writing), believes the sentence 'G6 wrote Word and Object'.
The problem with Quine's argument as it stands, then, is in the first instance a use-mention confusion. (None of the papers cited makes anything of this point.) We have now seen that the problem cannot be fixed by expanding the argument to contain a quotational and a disquotational step; the quotational step is invalid. Can it be fixed by rephrasing the acumen hypothesis as a schema containing placeholders for unquoted sentences?
It cannot. Such a schema would run: 'John believes that #p = 1 when and only when he believes that p'. The dilemma here is that, if 'believes' is taken transparently, the schema is not a defensible principle of rationality (even for logicians), and if it is taken opaquely, the principle doesn't apply to Quine's argument.
Finally, to return to the first step of the argument, namely (3)'s justification. The 'hypothesis' Quine cites here is, as far as I can tell, the acumen hypothesis. And so this step is just as invalid as Quine's inference to (4). For the case where 'p' is true, however, (3) will be true anyway, so long as Tom believes that 1 = 1.
Tristan Haze
The University of Sydney
References
- Quine, W. V. (1960). Word and Object. The MIT Press.
- Charles Sayward (2007). Quine and his Critics on Truth-Functionality and Extensionality. Logic and Logical Philosophy 16:45-63.
- R. C. Sleigh (1966). A note on an argument of Quine's. Philosophical Studies 17 (6):91 - 93.
- David Widerker (1977). Epistemic opacity again. Philosophical Studies 32 (4):355 - 358.
