Saturday, 17 December 2011

Against Quine's Argument in Sect. 31 of Word and Object

ADDED: Here is a followup post from 26 December 2015.

Section 31 of Quine's Word and Object contains an arresting fallacious argument. In 1966, R.C. Sleigh Jr. published an objection to it. In 1977, David Widerker published an objection to Sleigh's objection. More recently, in 2007, Charles Sayward has published a paper where Sleigh's objection is further criticized. (References below.)

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


- 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. 


  1. The following might be useful, might be rambling garbage:

    It is true that the transparent belief that p does not entail the (opaque) belief of every sentence expressing p.

    This even holds under Quine's behaviouristic view of beliefs (or at least belief-acription). Grant a simplified version of this view where Tom believes the sentence 'p' iff Tom assents to 'p'. Suppose Tom does this. Hold Tom's theory T fixed, and let 'q' be interchangeable with 'p' in T without changing the verification conditions of T. Does Tom believe that q? Well, if we make an opaque belief-ascription to Tom, then the answer is 'no'--that 'p' and 'q' are interchangeable in T does not mean that 'q' is in T. If we make a transparent belief-ascription to Tom, then the answer is 'yes'--it is a fact that 'p' and 'q' are indeed interchangeable in T. So transparent belief that q does not imply opaque belief of 'q'.

    But keep Quine's treatment of beliefs in mind and run through the steps of the argument again.

    Let 'Dcc' = 'Cicero denounced Catiline'. Firstly, Tom transparently believes that #(Dcc) = 1. What does this mean? It is consistent with Tom having only the opaque belief that 1 = 1, since Dcc is in fact true, meaning that '#(Dcc)' is interchangeable with '1'. In other words, it is consistent with Tom assenting to the sentence '1 = 1' but NOT the sentence '#(Dcc) = 1'. But then the 'acumen' hypothesis is just too silly. If Tom assents to '1 = 1', why would we predict that he would also assent to '#(Dcc) = 1', let alone 'Dcc'?

    Dusting off the principle of charity, it seems that the only way to make the 'acumen hypothesis' less silly is by taking it to stipulate that(i) Tom has the OPAQUE belief of 'if p then #p = 1 & if ~p then #p = 0' (if this is a transparent belief, then Tom may only opaquely believe the sentence 'if p then 1 = 1 & if ~p then 0 = 0', which is not very informative!), and also that(ii)Tom has the OPAQUE belief of '#(Dcc) = 1'.

    This last stipulation, (ii), is central to your counter-argument as I see it. I.e., a transparent belief that #(Dcc) = 1 does not imply the opaque belief of '#(Dcc) = 1'. As I have said, it is consistent with Tom only having the opaque belief that '1 = 1'. So if there is reason to believe that Quine's 'acumen hypothesis' involves stipulating (ii), then his argument is not invalid (even though it might not be very interesting).

    But consider that for Quine, when we ascribe a belief of any kind to Tom, we do so on the basis of his behaviour--in my simplified version, his assent to sentences. So if we are to say that Tom TRANSPARENTLY believes that #(Dcc) = 1, then we do so by virtue of (inter alia) his assenting to some sentence or other (which could just be '1 = 1'), and then deciding on interpreting this behaviour in line with transparent belief (similar, it should be said, to how some people use 'de re' beliefs--this perhaps not being as appropriate for Quine). We could just as easily have decided to interpret it in line with opaque belief.

    But if we grant stipulation (i), why would we then have any reason to think that Tom would believe 'Dcc' just because (i) holds (he opaquely believes the sentence expressing the conditions of #p), together with him opaquely believing '1 =1'. We don't have any such reason. It seems to me that in order to avoid an absurdity that must have been just as obvious to Quine, we must assume that the sentence Tom assents to (and thus opaquely believes), such that we subsequently interpret that assent behaviour in line with transparent belief-ascription, is the sentence '#(Dcc) = 1'. If this seems reasonable (or charitable), then the argument does not stumble into invalidity on the way to a supressed quotational step.

  2. Adam,

    Thanks for commenting.

    You write: 'Dusting off the principle of charity, it seems that the only way to make the 'acumen hypothesis' less silly is by taking it to stipulate that(i) Tom has the OPAQUE belief of 'if p then #p = 1 & if ~p then #p = 0' (if this is a transparent belief, then Tom may only opaquely believe the sentence 'if p then 1 = 1 & if ~p then 0 = 0', which is not very informative!), and also that(ii)Tom has the OPAQUE belief of '#(Dcc) = 1'.'

    Firstly, I think this goes beyond what one could call a charitable reading of what Quine meant - it seems like an amendation of Quine's argument.

    Anyway, what of this new argument? I'm not sure I follow it completely, but I take your point that assuming that Tom opaquely believes that #(Dcc) = 1, we can indeed make a quotational step, to: Tom believes '#(Dcc) = 1'.

    But I don't see how that enables one to run an argument which shows what Quine was trying to show with his invalid argument. Can you see a way?

  3. Hey Tristan, I tried to clear up my thoughts on this. It still ended up being a long ramble, and I think we pretty much agree actually. If you have the time, I posted it on my blog because of its length:

  4. Thanks for the link and for your interest!