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

ADDED: Here is a followup post from 26 December 2015.
UPDATE (Nov 2019): I have recently published a paper on this topic, 'Quine's Poor Tom', in the European Journal of Analytic Philosophy.

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

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. 

An Analysis of Davidson's Slingshot Argument

There is a peculiar kind of logical fallacy which, ironically, is only committed by people who have an acquaintance with formal logical theory. Fallacies of this kind arise when principles of inference from formal logic are applied inappropriately to arguments carried out in a natural language.

Here I make a case-study of Donald Davidson's famous version of the Slingshot argument against facts. The argument, in its dialectical context, is meant to show that if true statements correspond to facts, then every true statement corresponds to every fact. Davidson tries to demonstrate this conditional in order to motivate us to give up its antecedent (that true statements correspond to facts). Here is the argument:

The confirming argument is this. Let ‘p’ abbreviate some true sentence. Then surely the statement that p corresponds to the fact that p. But we may substitute for the second ‘p’ the logically equivalent “(the x such that x is identical with Diogenes and p) is identical with (the x such that x is identical with Diogenes)”. Applying the principle that we may substitute coextensive singular terms, we can substitute ‘q’ for ‘p’ in the last quoted sentence, provided ‘q’ is true. Finally, reversing the first step we conclude that the statement that p corresponds to the fact that q, where ‘p’ and ‘q’ are any true sentences. (Davidson 1969, p. 753.)

Let us go through it bit by bit.

The first apparent inference in the argument is curious. 'Then surely' suggests that reasoning is taking place here, but from what? Apparently:

Let ‘p’ abbreviate some true sentence.

But that is an instruction, not something we can infer from at all. This shows that what Davidson has supplied is not an argument, so much as a recipe for making one. And since this first step is not an inference, it can't be a fallacious inference. Still, in its slightly confusing use of a technique from logic (in this case, schematization) it gives us a small taste of things to come.

Now, following Davidson's recipe, we shall let 'p' abbreviate 'snow is white'. For perspicuity, we shall not use these abbreviations in our writings-out of the steps of the argument. (Surely this could not affect validity.) Thus our first real premise is:

The statement that snow is white corresponds to the fact that snow is white.

Now we are told we may make a substitution, yielding:

The statement that snow is white corresponds to the fact that (the x such that x is identical with Diogenes and snow is white) is identical with (the x such that x is identical with Diogenes).

The first thing to note about the above is that it doesn't obviously mean anything. This should make us suspicious. After all, we are not supposed to be merely calculating with signs here. This is supposed to be an argument - a reasoned chain of statements leading to a conclusion. How did we get to the above sentence, then? There are two things Davidson needs us to accept if we are to go along with this inference:

(1) That it is valid when arguing in English to substitute, for a sentence, a logically equivalent sentence - even when this sentence is embedded in a larger one.

(2) That 'snow is white' is logically equivalent to '(the x such that x is identical with Diogenes and snow is white) is identical with (the x such that x is identical with Diogenes)'.

In trying to assess these claims, we face a stumbling block: the lack of a clear, agreed upon notion of logical equivalence as a relation between sentences of natural languages. Some would say that 'snow is white' is logically equivalent to 'snow is white and Socrates is either mortal or not mortal'. Others would deny this, on the grounds that Socrates' existence is not implied by the original sentence. Some would say that 'John is a bachelor' is logically equivalent to 'John is an unmarried man', by the logics of bachelorhood, gender and marriage. Others would say these are perhaps analytically, but not logically, equivalent, because the equivalence does not turn on the use of "logical vocabulary".

Having made due note of this difficulty, let us observe that Davidson has no problem bringing in, out of the blue, mention of Diogenes. This gives us some handle on Davidson's intended notion of logical equivalence - enough, I think, to justify us in sweeping the difficulty under the carpet so that we may proceed to ask if (1) might be true.

That the answer is 'no' can be seen from these invalid instances:

(i) It is obvious that snow is white. Therefore, it is obvious that snow is white and [some elaborate and opaque tautology].

(ii) If you assume that the square root of two is rational, it is easy to derive a contradiction. Therefore, if you assume that [some elaborate and opaque logical equivalent to 'the square root of two is rational'], it is easy to derive a contradiction.

(iii) The statement that snow is white involves no semantic concepts. Therefore, the statement that snow is white and "grass" either refers to grass or does not refer to grass, involves no semantic concepts.

(In classical formal logic, the range of possibilities for sentential embedding is far narrower than in natural languages, and therefore no analogous counterexamples arise.)

How about (2)? For a start, can we even understand '(the x such that x is identical with Diogenes and snow is white) is identical with (the x such that x is identical with Diogenes)'? The use of the variables and brackets is, in itself, not a deal-breaker, since we can understand '(the x such that x is identical with grass) is green'. But now: on this understanding, what is the role of that which comes after 'such that' in the bracketed construction? Intuitively, the construction as a whole is a referring term, and after the 'such that' ought to go conditions relating to the variable which are met by exactly one of its possible values, thus determining a unique referent.

But then what happens if, as well as conditions involving 'x', we insert closed sentences like 'snow is white'? Well, on the intuitive idea behind the bracketed construction, this just doesn't make sense. Nevertheless, "appropriate" reference-conditions come to mind: a bracketed 'the' construction refers iff the conditions relating to the variable are met by exactly one object and all constituent closed sentences are true. To complete the semantics, we can stipulate that if such a construction refers, it refers (of course) to the condition-meeting value of the variable.

Thus we can define a new kind of referring construction, albeit a strange one. Also, it does appear that our complicated identity sentence, in light of this definition, is logically equivalent (in some sense) to 'snow is white'. Of course, this is of no use to us, since the principle whose application we wanted the equivalence for is invalid.

Before we move on: the addition of this new referring construction to our language may render previously valid principles invalid, so we must now be extra careful. (If, earlier, we had decided that (1) was true - that the unrestricted substitution of logical equivalents was valid - we would now have to go back and reconsider.)

Now, despite the fact that things aren't going very well for our argument, let us press on. We have gotten as far as:

The statement that snow is white corresponds to the fact that (the x such that x is identical with Diogenes and snow is white) is identical with (the x such that x is identical with Diogenes).

And now, citing the principle that we may substitute coextensive singular terms, Davidson has us substitute some true sentence - let us pick 'grass is green' - for 'snow is white'. (This then yields a new 'singular term', '(the x such that x is identical with Diogenes and grass is green)'.) Thus we get:

The statement that snow is white corresponds to the fact that (the x such that x is identical with Diogenes and grass is green) is identical with (the x such that x is identical with Diogenes).

And now we must ask: does the principle of substitution of coextensive singular terms hold in natural language? Notoriously, and as anyone familiar with twentieth-century philosophy of language will know, it (very arguably) does not; there are numerous contexts where such substitutions (strongly seem to) fail. (Witness the existence of intensional logics.) Here is an example of one kind of invalid instance:

Lois Lane knows that Clark Kent is Clark Kent. Therefore, Lois Lane knows that Clark Kent is Superman.

There are also well-known problems with substitution into modal contexts. Furthermore, and closer to our current context: 'the fact that Clark Kent is Clark Kent' does not obviously have the same reference as 'the fact that Clark Kent is Superman', even though the differing embedded singular terms are coextensive. And certainly the statement that Clark Kent is Clark Kent is not identical to the statement that Clark Kent is Superman. For all these reasons, we can not accept an unrestricted principle of substitution of co-extensive singular terms. Thus our last inference was invalid.

Since the final inference is a reversal of the first substitution, that concludes our step-by-step evaluation.

If there be any residual doubt about the invalidity of Davidson's argument (recipe): note that no special properties of the sentence 'The statement that snow is white corresponds to the fact that snow is white', beyond its embedding 'snow is white', are drawn upon in the derivation of 'The statement that snow is white corresponds to the fact that grass is green'. If this were really a valid way of arguing, we would also have to accept the following:

Suppose there is a chameleon, Euclid, who lives in a field of grass. Suppose further that Euclid is green because grass is green. Using Davidson's form of argument. we can infer from this supposition first:

Euclid is green because (the x such that x is identical with Diogenes and grass is green) is identical with (the x such that x is identical with Diogenes).

Then:

Euclid is green because (the x such that x is identical with Diogenes and Davidson is the author of 'True to the facts') is identical with (the x such that x is identical with Diogenes).

And finally:

Euclid is green because Davidson is the author of 'True to the facts'.

Tristan Haze

Reference

Donald Davidson. True to the facts. The Journal of Philosophy, 66(21):748–64, November 1969.