Wednesday, 21 September 2011

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):74864, November 1969.

13 comments:

  1. Maybe all the trouble is caused by Davidson's swapping out Tarski's (admittedly problematic) 'iff's for 'corresponds to'. If we are really careful about what we are saying, there doesn't seem to be anything controversial going on (or at least not in the way Davidson is suggesting).

    (1) The sentence 's' is true if and only if the fact expressed by 's' is the case.
    (2) The fact expressed by 's' is the case if and only if the fact expressed by 't' is the case.
    (3) Therefore, the sentence 's' is true if and only if the fact expressed by 't' is the case.

    I have no idea why all this Diogenes business is relevant (although I have probably missed something crucial).

    But notice that we could just as easily conclude that: The sentence 's' is true if and only if the sentence 't' is true. We have said nothing about what either sentence corresponds to which, as it were, CAUSES it to be true. All we have said is that, since 's' is true where ever s, and s holds where ever t, the truth of 's' covaries with t just as much as it does with s. But (if I am permitted a modal SHORTHAND), it might further be the case that there are possible worlds on which the facts s and t are not perfectly correlated, and on which, as a consequence, 's' is not true if and only if t, but only true if and only if s. Thus, there is a special correspondence that 's' has with s which is germaine to its truth, and which 's' does not have with t. The original Tarskian notion says nothing about this relationship.... and it is this relationship that Davidson erroneously appeals to.

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  2. Tristan: I noticed that you challenge the use of substitutivity of co-referential terms on grounds that it does not hold in natural language (since natural language is partly intensional) -- fair enough. But that doesn't appear to go to your very first claim regarding the misapplication of principles of formal logic to natural language; it's more indicative of a conflation of extensional context with intensional context. Have I missed something there? Or did you just intend to reject the sub of co-ref principle to show that the argument doesn't go through?

    Adam: reply to follow!

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  3. Adam,

    I'm not sure why you mentioned Tarski's T-schema?

    Davidson is arguing against the correspondence/truthmaker/etc. analysis of 'is true' in terms of being true to the fact(s). On one version of the slingshot argument, we are led to the conclusion that for any true thing, it corresponds to every fact -- or one monster fact, if you like. This is meant to be an absurdity. I take it that this is common knowledge. Now to your argument.

    (1) I'm not sure whether being true to the facts is meant to be a relation of causation -- probably no. Correct me otherwise. That's neither here nor there anyway, just thought I'd clarify.

    (2) Your modal suggestion works both ways. If the slingshot argument goes through then, I think, there are also worlds in which 's' will (unfortunately for the correspondence theorist) "covary" -- to use your term -- with the fact that t, rather than the fact that s. In short, the argument shows that there's no privileging s over t or vice-versa as to the truth of 's', so you need an independent account of why we can privelege s over t as to the truth of 's' in other worlds.

    (3) 'The original Tarskian...'

    How is it that Davidson is "erroneously" appealing to the 'true to the facts' notion? It is a notion that has been in philosophy since Aristotle, or even further back, and Tarski's T-schema is, ultimately, irrelevant to the notion. The T-schema does different philosophical work than an analysis of truth in terms of correspondence does. What do you think?

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  4. Adam, hi, thanks for commenting! I pretty much second Johann's response, with one exception which isn't crucial to the present issue: I'm not so sure that Tarski's schema has nothing to do with truth as correspondence. In fact, I think it's meant to capture that idea in a precise way. (Whether it succeeds is another question.)

    Anyway, it seems to me that Davidson could have run his argument just as well - or as badly! - on an instance of the T-schema. (If you'd like me to clarify this, let me know and I'll spell it out - but see the "Euclid" example at the end.)

    You give a quite different argument for a similar sort of conclusion (namely, some sort of collapse of truthmaking into a many-one or many-many relation). I'm not sure what principles you're invoking, and don't think that that argument is valid either, but in any case it's quite different from Davidson's, and the point of my post was to analyze and refute Davidson's argument.

    I'm probably missing the point of your comment, so don't hesitate to make it again!

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  5. Johann,

    These are good questions - you've missed nothing! As I see it, there are two issues raised by what you say. (The second one is maybe not explicit in your comment, but it came up for me as a result.)

    (1) Why do I claim at the start of the article that the fallacies in this argument are of a sort that are only committed when formal logical principles are misapplied to natural language?

    (2) Is it really enough, for my purpose, to reject substitutivity of singular terms as applying across the board in natural language?

    Regarding (1): There is a bit of a leak here I guess. It is conceivable that someone who has no contact with formal logic would spontaneously argue using the principles invoked by Davidson, and I haven't given any argument that there was an influence here. But the argument is so bizarre - I'm pretty sure it's impossible to understand how it's meant to work without knowing some philosophy/logic - that I took this as a safe bet, and so put that spin on it. But feel free to ignore that. The analysis itself doesn't depend on it.

    Regarding (2): This is slightly tricky: given that Davidson doesn't cite a restricted or qualified principle, showing that it doesn't apply *throughout* natural language does show that the very principle he cited isn't acceptable, and thus his argument as given isn't valid. However, it might be objected that this is uncharitable; maybe the principle could be restricted, and maybe Davidson's use here falls within those restrictions. (This is something I should have been more explicit about.)

    However, I think that wouldn't work. As hinted at toward the end, it seems like 'The fact that' may well be an intensional context (I think it is, but sophisticated views about facts might motivate a denial of this), and 'The statement that' seems clearly to be one. Thus if Davidson were applying, say, a substitutivity principle restricted to extensional contexts, this wouldn't be a legitimate application.

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  6. Ah, I now catch your point regarding (1). At any rate I think we're on the same page there.

    As to (2), I was thinking about that very thing before I posted my original comment. I came to the working conclusion that, given an unconstroversial characterisation of the intension-extension distinction, Davidson's use violates the restrictions on the employment of Substitutivity -- a demonstrably extensional principle. You share this conclusion.

    I then thought that surely a person as clever as Davidson (who was mentored by none other than Quine!) could not have made such an error, so it must be that I misunderstand how he is deploying Sub.

    In other words, I conditionally agree with everything you've just said!

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  7. (...the condition being that I (we) have the correct understanding of Davidson's use of Substitutivity.)

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  8. Cheers - yep, the argument tends to have that effect on people (indeed, a cynic might suggest that this was the intention). It's breathtakingly invalid!

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  9. I need to read your replies a bit more carefully, but before I do, I should clarify some things...

    My mention of Tarski comes from Davidson's own invoking of Tarski's truth theory in 'true to the facts'. Tristan - while I agree that Tarski thought of his theory as supporting the intuitive correspondence notion of truth, when it is interpreted in purely formal terms it gives us no particular reason to align ourselves with correspondence theories at all (although without modifications in that direction, it seems a bit circular).

    A further point of clarification is that, since Davidson pays lip service to Tarski's theory of formal truth, it is worth mentioning that in such a formal context, the applications of logic that Davidson uses are not inappropriate. It simply occured to me that Davidson slides between the formal assumptions and considerations of natural language, which he really shouldn't be doing, and that part of the problem is in replacing 'iff' with 'corresponds to'. The first is an operator in a formal metalanguage. The second is a metasemantic concept which concerns the relationship between sentences and facts by virtue of which those sentences are true.

    Finally, Johan - with respect to the modal shorthand I helped myself to: The idea would be that at every possible world at which 's' is true, s is the case. This is like the necessitation version of truthmaker theory, I guess. If it is the case that, in actuality, s and t always hold together, and so that 's' is true iff s or t, this is consistent with it being necessarily the case that 's' is true iff s, but possible that 's' is true and not-t. Correspondence would relate to the latter, modal formulation, and thus be an intentional matter. (But I'm not so sure I buy this, just food for thought).

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  10. Adam,

    'My mention of Tarski comes from Davidson's own invoking of Tarski's truth theory in 'true to the facts'.' - Ah, I'd forgotten about that. I actually wrote the above piece nearly two years ago!

    I'm a little hazy on what you mean by 'interpreted in formal terms', in particular, whether you're pointing to the difference between reading the 'iff' as a material equivalence or a real biconditional, or something more like the difference between a strict interpretation and a gloss. Incidentally, I think the material equivalence reading is just a misreading (following Adrian Heathcote). What do you think on that point?

    Regarding the second thing, that the theory so interpreted gives us no reason to align ourselves with correspondence theories at all, I think you're onto something there, but it's a very tricky subject. 'Correspondence', especially, seems to get used in many ways and at many thicknesses. (This is fairly loose and evasive talk; I need better ideas about this.)

    On 'correspondence', cf. this article: Lewis, David (2001). Forget about the ‘correspondence theory of truth’. Analysis 61 (272):275–280. Furthermore, the very idea of rival theories of truth, and of there being a task for something called a theory of truth, while clear on some levels, puzzles me.

    I'm inclined to think that no non-circular analysis can be given of the concept of truth (although that statement may be problematic or vague). Perhaps that *is* a theory of truth. I think Moore and Russell both held some view like this fairly early in their careers, but presented more ontologically (truth as an 'indefinable quality' or something like that).

    Also, I wonder what James and Peirce would have thought of the T-schema. I think it's quite likely that they would have accepted it - as true, I mean; not as an account of truth. James famously accepts, in a reply to Russell, that truth consists in correspondence to fact. He just doesn't think that says much.

    One last thing: 'part of the problem is in replacing "iff" with "corresponds to". The first is an operator in a formal metalanguage.'

    This seems like another tricky issue. Are you taking the 'iff' to indicate material equivalence here? In which case, there's no problem about having a formal metalanguage: one can use classical logic. But then that's arguably going to fail to capture the correspondence idea, and the idea that if things were otherwise, the truths would be otherwise. If, on the other hand, 'iff' is to be interpreted as a real biconditional, or something like that, then it's not clear what the formal metalanguage might be. But there may be serious candidates! Probably proposals have been made about this, but I don't know the area well enough - suggestions are welcome.

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  11. Above I wrote: 'James famously accepts, in a reply to Russell, that truth consists in correspondence to fact.'

    I should say that this is not a quotation, but an unscholarly paraphrase from memory. James may not have explicitly accepted that formulation ('consists in').

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  12. Another second thought: I say that the material equivalence reading of Tarski's T-schema is just a misreading, but Tarski himself may have encouraged, or even ensured, that reading in some writings. In 'The Semantic Conception of Truth' of 1944, however, it's always 'if, and only if'. It seems likely that his views developed and got clearer on this point.

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  13. Scratch 'interpreted in formal terms'. I just mean that Tarski's theory is couched in a formalised language. If we do not interpret the 'if and only if' as material equivalence, we can still distinguish two difference readings:

    (i) 'if and only if' as establishing a truthmaking relation between sentence and fact, or a dependence relation between the truth of a sentence and the fact it expresses.

    (ii) 'if and only if' as expressing what I will infuriatingly call a 'minimal perfect co-occurance' of the truth of a sentence with the fact it expresses.

    Interpreted in the first way, Davidson is perhaps saying something important. Without an extra term in the formulation which expresses a special relationship between a sentence 's' and the fact s, the slingshot argument may seem convincing.

    But given that there is no such extra term expressing a special correspondence, it seems more likely that (ii) is the best interpretation of Tarski's theory, especially in light of its only needing to satisfy material adequacy. I.e., it does indeed seem to be the case that every actual fact also perfectly co-occurs with the truth of 's'. The point is that the only solitary fact which tells us anything about the truth value of 's' is s itself. If we find that not-s, we know that 's' isn't true. If s, then 's' is true. Yet if t is also an actual fact--and it further happens to be the case that 's' is true--then it t also perfectly co-occurs with the truth of 's', but delivers no verdict on the truth value of 's' on its own.

    If we only know that the fact t holds (or any other fact which is not s), we cannot predict anything about the truth value of 's'. 's' might be true or false. We only know that every actual fact co-occurs with the truth of 's' once we know that s, and hence from the fact of s that 's' is true and not false.

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