Tainted Love

 

Love of the truth is a particularly philosophical virtue, whence analytic philosophers are such fans of science. “Science is – or, better, sciences (in the plural) are – communally organized efforts of real people to find their way in some section of the real world.” (Grene and Depew 2004: 352) And truth is simply a matter of our words representing sufficiently accurately the relevant aspects of the world, much as a good map maps the world (a good introduction to the modern philosophy of truth is Blackburn 2005; and for a few more details see Candlish and Damnjanovic 2007). Sounds simple enough, and it should because such representation is of course the primary function of our words.

Now, what we say may contain only analogical elements of the truth, rather fuzzily, and often there are all sorts of other nuances to what we say, but still, truths are what our sciences seek, and what we especially wish to avoid is that our words are false, that they misrepresent the world—that we say of what is, that it is not, or of what is not that it is (as Aristotle put it). So far so good, but many philosophers believe (falsely, according to the following) that the Liar paradoxes show that ideal to be naïve (a good introduction to such paradoxes is Clark 2002: 99-106; and for their more recent consideration see Jacquette 2007: 137-46). The simplest Liar sentence is of course “I’m lying,” but in what follows I shall focus upon the logically simpler “This assertion is not true,” which I shall refer to by its acronym, Taint.

 

Consider someone—a Liar—saying Taint. Clearly, in saying Taint she says that Taint is not true. If she speaks truthfully then Taint is indeed not true. But that would mean that what she says—Taint—is not true, i.e. that she does not speak truthfully. So, if she speaks truthfully then she does not speak truthfully, and of course either she speaks truthfully (and so does not) or else she does not, and so in short she does not. What she says—Taint—is not true. The paradox is that, since that is precisely what she was saying (i.e. that Taint is not true) she seems after all to have been speaking truthfully.

On the other hand, the Liar does not seem to be saying so much—cf. the Truth-teller, who states only that her statement is true, e.g. “This assertion is true,” which I shall call ‘Tait.’ The Truth-teller is saying that she is truthfully saying… what? No more than that! But surely truth is more important, more substantial than that. So, I would say that Tait is too trivial, too vacuous to be true. Furthermore since the Truth-teller is hardly asserting anything, she is hardly making an assertion, whence her “This assertion” is not referring to anything, but is rather an empty name. Tait is not true, and the reason is that it is not even an assertion. It is too meaningless, too senseless to be true.

Similarly the Liar is saying only that she is lying, when she says that she is lying. So by analogy with Tait, Taint might also be too vacuous, too lacking in propositional content (or sense) to be an assertion. If so then, although Taint would not be true, the Liar could not (contrary to appearances) have been asserting that fact.

 

You can’t say you are talking nonsense by talking nonsense, since to talk nonsense is not to say anything. But having talked nonsense, you can go on to say that it was nonsense, and now you are talking sense. (Clark 2002: 102-3)

 

Of course, the Liar does seem to be asserting that Taint is not true, and were that appearance accurate we would therefore have a contradiction (that the Liar would be telling the truth), implying that Taint could not be too vacuous to be true (and nor presumably would Tait be). Indeed, that appearance may well be stronger with Taint than it was with Tait, because as we go through the reasoning of the Liar paradox so much seems to be said by the Liar. Nonetheless it is no less reasonable to think of such appearances as being deceptive as follows, because the Liar presents herself as misrepresenting herself—she is not just straightforwardly asserting the fact that Taint is not true, she is simultaneously asserting that she is lying about precisely that.

 

Jacquette (2007: 143) thinks that Taint is not so much paradoxical as a “disguised contradiction” because the Liar does indeed seem to be asserting both P and, less obviously, not-P (for some P). The Liar seems to be asserting not only that Taint is not true (obviously), but also that Taint is true (via her self-reference), so that overall her assertion would be false (via the latter conjunct), whilst seeming true (via its relative obscurity). But while Taint is indeed not true, things are I think more complicated when the Liar says that, because it is that very self-reference which seems to result in the additional untruth (insofar as the former was a truth, of course).

Perhaps the Liar only seems to be asserting something identical to my assertion that Taint is not true. If so, then (since nothing else appears to be asserted by the Liar) perhaps Taint is too vacuous to be true after all, despite appearances—the primary appearance there was of Taint being an assertion. Taint clearly presupposes that there is an assertion (with “This assertion”) and it does indeed continue in something like an assertion (for all that it is being self-referential, and fairly vacuous). But the secondary appearance was of that assertion being a conjunction of two (inconsistent) assertions. Furthermore one of those (contradictory) conjuncts appears to be the same as that original assertion, and that is why it was reasonable to think that original appearance was deceptive.

Cf. how in maths, if we have x = x – x, then we may usually deduce x = 0. Similarly it is reasonable to think that Taint had no propositional content, that it was nonsense. Taint is not so much a disguised contradiction as nonsense disguised as a contradiction and disguised again as just one of the inconsistent conjuncts. As we run through our paradoxical reasoning about Taint, our equivocation naturally goes unnoticed because the words we naturally use to assert the truth that Taint is not true are essentially the same words that the Liar uses to utter nonsense (and we naturally assume an interpretation of the words of others that makes them meaningful, see below).

Jacquette (2007: 144) regards the negation of the necessarily false Taint as the tautological Tait. But prima facie the negation of Taint (“This assertion is not true”) ought to be about that assertion (if that is an assertion), not a different one. Of course, what Taint and Tait both concern appears to be nothing; what they share is their vacuity. (This appears, then, to be relatively complicated territory, and I shall glance again at negation, and equivocation, below.) But what is relatively clear is that there is something deceptive about Taint. And since Tait is akin to Taint, and is clearly vacuous, I therefore think it reasonable to suppose that, notwithstanding first appearances, Taint is similarly too vacuous, too meaningless to be true.

 

Nevertheless you may well be thinking that if Taint is not an assertion then (since Taint could not then be referring to itself as “This assertion” successfully) the paradox would have been too easily avoided, that it would return if the Liar referred to a sentence, e.g. by saying “This sentence is not being used to assert a truth.” And perhaps we would then have our two assertions, since sentences can be used to say contradictory things.

So let us run through our paradoxical reasoning again. Could that sentence have been used to assert a truth? If so then, according to what it would apparently have been used to assert, it would not have been. And if it was not, and if we look again at that sentence, we want to say that’s right, it was not being used to assert a truth—but then it appears to have been used to assert a truth after all. And by ‘it’ is certainly meant the same object named (non-emptily) by the phrase “This sentence” (in the Liar’s new words).

Now, if any utterance of that sentence could only have expressed nonsense, then clearly that sentence was not being used to assert a truth—which is just what an utterance of it would seem to be saying. What would be asserted of that sentence, by any utterance of it, would be (if anything) that it was not being used to assert a truth. But therefore there would have to be, upon any sensible use of that sentence, a vicious self-reference to that very use of it—not to another use of it, as equivocation would require—so that what it was being used to assert (if anything) would have to be essentially the same as Taint. Moving from Taint to a use of that sentence has therefore achieved no more than moving from “That round square is round,” to “ ‘That round square’ is being used to refer to something round,” as follows.

As mentioned above, it is part and parcel of our natural language-use (with our imperfect grasp of its flexible, not to say indefinite, words) that we are naturally quite charitable in how we usually interpret other people’s words. It is only natural, when we are initially unsure of their meaning, for us to veer towards giving them an interpretation (unconsciously) that will make them meaningful (at the very least; and often, if that would not be too implausible, true).

So, suppose that we have got as far as working out that what the Liar says is not true (e.g. because if it is true then it is not, and if it is not then it is not, and it is either one or the other, whence it is not), so that we think it true that that sentence was not being used to assert a truth. Then we naturally see that sentence (upon our reading of it, as though it was being uttered) as asserting a truth. We have to think our way through to the conclusion that it cannot be (e.g. because the truth we saw it asserting was precisely that it was not being used to assert a truth, whence it must have been nonsense). And that latter act of thinking it through will hardly seem as compelling as the former act of seeing that sentence as being right (even though in this particular case it ought to be more compelling).

The problem with what the Liar and the Truth-teller say is not syntactic but semantic, which is why the paradox will not return even if we go to syntactically non-self-referential forms, such as a pair of sentences A and B, where A says that B is not being used to assert a truth and B may or may not (depending upon some contingencies) say that A is being used to assert a truth. When we don’t even know what B is, A seems to be far from nonsensical. But still, if it did so happen that B was “A is being used to assert a truth” for example, then any utterance of A would happen to have no more sense to it than Taint—the meaning of A would then be something like “ ‘A is being used to assert a truth’ is not being used to assert a truth,” or “A is not being used to assert a truth.” So the meaning of an utterance of A would be just like the meaning of an utterance of “This sentence is not being used to assert a truth,” i.e. any utterance of A would then be meaningless (senseless, nonsensical or vacuous).

 

Incidentally (and independently of the above disagreement) Jacquette’s (2007) metaparadox also shows that Liar sentences do not threaten our naturally realistic concept of truth. And after all, the most puzzling aspect of truth is not the Liar’s sayings, but how our words could represent the world—how do mere clumps of ink on paper, mere vibrations of the air, come to have semantic content? Compared to that, the Liar’s sayings are trivial, and ought to have a trivial resolution.

The common initial reaction to Taint—that it was wrong somehow—is correct. What was incorrect was our tying ourselves in logical knots, deceived as we were by the assertive appearance of Taint. Cf. how superfluids don’t refute the basic presupposition of physics—that there is a physical world to investigate—for all that they can complicate our mathematical models. Similarly the Liar’s sayings don’t refute the basic presupposition of philosophy—that the truth is to be discovered—for all that they can complicate our symbolic logics.

Consequently, the set-theoretical foundations of our current scientific theories, which first made the existence of such paradoxes seem so natural (by analogy with Russell’s paradox) and which then facilitated their formal avoidance (in Tarskian and Kripkean ways), have not been as useful as is often thought—and if I am right about the potential infinitude of simple infinities then that is because they are fundamentally flawed.

But I shall end by looking again at negating Taint (for there are always many ways to look at something, and people vary in their preferences). Negating self-referential statements is not necessarily problematic, e.g. the negation of the true “It is true that these are eight words” is just the false “It is false that those were eight words” (or the false “It is true that those were not eight words”), not the false “It is false that these are eight words” because those words are slightly different.

 

The truth of “It is true that these are eight words” followed from ‘It’, ‘is’, ‘true’, ‘that’, ‘these’, ‘are’, ‘eight’ and ‘words’ being eight words, the third of which is ‘true’ not ‘false’. It followed from repeated applications of the following (essentially Tarski’s) rule T, which it is impossible to imagine truth disobeying: It is true that S is (not) P if and only if S is (not) P.

In that rule, ‘S’ refers to some existing thing, ‘P’ refers to some definite property, and a particular statement (“S is P”) is being said to be true if and only if it describes the apposite facts (i.e. that S is P) adequately. Note that it follows from T (plus classical negation) that it is false that S is P iff (i.e. if and only if) S isn’t P, which I shall refer to as F.

A property P is definite iff for any existing thing X, either X is P or else X is not P, exclusively and exhaustively. Of course we usually require only relatively definite kinds of P, because we need only be as definite as our universe of discourse requires. If we get problems (e.g. if we discover that S is both P and not P) then we may need to introduce more definite predicates (e.g. had we been equivocating). The problem, of course, is that since the univocal ideal of truth is precisely what we aim towards when we thus make our words better fit the world (when we improve our linguistic map of the world, cf. Blackburn 2005) we don’t want to discover, in that way, that (as the Liars may seem to imply) ‘true’ is not definite.

Incidentally, note that the form of rule T (with its “S is P”) is quite exceptional, not just because of P being usually only as definite as our evolving universe of discourse requires, but also because of the niceties of proper names, which indicate that to talk of S is usually to be involved in some degree of deceptiveness. And rule T is also a little misleading because “Grass is green” being true iff grass is green can look too much like two photographs in a book on photography, illustrating the connection between a photograph and its subject by showing borders on the former, whereas the only real way to see that connection is to actually take some photographs.

When we communicate we have to follow natural language-rules, but those rules do allow natural clarification procedures to come into play when we meet a contradiction. I might assert truthfully “Grass is not green” as well as “Grass is green” for example, in order to convey the idea that something is wrong; e.g. although the grass that we find in lawns is ideally green, the lawn is actually yellow. By giving me the benefit of the doubt (i.e. by assuming that both statements were true), you might deduce (assuming logical bivalence of course) that I must have had in mind two different kinds of something and reply, “Yes, we need more rain.” The contradiction of “Grass is not green” with “Grass is green” leaves us many options, and our choice is therefore context-sensitive—with those same words I might have meant the puzzle of perception (that colours are both subjective and are attributed correctly to objects), or I may have been referring to a fuzzy range of grasses, or have had a borderline shade of green in mind, and so forth.

 

So the contradiction at the heart of the Liar paradox is quite naturally taken to be revealing an equivocation, an effective difference in the meaning of at least two of the expressions that are repeated in the derivation of the contradiction, which was that if Taint is true then (by definition of Taint) Taint is not true, and if Taint is not true then (by rule T) Taint is true. One reason why the obvious candidate for equivocation is therefore “Taint is not true” is as follows.

Since the phrase “This assertion” in Taint refers to the whole of Taint, so Taint is (by definition of Taint) the assertion that Taint is not true, i.e. the meaning of Taint is that Taint is not true. Note that we have just copied Taint into itself—that last phrase “is not true” is identical to the same phrase within the preceding Taint.

But rule T, in the form of F, supplies a new phrase “is not true,” giving us that “This assertion is not true” (i.e. Taint) is false (i.e. not true) iff that assertion isn’t not true—i.e., that Taint is not true iff Taint (is true). So rule T also gives us “Taint is not true” from Taint, but that phrase “is not true” is not now quite so identical to the corresponding phrase within the preceding Taint.

When we say that Taint is not true, we are referring to it from outside of it, and so our statement might be negated straightforwardly to “Taint is true.” But that is not to negate Taint, which the Liar says. Since the phrase “is not true” to be negated by her would also be occurring within the preceding Taint to be negated, she ought indeed to say something more like Tait (“This assertion is true”), in which the subject has also changed, in what appears to be the right kind of way (although if Tait and Taint are indeed nonsense, then there are no such subjects, and no negations).

Anyway, the ‘true’ introduced by T must have its usual meaning, and presumably the ‘true’ within Taint is supposed to have that meaning too, for otherwise the Liar paradoxes would not have us worrying about the nature of truth (and similarly we want ‘is’ and ‘not’ to be univocal), so we want to find the equivocation with “Taint is not true” (if there is one) centred upon ‘Taint.’ But prima facie it cannot be, since that is just our name for what the Liar said, whence we are quite naturally led back towards locating the equivocation at ‘true’ after al (and thence to positing the usual infinite hierarchies of truth-predicates, etc.). The equivocation is with ‘Taint’ however because (as argued above) Taint is not an assertion.

Insofar as ‘Taint’ is just our name for what the Liar said, “Taint is not true” is a true assertion. But insofar as ‘Taint’ occurs within the Taint that is “Taint is not true,” it makes that last expression the very thing (the nonsense) that we were truly saying (in so many words) was not true. Paradoxically, that is perhaps easier to see if simpler symbols are introduced. So let my saying that L is not true be M = “L is not true,” where L is the Liar’s saying of those same words, i.e. L = “L is not true.” Clearly ‘M’ is an ordinary sort of name, whereas ‘L’ actually occurs within its named object, so it is doing more than just naming something; ‘L’ is clearly involved in the construction of the very object that it names, whence we would hardly expect M and L to be the same. Indeed, we may prefer not to regard ‘L’ as a name at all, and insist that names be well-founded.

[Comments]

 

References

 

Blackburn, S. (2005) Truth: A Guide for the Perplexed, London: Allen Lane.

Candlish, S. and N. Damnjanovic (2007) ‘A Brief History of Truth,’ in D. Jacquette’s (ed.) Philosophy of Logic, Amsterdam: North-Holland, 227-323.

Clark, M. (2002) Paradoxes from A to Z, London: Routledge.

Grene, M. and D. Depew (2004) The Philosophy of Biology: An Episodic History, Cambridge University Press.

Jacquette, D. (2007) ‘On the Relation of Informal to Symbolic Logic,’ in his (ed.) Philosophy of Logic, Amsterdam: North-Holland, 131-54.