Saturday, January 01, 2011

Liars Are Fairly True

Suppose I say “what I’m saying isn’t true.” If what I said was true, then as I said, what I said wasn’t true. Does it follow that my words weren’t true? The famous paradox is that if so, then since that’s what I seem to have said, I seem to have said something true. A fairly popular resolution takes my words to have been meaningless, so that I didn’t say anything. But if my words had been meaningless, you could hardly have known what they would have meant had they been true. Is our ordinary conception of truth shown by such Liar-style sentences to be deficient? Let’s see why not.
......To begin with, such sentences are in some ways like Truth-teller-style sentences. If I said “what I’m saying is true,” for example, what would I be saying? Not much. Questions of truth are essentially questions of how well our words describe the world, and “this is a good description” isn’t much of a description. Still, it might not be too bad a self-description, precisely because there isn’t much to describe. If someone saying “what I’m saying is true” intended to be speaking the truth, should we deny that she was telling the truth? It may be hard to say, but therefore it might be that such sentences are not so much vacuous as vague. Since “what I’m saying isn’t true” also addresses nothing but its own descriptive power, might it also be, in its own way, rather vague? Consider the following analogy.
......If I said of some colour, “I wouldn’t say that it’s blue,” I might not be saying that it wasn’t blue, because colours don’t divide into those that are blue and those that aren’t. To see that, consider a spectrum: On the two sides of any such line, between the blue and the other colours, there would be colours that were indistinguishable. So there’s no such division; so there’s some colour of which, rather than saying it was blue, or that it wasn’t, I’d prefer to say, more precisely, that it was bluish but not very blue. (Since the perception of colour is subjective, you might say it was blue, or that it wasn’t.) Our perception of colour is also context-sensitive, e.g. it’s affected by surrounding colours, and by our preconceptions. So if I wondered if our colour really was blue, I might thereby see it as not blue, while if I then wondered if it was therefore not blue, it might seem pretty blue (even to me).
......And similarly, it’s when “what I’m saying isn’t true” has been thought of as definitely not true that it seems most clearly to be true. More precisely, while those words aren’t giving us a very good description of their own meaning—they’re self-contradictory—we therefore have a description that isn’t too bad, insofar as it’s saying that it’s not a very good description. In short, they’re rather nonsensical (and false), but therefore fairly true (and false). And that’s basically how Liar-style sentences are compatible with our ordinary conception of truth. We need a bit more clarification, but it should soon become clear that while we can always be more precise, there’s no threat to truth here.
......What is truth, if not a sufficiently accurate description? Usually we describe things accurately enough for some obvious purpose, or else we don’t, so we tend to assume that truth is black-or-white. But it’s really a matter of degree, in a context-sensitive way. E.g. the table at which I’m writing this is flat enough for that purpose, so “this table is flat” is true enough, but might be false were I writing about geometry. And in general, our words tend not to be much better defined than our purposes have required them to be. So natural language has a ubiquitous—since ordinarily unobtrusive—vagueness (whence the way to resolve paradoxes, and uncover other fallacies, usually involves clarifying some terms). Of course, the words of “what I’m saying isn’t true” have clear enough meanings, so there’s no simple equivocation to discover. But it should help us to resolve the paradox if we don’t demand anything too unrealistic. (Similarly, we shouldn’t demand that colours be either blue or else not blue.)
......Liar-style sentences present themselves as misrepresenting themselves, so their meaning is self-undermining. And they can be read (or heard) in two basic ways—each a necessary part of the other’s context—because their meaning self-undermines in a loopy sort of way. Insofar as Liar-style sentences are true they’re also false, and they need concern nothing but their own truth, so they can certainly be read as nonsensical. But they’re not just senseless, and hence not at all true, because insofar as they’re not true they’re easily read as true. So they also have that sense. But they can’t be nothing but partly true and hence partly false, because that would leave nothing for them to be true or false about.
......This resolution—that Liar-style sentences are fairly true, in that loopy way (they’re fairly true because they’re rather nonsensical, and they’re rather nonsensical because insofar as they’re true they’re also false)—is a strengthened version of the resolution that takes them to be nonsensical. So for those who believe that an omniscient being is logically possible, it allows a similar reply to Divine-Liar-style sentences. E.g. the problem with “no omniscient being knows this” is that it can’t be true if there’s an omniscient being, but if it isn’t true then, since no one could then know it, it would seem to be true. My new reply is that if it’s only fairly true (in this loopy way) then no epistemically perfect being would have to know it, except to know it for what it is. And note that “no omniscient being knows any of this” is simply false, e.g. such a being would know those words. (Similarly, “what I’m saying isn’t at all true” is fairly false.)


Xamuel said...

Truth, as we intuitively understand it, is a semantic notion and exists in the meta-language. The Liar, with his Sentence, attempts to make syntax out of semantics, and that's the deeper reason behind the paradox.

In order to actually formalize the Liar's Sentence, we have to augment our language with a truth predicate. Even then, we do not obtain a sentence L which says "L is not true": instead we get a sentence L such that basic arithmetic proves "L iff L is not true", where "L is not true" means "the interpretation of the truth predicate doesn't contain the Godel number of L". This by itself is not paradoxical, only seeming paradoxical if we (mistakenly) identify the syntactic truth predicate with actual semantic truth. If we make reasonable-seeming assumptions based on that identification, such as T(phi)->phi and phi->T(phi), those assumptions are inconsistent with basic arithmetic. There's nothing deep or profound about this: it just demonstrates the futility of trying to syntax-ize semantics.

enigMan said...

The notion of a meta-language only makes sense against a context of formal languages that need it in order for them to carry meanings. These words, by contrast, are in English, and have the meanings they have. A word is a shape and a noise that has some meaning. (The meaning isn't added to the word. We're naturally inclined to learn a language like English. It's a fascinating topic of study, how words come to have the meanings they have.)

The Liar Paradox exists in a natural language like English. It's a paradox because such sentences seem to be true because they're not true. Some people took the notion of truth to be suspect. The resolution makes clear what's going on. There's no threat to the notion of truth; and no making syntax out of semantics: English isn't a formal language. And for all it's faults, we can't replace it with a formal language, or a set of formal languages, because such languages need a meta-language, if not English then something like it.

If we try to formalise Liar-style sentences, most formal languages won't let us get very far. But so what? We don't invent formal languages to replicate in them the messiest bits of natural language, but to help us with mathematical modelling. And note that it's not just the truth predicate that doesn't exist in the formal language. In a formal language, all the predicates are just models of predicates. Indeed, predication doesn't exist in formal languages, only something like predication, something that can be used to model predication.

Xamuel said...

Good point. Hmm...

Maybe it is like this. In the formal language, the liar sentence is not a statement about itself, but about its Godel number. Maybe we can use that to understand the English paradox. The English liar sentence "this sentence is untrue" has two forms: it is a set of sounds/Roman characters, and it is the "meaning" we ascribe to those sounds/characters. The "meaning" is analogous to the formal formula, and the "sounds/characters" are analogous to the formal Godel number. Thus, when the Liar says "this sentence is untrue", "this sentence" means "this set of sounds/characters", which brings up the question of what it means for a set of sounds/characters (rather than a "meaning") to be true. It seems that trying to establish the truth of sounds/characters in English is no easier than adding a truth predicate to PA... In fact the two endeavors are almost the same...

enigMan said...

I don't think it is like that (although the last century of positivistic philosophy may well make it look that way to most people who meet the Liar Paradox after Logic 101).

In "this sentence is untrue," "this sentence" refers to that sentence, where sentences are things that can be true or not. And truth in English is very unlike adding functions to formal languages. Truth is not added to a language. The whole point of a language is to help us to communicate about our world, and when we are describing the world accurately enough, that is what it means for our words to be true. As part of our pursuit of truth, science includes mathematical models of parts of reality. A formal language is a mathematical model of part of language (often a scientifically important part). A formal logic is a model of good reasoning. But we always, inevitably, reason in ways best expressed in natural language.

Perhaps formal models of the Liar Paradox can show us some of the limitations of purely formal calculations (e.g. Godel's refutation of Formalism), or perhaps our best formal language would just not allow us to make such a model. We have such negative results; but do such formal models tell us anything about "trying to establish the truth of sounds/characters in English"? Who establishes such truth? I decide that "this table is flat" is true because this table is flat enough for me to type on. If you wish to disagree you must address that context. In the face of such argument I may concede that this table is not exactly flat but it's flat enough.

Or by "truth" are you thinking of a function from all possible sentences to a pair-set? But why think of truth as such a function? Maybe the Liar Paradox does reveal one difficulty with defining such a function, in such a way that it is much like what we intuitively know truth to be. But other problems are that we don't mean by "truth" such a function anyway. Is it that we want such a function to do the job of truth in some scientific theory? But I don't know of any scientists who need such a formal truth function. And note that even modern logicians can choose to modify a propositional logic when it conflicts with our natural reasoning (the decision whether or not to do so being best expressed in natural language).

enigMan said...

Incidentally, liars are half-truths according to fuzzy logic, as I found out via this old post at TAR.