Friday, April 01, 2011

Introduction (via Grelling’s paradox)

This post is the first part of Vaguely True Liars.

To begin very simply, ‘is long’ is not long, not for a predicate expression. Kurt Grelling called it ‘heterological’. Heterological expressions don’t apply to themselves. Grelling asked, is ‘is heterological’ heterological? It is if it doesn’t apply to itself – that’s what ‘heterological’ means – but if it is, then it applies to itself – that’s what ‘applies’ means – and so it isn’t heterological. It is if it isn’t, and it isn’t if it is; that’s Grelling’s paradox [i]. Could we arbitrarily include ‘heterological’ in, or else exclude it from, the range of its application? But then ‘heterological’ would not, in that particular instance, mean what it should, intuitively, mean. Furthermore, Grelling’s paradox is of a kind with the Liar paradox (our main topic), and taking such an approach to the Liar would amount to denying the universal relevance of truth (or its coherence) [ii].

The approach I take to such self-descriptive paradoxes begins by noticing that descriptive accuracy is in general a matter of degree. I would say, for example, that because ‘is pretty’ makes us think of prettiness, it’s vaguely pretty, but only vaguely pretty because it’s only a word (outside of calligraphy or song). It’s a matter of opinion, of course; but what about ‘is a bit long’, or ‘is a little lengthy’, or ‘is only slightly lengthy’? Anyway, if descriptive accuracy is a matter of degree, in general, then I should have said that a predicate expression is heterological insofar as it doesn’t apply to itself. And if that is – or should be – what ‘heterological’ means, then ‘is heterological’ is heterological only insofar as it isn’t. That is, it’s as heterological as not. So we might say that it’s only vaguely heterological.

And descriptive accuracy is likely to be a matter of degree, in general (in various context-sensitive ways). Our words are unlikely to be much better defined than our purposes have required them to be, and so we should expect a ubiquitous – since ordinarily unobtrusive – imprecision throughout our natural languages. Such imprecision is not usually important – that’s why it’s there – and even when it does matter, we can always clarify what we mean, because it has enabled our languages to be the versatile tools that they needed to be. Now, descriptions can become more accurate by becoming more detailed, so long as they remain true. Can they also become more accurate by becoming truer?

If you said ‘that’s blue’ of a fading print, for example, would your words become truer as the print became bluer? Certainly, a description can become truer by becoming more detailed, as when we bring out the element of truth from some half-truth. So perhaps we should say that, in general, our words are true insofar as they describe the world. E.g. ‘snow is white’ is true insofar as snow is white. The famous biconditional – ‘snow is white’ is true if, and only if, snow is white – is either a special case of that, or else it presumes that the elements of truth and falsity can always be effectively isolated. (Clearly ‘snow is white’ is usually true enough, but snow can also be faintly blue, discoloured, transparent, or sparkling all the colours of the rainbow.) Now, the idea that there are degrees of truth is nothing new. That it’s only common sense is shown by such common phrases as ‘true enough’ and ‘very true’; and it has been formally explored by the so-called ‘fuzzy’ logicians [iii]. But I want to emphasise its prima facie plausibility here, because if statements can be, not just true or not, but also vaguely true – about as true as not – then the Liar paradox is easily resolved.

A simple Liar statement is ‘this is false’, which is false if true, but if false then not false (a statement is false when its negation is true). Since statements might be neither true nor false, a better example may be ‘this is not true’. And since sentences can mean different things in different contexts, an utterance of ‘what I’m now saying isn’t true’ is what we shall examine below. However, whereas modern introductions to the Liar paradox tend to be rather formal [iv], I shall be quite informal. Philosophers are in the business of clarifying things, and imprecision can of course lead us astray, so it’s unsurprising that many modern philosophers are fond of formal precision. But formal languages must get their meanings from natural ones. And an obscure formalism might also lead us astray. E.g., the use of biconditionals to define truth might go too easily unquestioned on a page of mathematical symbols; and for another example, see Curry’s paradox (below). So, let’s get back to informal alethic basics. (To be continued.)

[i] For a class of paradoxes that includes Grelling’s, see Thomas Bolander, ‘Self-Reference’, in Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy.

[ii] Alfred Tarski took that approach to formal languages (taking natural languages to be inconsistent). For details, see Wilfrid Hodges, ‘Tarski’s Truth Definitions’, in Zalta, op. cit.

[iii] Petr Hajek, ‘Fuzzy Logic’, in Zalta, op. cit.

[iv] J. C. Beall and Michael Glanzberg, ‘Liar Paradox’, in Zalta, op. cit.

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