Sunday, April 03, 2011

My ‘revenge’ paradox (and Yablo’s paradox)

This post is the third part of Vaguely True Liars.

In my previous post I resolved a Liar utterance; can something similar be said of the other Liar sentences? Since there are many such sentences, let’s just see if we can resolve the trickiest. In general, the most difficult sentences for any putative resolution are those that threaten so-called ‘revenge’ paradoxes, which is when the terminology of the resolution – in our case, ‘vaguely’ – is used to make a new Liar sentence that resists resolution along the same lines. So suppose I said ‘what I’m now saying is not even vaguely true’. Something that’s not even vaguely true is, prima facie, something untrue. And if this new statement is asserting its own untruth then, like my previous utterance, it should be vaguely true. But then what I said – that it wasn’t vaguely true – would seem to have been false, not just vaguely untrue. And it would then, if false, seem to be true, rather paradoxically. So this does seem to be my ‘revenge’ statement.

Was it asserting its own untruth? Something that’s not even vaguely true (in our sense) is something that’s at least a little less true than not. And that’s compatible with it being more vaguely true, in the sense of more roughly as true as not. (It’s also compatible with it being false, of course.) Now, because of that compatibility, my ‘revenge’ statement seems true if more vaguely true, as well as false if vaguely true. Is there something in between the vaguely and the more vaguely true? Well, what about the vaguely more vaguely true? That’s a clumsy turn of phrase, but it is describing an unusual statement. And perhaps we could more loosely say that my statement was vaguely true, and then qualify that by adding that, more precisely, it’s vaguely more vaguely true than the vaguely true it refers to. (How vague the latter was would depend upon what counted as true when I uttered my ‘revenge’ statement.)

That’s a bit obscure, but such obscurity would at least explain how this resolution could have been overlooked, were it correct. And we can get some clarification by glancing at the related problem of higher-order vagueness [i]. That problem might, for example, arise with vaguely bluish colours – those that are roughly as blue as not (in some context) – were we tempted to regard them as neither blue nor not blue (in that context). If they weren’t blue and yet were blue, in the same context, then they would be contradictory, so we should resist that temptation. However, being so tempted we might take them to be, for example, neither definitely blue nor definitely not blue. And then we would face a ‘revenge’ problem, via the question of what happens between the definitely blue and the vaguely bluish colours. Were these colours neither definitely blue nor vaguely bluish, and the latter not definitely blue (nor definitely not blue), then these colours would be neither definitely blue nor not definitely blue, in the same context.

That’s a problem of higher-order vagueness. But such problems don’t affect the common-sense approach that I’ve been taking. While I don’t want to call vaguely bluish colours ‘blue’ or ‘not blue’, that’s only because calling them either would be only vaguely true, not because it would be false. Blue shades smoothly into blue-green, in reality. And a colour that’s roughly as blue-green as it is blue might be described quite accurately as ‘blue’ (since it’s a faintly greenish blue). Similarly, a good description of my ‘revenge’ statement could be ‘vaguely true’ (qualified as required). After all, the term ‘vaguely’ is an especially imprecise term, and highly context-sensitive, being generally used to gesture beyond the other adjectives in use, or made apposite by such use.

Things can be made more complicated, of course. E.g. a more awkward ‘revenge’ paradox might be based on Yablo’s paradox [ii]. Imagine a place where ‘no previous utterance here was even vaguely true’ was said by someone, once a year – every year, throughout the infinite past – with nothing else ever being said there. Now, it seems to me that all those utterances would have been (vaguely more) vaguely true. But it would certainly be more awkward to justify that view in this case. Still, suppose my approach failed here; that might only mean that this paradox was more like the paradoxes of infinity than the Liar. Certainly, this paradox concerns an infinite set of semantically ungrounded sentences, rather than self-description (or circular description [iii]). And the paradoxes of infinity are not our topic. (To be continued.)

[i] That problems of higher-order vagueness are akin to ‘revenge’ paradoxes was noted by Mark Colyvan, ‘Vagueness and Truth’, in Heather Dyke (ed.), From Truth to Reality: New Essays in Logic and Metaphysics (Abingdon: Routledge, 2009), 29–42.

[ii] Stephen Yablo, ‘Paradox without Self-Reference’, Analysis 53 (1993), 251–2.

[iii] A simple example of a circular Liar is the following pair of sentences. The next sentence is true. The previous sentence was false. I regard them both as vaguely true, when read or said in the obvious way.

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