Thursday, March 31, 2011

Vaguely True Liars

My next 4 posts explore, informally and briefly, the possibility that Liar utterances are about as true as not. Abstract:
I suggest that Liar statements – e.g. ‘this is false’ – are about as true as not. In other words, they are vaguely true, and vaguely false. And truth does seem to come in degrees (e.g. if a colour is about as blue as not, calling it ‘blue’ would be about as true as not). I also suggest that ‘this is not even vaguely true’ can be called ‘vaguely true’, even though it may then seem false, not just vaguely false. That’s because it is, more precisely, vaguely more vaguely true (similarly, a colour that’s roughly as blue-green as it is blue would usually be called ‘blue’ becaue it is a faintly greenish blue). I give similar resolutions to the paradoxes of Kurt Grelling and Keith Simmons, while I regard as of a different kind those of Bertrand Russell and Haskell Curry.
Links to the 4 posts:

......Introduction (via Grelling’s paradox)

......Liar statements are about as true as not

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

......Russell, Cantor, Curry, and Simmons’ paradox

An application of the above is a common sense refutation of the Divine Liar argument (against omniscience), see my Liars, Divine Liars, and Semantics revisited (in April’s issue of The Reasoner).

Friday, March 18, 2011

Curry's paradoxes

The Liar paradox concerns utterances such as ‘what I’m saying isn’t true,’ which is, if true, not true, and which seems true if not. Another way of saying the same thing would seem to be to say ‘if what I’m saying is true, then pigs fly.’ Yet that utterance is paradoxical in a way so different that not only has it a different name – Curry’s paradox – it’s debatable whether its resolution should even resemble that of the Liar.
......Suppose I say ‘if what I’m saying is true, then P,’ where ‘P’ stands for any proposition. For simplicity, let’s say that C is the assertion that C implies P. Suppose, just suppose, that C is true. We are thereby supposing that C implies P. So we would also have P. That much is simple enough. Given C, and that if C then P, we get P. And yet that much is too much. If it’s true that by supposing C we also get P, then C really is true, and hence P is true, even though P could be asserting anything at all (even that pigs fly).
......That seems quite unlike the Liar. E.g. there was no ‘not’ in the previous paragraph: We didn’t consider C being either true or else not, and find both possibilities inadequate; nor did we see C seeming to say that it wasn’t something that C did indeed seem not to be. Rather, just by wondering what C was – in particular, whether C might be true – we seemed to get P. And so by deriving actual being from mere possibility, Curry’s paradox seems to be more like the Ontological Argument than the Liar.
......On the other hand, if it follows from the meaning of ‘not’ that either A or not-A, where ‘A’ stands for any proposition (e.g. that it’s raining), then it seems that if A implies B (e.g. that I’m carrying an umbrella), then either B or not-A (either I’m carrying an umbrella or it’s not raining). And conversely, if it’s the case that either not-A or B, then if it’s A (if it isn’t not-A), it must be B. So in short, C could seem to be asserting that either not-C or P. And then P effectively disappears if it’s false, leaving C asserting not-C.

Wednesday, March 16, 2011

Simmons' paradox

The following, which is akin to the Liar paradox, is a paradox of self-reference by Keith Simmons, described on p. 231 of his ‘Reference and Paradox’ in JC Beall (ed.) Liars and Heaps (Oxford 2003):
Suppose I’ve just passed by a colleague’s office, and I see denoting phrases on the board there. That puts me in the mood to write denoting phrases of my own, and so I enter an adjacent room, and write on the board the following expressions:

......pi
......six
......the sum of the numbers denoted by expressions on the board in room 213.

Now I am in fact in room 213, though I believe that room 213 is my colleague's office. I set you the task of providing the denotations of these expressions.
Simmons goes through your (fictional) reasoning; here is what I think:
......The first expression, ‘pi’, referred to 3.14159..., and the second to 6. So if the third expression does denote a number, say N, then N = N + 9.14159... Given that by ‘number’ we mean finite number, it seems that the third expression can’t denote a number. So those three expressions denote only pi and six. But then the sum of the numbers denoted by those three expressions is 9.14159..., and so the third sentence does seem to denote a number after all. Or rather, it does because it doesn’t; and furthermore, it seems to denote 9.14159..., but therefore it seems to denote 18.283..., or rather, 27 and a bit, etc.
......I have, however, been implicitly assuming that reference is an all-or-nothing affair. Usually we can – and indeed, should – take it to be so, but is it so in general? Imagine, for example, a man staggering through a desert. He sees a mirage, which he takes to be a pool, and as it happens there is a pool, just where he takes one to be, but it’s obscured from his view by the mirage. As he staggers towards it, he’s constantly thinking ‘that pool looks cool’. As he nears the pool, its image gradually replaces the illusory one, without him noticing, so that the referent of ‘that pool’ gradually changes to the pool. And so at some point he may have referred only vaguely to it.
......Such a case would of course be exceptional, but so are scenarios designed to be paradoxical. And it does at least seem possible that the third expression of Simmons’ paradox referred only vaguely to 9.14159... It would also have referred, even more vaguely, to 18.283 (and so on), but while that’s even odder, it too seems possible. And if the alternative is paradoxical, vague reference may not be too odd. And such a resolution would cohere with ‘this is not true’ being vaguely true (about as true as not), and ‘is heterological’ being as heterological as not (see my previous posts this year).

Sunday, March 06, 2011

God is Timeless

When people say that God is timeless, they may mean many things. They may mean that He is above and beyond the mundane world, like the truths of mathematics, for example. But they would not then be disagreeing with Open Theism:
......Under Open Theism, God is certainly above and beyond His creation, a bit like a dreamer and the dream he finds himself within. And one of the few coherent philosophies of mathematics is Open Theistic Constructivism, in which God, being omnipotent, creates the truths of mathematics—from the basic concepts of a thing and of possibility (the latter grounded in His omnipotence)—doing so endlessly because such is the nature of the former concept, according to the resolution of Cantor’s paradox that takes it to be showing that cardinal numbers are collectively indefinitely extensible (one of the few coherent resolutions).
......Of course, they may instead mean that God is not changeable. But even that isn’t incompatible with Open Theism, under which God cannot change his essential properties. Nor can you, of course. You can’t become me, for example. You might change by becoming in a manner of speaking a different person (i.e. your character might change, for better or for worse), but under Open Theism God’s character remains perfect. So the only disagreement with Open Theism could be that such philosophers are denying that God could choose to cause any real change in anything; and not only is that clearly not what most believers mean when they say that God is timeless (is rather more like an absurd denial of the reality of change), such philosophers would be denying God’s omnipotence:
......The deliberate creation of anything contingent surely requires several real possibilities to choose between, as well as a single actuality amidst counterfactuals, and hence some sort of change (not necessarily one that takes place within spacetime).

Tuesday, March 01, 2011

Only the unfit evolve

From Philosophy Now, I learn 'that people and organisations that miss their goals disastrously perform better in the long run.'
Professor Desai, who led the study, said “knowledge gained from success was often fleeting while knowledge from failure often stuck around for years.” [...] He says failure causes a company to search for solutions and it puts the executives in a more open mindset. He doesn’t recommend seeking out failure in order to learn.
...which makes me think of the merchant bankers. Will they perform better now? Only if we make them (despite our politics being dominated by the short-term), I think. You know, their performance was always chaotic, even in the good old days, as the new maths of chaos showed us in the 80's. But they took past success to be indicate a propensity for success; ironically, they were bad at applied math.
......I find that ironic because it was pointed out to me, when I was reading physics (in the 80's), that investment banking was the sort of 'good job' that a physics degree qualified one for. (That was what made physics such a good degree.) At the time, I wondered whether we really would've been a better society had more women wanted to play such games. (Margaret Thatcher, with her Chemistry degree, hardly thought of investment banking as a waste of a physics degree.)
......What can you do? The young are always with us. And we can hardly want to dilute democracy, but, why not give more experienced voters additional votes? Perhaps we shouldn't take the vote away from those who've shown themselves to be very selfish (unless we've locked them up and thrown away the key), but why not give an extra vote to all those who haven't (yet) shown themselves up; indeed, why not give extra votes to those who deserve honours? Wouldn't that be fairer? (It would certainly be safer.)