Sunday, August 26, 2018

Sequence and Consequence


I add grains of sand to the same place, one by one, and eventually I have a heap of sand.

Originally I did not have a heap of sand, of course, and in between having such numbers of grains – just a few originally, and later on lots of them – there are numbers of grains with which I do not obviously have a heap, and do not obviously not have a heap.

Let n be some such number. Perhaps it is the case that when I have n grains of sand I have a heap of sand; but certainly, if that is not the case, then n grains is not a heap. So for all n, either n grains is a heap, or else n grains is not a heap. It follows logically that there must be some n, say N, such that N is large enough for N grains of sand to be a heap of sand, but (N – 1) is not large enough for (N – 1) grains of sand to be a heap of sand.

The problem is that the rather vague meaning of “heap” does not allow there to be such a number. Given any heap of sand, if I take just one grain away from it, then I would still have a heap of sand. Quite generally, if you take one unit away from any very large number of units, then you still have a very large number of units. So it must, after all, be false that for all n either n grains is a heap or else not. That is surprising, but at least we have a picture of how that can be false, with this picture of my adding grains of sand one by one. So while it is surprising, it is not beyond belief; it is not paradoxical (despite what many analytic philosophers seem to think).

It is not, then, always the case that, given some description, either that description is true or else, if that is not the case, the description is not true. It could be that there is no fact of the matter of whether the description is true or not, as when meanings are a little vague and we have a borderline case. Now, in such cases the description cannot simply be neither true nor not true, as that is just to say that it is not true while ruling out its being not true. The only thing left for us to say is that it is about as true as not. In the picture above, the sand changes from not being a heap, to being about as much a heap as not, and then more a heap than not. That is just a fact, of such matters.

One consequence of the possibility of assertions being about as true as not is that there is a relatively simple resolution of the Liar paradox.
Note that it can indeed make sense to say that a proposition is about as true as not. Consider, for another example, how if I say of some artwork that I think good “That is not good” then I am lying. I am saying something false. What would be true would be for me to say that it was good. And if some artwork seems to me to be about as good as not – and you must allow me such a possibility, because such matters are matters of opinion – then it would be true for me to say that it was about as good as not. In such a case, it might make sense (as follows) for it to be about as true as not for me to say that it was good. And if so, and if we all agreed that a particular artwork, say Z, was about as good as not, then it would make sense for “Z is good” to be about as true as not. Does that make sense? Well, were it simply true to say that Z was good, then were “Z is not good” true too, it would follow that Z was good and not good, whereas the symmetry of Z being about as good as not means that we could hardly have one true and the other not true. And if it was instead not true to say that Z was good, so that it would not be the case that Z was good, then there would be a problem with it being not true to say that Z was not good, because that would mean that Z was good.

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