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.
Note that we do not need to model such changes with real numbers. (Cf. when we know nothing about a situation except that it is either A or B. We might think that we have 50% credence for each, but the next thing that we find out might be that B can be divided into Bi and Bii; do we then have 50%, 25% and 25%, for A, Bi and Bii, or do we have 34%, 33% and 33%?)
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.

Saturday, August 18, 2018

The Modal Paradox

Here in the actual world A we have a ship, let us name it the good ship Theseus, made of 1000 planks. Our first intuition X is that the same ship could have been made of 999 of these planks plus a replacement for plank #473. In possible-worlds terms that means there is another world B where the same good ship Theseus exists with all but one plank the same as in our world A, and only plank #473 different. But then in world B one has a good ship Theseus made of 1000 planks, and by the same sort of intuition, there must another world C where the same good ship Theseus exists with all but one plank the same as in the world B, but with plank #692 different. That means for us back in world A there is another world C where the good ship Theseus exists with all but two planks the same as in our world A, but with planks #473 and #692 different, so one could have two planks different and still have the same ship. The same sort of considerations can then be used to argue that one could have three planks different, or four, or five, or all 1000. But that is contrary to our other intuition Y [a ship made of a thousand different planks would have been a different ship].
     The modal paradox resembles well-known paradoxes of vagueness, such as the heap and the bald one, for which proposed solutions are a dime a dozen — except that here what seems to be vague is the relation of identity. And the idea that ‘is the very same thing as’ could be vague is for many a far more troubling idea than the idea that ‘heap’ or “bald’ is vague. Indeed, according to many, it is an outright incoherent idea.
From John P. Burgess, "Modal Logic, In the Modal Sense of Modality" pp. 40-1.
     In the fourth line Burgess says "by the same sort of intuition," which is a relatively weak sort of thing to say, and so it might be where the paradoxical reasoning started to go wrong. Our original intuition X was that in the actual world A, Theseus could have had one plank different and still have been the same ship. We know, equally intuitively, that X coheres perfectly well (somehow) with the intuition, Y, that in the actual world A, Theseus could not have a thousand planks different and remain the same ship. Whereas the ship in B is not exactly the same as the ship in A. And the meanings of all our words are rooted in A. The further we get from A, the more vague we might expect our meanings to become.
     In the first line of the second paragraph Burgess observes that proposed solutions to the paradoxes of vagueness are "a dime a dozen" and that means, I think, that anything I might say is bound to pointlessness; but, onward and upwards. And it is certainly the case that "the very same thing" often equivocates, as in the case of the famous clay statue: if that thing is squashed then, while it is the same lump of clay, it is no longer a statue at all. So let us look at a very different formulation of the ship paradox, one that does not involve modality at all. The following is by Ryan Wasserman, "Material Constitution", §1:
[...] the story of the famous ship of Theseus, which was displayed in Athens for many centuries. Over time, the ship’s planks wore down and were gradually replaced. [...] Suppose that a custodian collects the original planks as they are removed from the ship and later puts them back together in the original arrangement. In this version of the story, we are left with two seafaring vessels, one on display in Athens and one in the possession of the custodian. But where is the famous Ship of Theseus? Some will say that the ship is with the museum, since ships can survive the complete replacement of parts, provided that the change is sufficiently gradual. Others will say that the ship is with the custodian, since ships can survive being disassembled and reassembled. Both answers seems right, but this leads to the surprising conclusion that, at the end of the story, the ship of Theseus is in two places at once. More generally, the argument suggests that it is possible for one material object to exist in two places at the same time. We get an equally implausible result by working backwards: There are clearly two ships at the end of the story. Each of those ships was also around at the beginning of the story, for the reasons just given. So, at the beginning of the story, there were actually two ships of Theseus occupying the same place at the same time, one of which would go on to the museum and one of which would enter into the care of the custodian.
For myself, I do not think that the ship in the museum was the famous Ship of Theseus, I think that what was left of that ship is now the custodian's ship. But I concede that it could be that the museum ship is legally the ship of Theseus. It would then follow that the custodian's ship was not, for legal purposes, the ship of Theseus. So I think that there are at least two senses of "ship of Theseus" in play. What we can say about those senses is another matter. Our language is inextricably rooted in the usual events of the actual world. But it could be scientific to know that there are those two senses even before our theories of such senses have become a dime a dozen. And we might find clues as to what we should be saying from related puzzles.
     There are many intuitive puzzles about identity. For just one example, suppose that the world that you are in splits into two worlds, so that you are in one while an identical person is in the other. You are the same person as the original you, of course, but so is the person in the other world. So, that other person is the very same person as the original you, who is the very same person as you, and yet that other person is not the very same person as you. So, either personal identity is not always a transitive relation, or else this scenario is impossible. And while the latter is certainly plausible, it seems to me to be the wrong sort of conclusion to draw from this scenario. Knowing whether it is the wrong sort of conclusion to draw or not may well be a prerequisite for getting anywhere with such puzzles as the modal paradox, because intuitions for "is the very same thing as" not ever being at all vague seem to be very similar to intuitions that it must always be transitive.

Tuesday, August 14, 2018

Force and Foreseeability

Some thinkers think that if there is a God, then God will know all about the future, because otherwise bad things might happen. About ten years ago I spent a few years trying to refute one such view as neatly as possible (see my result here), during which attempt I found a new theodicy (which I called "The Odyssey Theodicy" for no good reason) and discovered the mathematical proof that there is a God (who is not immutable) that I have recently been tidying up. Today I thought of this title to go with my original refutation; basically, my original thought was that God's power over God's creation gives God plenty of ability to know that good will definitely happen, without God needing to know all about the future. However, despite now having the sort of title that I like, for my thought, I find that I now have little interest in expressing as neatly as possible such academic thoughts. That is because my thought is so obvious that the view that I was refuting must have existed for some other reason than simply not knowing that thought. Could that view not have been clearer about its reasons, I wonder. But, that is just the academic way, it seems. I also now think that finding new theodicies is pretty pointless, though; consider this analogy: it is the first day of school, and things do not go well. And of course, you learn very little; but of course, that is no reason to have no first day of school. And the evidence that, if there is a Creator of all things, then it is an evil Creator, is a bit like that: if all of this was created by such a power, then there is very likely to be life after death (like further school days after the first, and then life after school, a life enhanced by prior schooling) because that would be better, and no less possible than this life; and so the worse this life is, the more likely there is to be life after death, if there is a God. The logic of such arguments is simple, and undeniable, and so the way the problem of evil is hyped up by mainstream analytic philosophers of religion is, clearly, pure rhetoric.