Entropy mod p - What is entropy modulo a prime number?
1 hour ago
a 'Meaning'-full name
no man might buy or sell, save he that had the mark, or the name of the beast, or the number of his nameEven the BBC (who are so not supposed to advertise that they say "sticky backed plastic," instead of sellotape) show us the Facebook symbol (f ).
Suppose that your friend tells you: “If what I’m saying using this very sentence is true, then time is infinite”. It turns out that there is a short and seemingly compelling argument for the following conclusion:That may look rather formal to you, but formal logic is not even logic (it is mathematics); the above is just very well laid out. Note the two uses of modus ponens, the two sets of three steps, with the first three steps, (1), (2) and (3), all beginning “Under the supposition that”. You should note that because we cannot always use modus ponens within the scope of a supposition, e.g.:
(P) The mere existence of your friend’s assertion entails (or has as a consequence) that time is infinite.
Many hold that (P) is beyond belief (and, in that sense, paradoxical), even if time is indeed infinite.
Here is the argument for (P). Let k be the self-referential sentence your friend uttered, simplified somewhat so that it reads “If k is true then time is infinite”. In view of what k says, we know this much:
(1) Under the supposition that k is true, it is the case that if k is true then time is infinite.
But, of course, we also have
(2) Under the supposition that k is true, it is the case that k is true.
Under the supposition that k is true, we have thus derived a conditional together with its antecedent. Using modus ponens within the scope of the supposition, we now derive the conditional’s consequent under that same supposition:
(3) Under the supposition that k is true, it is the case that time is infinite.
The rule of conditional proof now entitles us to affirm a conditional with our supposition as antecedent:
(4) If k is true then time is infinite.
But, since (4) just is k itself, we thus have
(5) k is true.
Finally, putting (4) and (5) together by modus ponens, we get
(6) Time is infinite.
We seem to have established that time is infinite using no assumptions beyond the existence of the self-referential sentence k, along with the seemingly obvious principles about truth that took us to (1) and also from (4) to (5).
Let your friend say instead: “If what I’m saying using this very sentence is true, then all numbers are prime”. Now, mutatis mutandis, the same short and seemingly compelling argument yields (Q):My suspicion is based on the fact that one could conceivably have a valid argument for
(Q) The mere existence of your friend’s assertion entails (or has as a consequence) that all numbers are prime.
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.
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].From John P. Burgess, "Modal Logic, In the Modal Sense of Modality" pp. 40-1.
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.
[...] 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.