Monday, March 30, 2009

Resolution of the "Unexpected Hanging Paradox"

The Unexpected Hanging Paradox:

Here is a short version of a paradox that has intrigued me since I was in college: A condemned prisoner is told by the warden on a Sunday evening that he will be executed at noon on one of the upcoming weekdays, i.e., M-Tu-W-Th or F, but he will not know in advance which day it is until the time comes. The day of his hanging will be a “surprise”. The prisoner reasons that if Thursday noon passes and he has not yet been hung, then he can rule out Friday since that is the only day left and then he would know what day he was to die, in violation of the conditions specified by the warden. But having crossed Friday off the list, he quickly realizes that he can cross Thursday off too, since once Wednesday noon is past Thursday is the only day left (since he has already eliminated Friday). Going on this way, he realizes that he can eliminate all 5 weekdays, one by one, and he is happy to realize that he cannot therefore be hung (let’s say that he strongly believes the warden to be a truthful man). But here is the (sad) paradox…the executioner comes to his cell on (say) that Wednesday, and informs him that he is now to be hung. His last thoughts before dying are that the warden indeed spoke the truth: that he, the prisoner, was indeed going to be hung, and he was certainly surprised by it too. Where did his reasoning go wrong? It seemed so logical that he could not be hung if the warden was telling the truth……and yet, in the end, the warden did tell the truth. Seemingly, a paradox!

The Resolution:

Here is my understanding of the resolution of the paradox (and I think I follow, essentially, Martin Gardner’s explanation from his book with the paradox as the title). The prisoners’ reasoning goes wrong on the very first step. He cannot rule out Friday even if Thursday noon has come and gone. In somewhat formal logic terms, I see it as follows. Let the wardens statement to the prisoner be written as C =A + B, where A = “you will be hung”, and B = “you will not know in advance of the day of the deed”. First, suppose A = true. That would mean that B = false, and hence C = A + B = false. Next, suppose instead that B = true. Then this would mean that either A or “not A” can be true; if A is true, then C = A + B is true, and if instead "not A" is true, then C = A + B is false. This makes sense because even if he doesn’t know the day of the hanging, then he could still either be hung or not hung. Thus, considering all of the above possibilities, we see that C can be true or false. It could just as well turn out to be true as it could turn out to be false. So, if it turns out after the fact that C is true, this does not really entail a contradiction. What the prisoner should have realized is that the warden’s statement could turn out to be true or false, and since he could not be sure ahead of time what the truth value was, he could not logically deduce the outcome from it.

Tuesday, March 24, 2009

More on the topic of concealed ovulation in humans

There was an entry in this blog a few weeks about what seemed to me to be a puzzle, namely, how can we understand “concealed ovulation” in humans from the perspective of evolutionary biology. I recently happened upon the book “The Third Chimpanzee” by evolutionary biologist Jared Diamond. He writes, on page 78 of my paperback copy, that “…the most hotly debated problem in the evolution of human reproduction is to explain why we ended up with concealed ovulation….”.

I suppose I must admit I found it gratifying that at least I wasn’t being a dunderhead in writing about this puzzle. That is, if distinguished professional biologists cannot agree on the reason for this trait, then a layman in the area (such as myself) certainly has the right to be puzzled by it.

At any rate, Diamond goes on to say that there are at least six current theories about why the trait evolved. As I understand his more lengthy explanations, it evolved for one of the following reasons:
1. To enhance cooperation among males (he considers this one to be rather male chauvinist for some reason)
2. To cement the bonds between mates
3. To enhance the probability of the female to receive her fair portion of meat from male hunters
4. To force the male into a permanent marriage bond (he considers this one to be “gender neutral” in a gender politics context)
5. To encourage seevral males to help her, and to at least not kill the infant (this one, by Sarah Hrdy, he applauds as a welcome “feminist” perspective)
6. Because a woman that had a tendency to conceal ovulation could not avoid the risk and pain of childbirth, and hence left more descendants that did those that did not conceal it (this is the theory that Nancy Burly suggested in about 1979, and the one that I mentioned as being a common sense explanation in my earlier post).

I find it rather regrettable that Diamond seems to place so much emphasis on being politically correct. For example, he approves of number 5 because it “overturns masculine sexism and transfers sexual power to women”. This seems an odd reason to like a hypothesis, for shouldn’t scientific truth be the only guide for favoring an explanation of any natural phenomena? Nevertheless, the several pages he devotes to this issue involve some intriguing speculation, and I find it quite interesting.

Sunday, March 8, 2009

Would you do it all over again?

A friend of mine recently asked me if I would live life all over again if I had a chance. Clearly an academic question, but it leads to some interesting speculations, and of course some juicy metaphysical questions.

He first phrased the question, or maybe I should say “posed the thought experiment”, in a way that suggested that I might be a baby again, identical to what I was at birth. Well, of course, right away that brings up the questions, “what is meant by identical”, and “would I know as a baby what I know now?”. Otherwise, I might note, I would probably repeat many of the mistakes and committed the same foolish acts that I have in this life. But what would it even mean, to have the knowledge that one has now? Surely, there is no way to model how a baby, in the true sense of that term, could have the conceptual knowledge that an adult has. A baby has to learn language and concepts, and so there just seems to be no way to make sense out of that idea.

It also raised the question of, “would the world go back to being the exact state it was in when I was a baby?” (we are having fun here, remember, not taking the possibility of this return to a baby thing seriously, but just dancing around some peculiar conceptual problems of identity that I want to return to in a minute). One might imagine, or for the present purposes model, one’s life as a movie, and this would amount to rewinding the film and having it played again. It seems like this would be a meaningless action to choose to do even if one could. What would be the point?…..wouldn’t everything just unfold again the same way (well, maybe, or maybe not, given the possibly chaotic nature of such a historical re-enacting)?

And one might even say, how do I know that hasn’t in fact been done, many times or an infinite number of times, already, in some sense (think of Nietzsche’s idea of “eternal recurrence”). But there would seem to be no point in it……and who would be behind it (what entity or being), and to what purpose? Improvement in one’s character? Hmmm---but without any memory, or at least faint recollections, of the previous cycles, how could one “learn from ones mistakes”?

Some of this territory was explored, or maybe I should say suggested to the philosophically minded viewer, in the film “Groundhog Day”, where a man (Bill Murray) lives one particular day over and over, only seemingly dimly aware that he has done it before, and has repeatedly botched it in some way (the idea was to get him to improve to a point where he was worthy of the female star that he hoped to win over, an unfortunately rather trivial spiritual “chick-flicky” motive for bettering oneself).

Now suppose one agreed to become a baby again, but without the amassed knowledge or memory of his life up through adulthood, and placed in the world in its present state. Well, now we move into a more practical area, as it is interesting to note that one could almost do that now (or probably would be able to in the not too distant future): one would simply have oneself cloned, then one would immediately commit suicide. The cloned cell would develop eventually into a baby, very nearly identical to what you looked like at birth (some inert gestational induced variations might be present, but these would presumably be very small). This illuminates the identity issue that of course your clone would not have your personal consciousness, and would not in any sense of the word, be “you”. But wouldn’t it be essentially the same kind of transformation of somehow magically returning the present you to a baby state, without implementing any of the acquired knowledge or character into the baby?

The same kind of issues arise with the concept of human teleportation, such as is depicted in “Star Trek” (“Beam me up Scotty”). How would one ever know that the teleported human had the essential consciousness, the same sense of being oneself as the original being had? This raises the question of what is consciousness, and what constitutes the peculiar fact of conscious identity. This even arises when we realize that our cells are replaced every so often, such that after several years we are not exactly the same person we were earlier. I suspect this is what bothered the ancient Greeks in their “ship paradox” (although they would have known nothing about cell development and replacement, they would have known that people change over time). In what sense do you retain your identity as your body sheds and acquires new “building blocks”?

These issues are explored very insightfully in Chapter 1 of the book “Riddles of Existence” (by Conee and Sider). In my opinion, they do not “solve” the myriad identity problems, but they do rather thoroughly explore the issues.