keithprosser2 wrote:Zeno's arrow and the unexpected hanging (or exam) are examples of paradoxes that seem to be based on water-tight logic but lead to incorrect conclusions. Two things strike me - the first is what exactly is wrong with the arguments (if anything) and how sure can we be that many other apparently logic proofs are in fact invalid.
I have thought about this a fair bit but not come up with anything much - can anyone do better?
Zeno's paradoxes are due to the fact that we tends to see the world in units when it is in fact analogue.
Even our perception of the analogue is wrought with arbitrary units that do not exist. And our entire understanding of maths is based in integers even when we are trying to represent values which have no exact numericality. The unit is a form of useful delusion that leads to Zeno's problems' intuition tells us that there is no problem - arrows always continue, Achilleus always beats the tortoise.
Last edited by chaz wyman on Sat Jan 07, 2012 12:06 am, edited 1 time in total.
Thundril wrote:We cannot have absolute certainty. But we do pretty well without it. Logic is one of our attempts to get as close as dammit to absolute certainty, and as such is does OK most of the time. The paradoxes are an example of where at some points in history we bump up against the limits of our understanding so far. Then we figure out other stuff that moves us on a bit. Eg calculus.
Logic does not get us nearer certainty. All it does is extend the range of our premises. We can be no more certain of them that when we start out. We can know that GIVEN X also Y. But that just begs the question X. and also asked the question Y.
Logic is by its nature circular.
This isn’t really a paradox for me, in the sense I usually understand “paradox”, so much as an ill-defined situation. Below, I explain a solution based on a reasonable or practical operational definition of what it means for the prisoner to be surprised. This approach leads to the unsurprising conclusion that the prisoner will fail to be surprised only if Friday of the fatal week is the day picked for his hanging.
Before getting to my solution, I want to point out that there are possible definitions of “surprise” that would lead to other conclusions. For example, if on Sunday before the fatal week the prisoner is unable to predict the day of the following week he’ll be hanged, and surprise means having on Sunday less than certain knowledge of the day of hanging, then the prisoner is surprised for sure. As another example, if the judge picks the day at random (each day gets probability 1/5 of being selected) and the prisoner is told this, and “surprise” means not having a even a clue, then the prisoner will not be “surprised” (at least not completely so).
Here’s my solution:
Assume the judge assigns the day of hanging by mixing up in a hat 5 marbles labeled Monday through Friday, one different day assigned to each, and drawing a marble out of the hat to determine the day. Of course the prisoner is assumed never to know the outcome of this drawing until the day of the hanging. Now define “prisoner surprise” as the prisoner not being able to predict with certainty by the end of day X that the next day is the fatal day.
Conclusion: The prisoner will be surprised if the hanging occurs any day but Friday. For if he gets through Thursday without being strung up, he knows by the end of Thursday that Friday is the day he’ll be hanged (barring having a fatal heart attack earlier in the week, in which case he might be "hung" up to frighten onlookers).
It’s true that if Thursday is the fatal day, by the end of Wednesday the prisoner knows that hanging day is either Thursday or Friday, so in a sense he’ll be only mildly surprised if the knock on the door comes on Thursday. This can be said for any of the other days, although the prisoner’s surprise becomes milder as the week progresses without being hanged. But my definition of “surprise” stipulates not having certain (absolute) knowledge of the day, so the prisoner is still surprised in these cases.
If the prisoner is innocent, the prisoner can only hope Friday was the day drawn from the hat, and that the judge stipulated that the prisoner is to be released if not surprised.
Without getting technical, the flaw in the prisoner’s thinking in the paradox as stated is that it was wishful thinking. The prisoner had no reasonable model for how his fateful day was picked, nor for what he could know about that day as the week progressed for him (if it did).