Tuesday, August 27, 2013

The Cookie Paradox

A friend of mine from graduate school, Dylan Dodd, discusses a new paradox (and how it relates to a similar paradox called the "Surprise Exam" paradox) in one of his new papers. You can find it at this link. Here's an abstract of his paper:
We've all been at parties where there's one cookie left on what was once a plate full of cookies, a cookie no one will eat simply because everyone is following a rule of etiquette, according to which you're not supposed to eat the last cookie. Or at least we think everyone is following this rule, but maybe not. In this paper I present a new paradox, the Cookie Paradox, which is an argument that seems to prove that in any situation in which everyone is truly following the rule, no one eats any cookies at all, no matter how many there are to be eaten. The `Cookie Argument' resemblance the more familiar argument that surprise exams are impossible, but it's not exactly the same. I argue that ultimately the biggest di difference is that, while every argument against the possibility of surprise exams (I discuss three) contains a subtle mistake, the Cookie Argument is actually sound! In addition to presenting this new paradox, I also present a new solution to a particularly difficult version of the Surprise Exam Paradox, the Conditionalized Exam Paradox.


Anonymous said...

Okay, I'll bite. And FCOL, leave the - it means we're supposed to share a laugh. If you don't eat the last cookie, your attention is focused on yourself and how you might be perceived. This is unhealthy at best, self-consumed (no pun intended) at worst. Eat the damn last cookie. It will go a long way to correcting the same sanctimonious sensiblities of your peers.

Anonymous said...

There seems to be a proclivity for logical paradoxes and a belief that scientific method (induction) aims at more than just predictability but reality. So, I suggest a treatment of confirmation: "The Raven Paradox" might, at the very least, entertain under this cloud of spirituality:

All ravens are black.
All non-black things are non-ravens (logically equivalent contraposition).

Most would tend to agree that discovery of a black raven confirms the proposition that "all ravens are black." Question: Does the discovery of a white piece of chalk, which seems to confirm the contraposed logical equivalent of "all ravens are black", do indeed confirm that "all ravens are black?"

Anonymous said...

To solve the raven paradox, you have to first come to terms with the idea that the proposition "all ravens are black" is a prediction and not a reality. Science models reality as a plurality - what Steven Hawking calls "model-dependant reality". The notion that the number of non-black things in a plurality is endless ( cf. infinite ) is conceptually misplaced. Infinity is not a number but a negative idea co-opted by mathematics to symbolize numbers as they grow very large. Modeling reality as atoms, quarks or strings will always result as a finite number of components and differ only by the sheer number of zeros. So, imagine a plurality comprised of only twenty discreet elements and you discover that one of them is a white piece of chalk. Regardless whether in this plurality ravens even exist, you have reduced the number of unknowns from twenty to nineteen; thereby, actually confirming - by not contradicting - the prediction "all ravens are black".