Also, if I were to tie your hand behind your back and then ask you whether you can touch your nose with it, that would be a peculiar question. And something similar is going on when one's asked whether one can defend all one's principles of reasoning. The whole practice of defending something assumes that principles of reasoning are in place. In fact, a cogito -like situation is indeed present: a state of affairs holds (thinking, defending) which demands presuppositions (existing, acceptance of rules) that make a certain doubt (about existence, about logic) self-stultifying.

Right, so it seems you think the argument is self-undermining. It assumes that you can get to the midpoint, C, and then it goes on to prove that motion from C to the endpoint B is impossible. Maybe we need to rethink our assumption that we could get to C! And indeed, other versions of this paradox of Zeno's work in that way. In order to get from A to B, this version runs, we need to get to the midpoint C. But in order to get from A to C, we need to that interval's midpoint, C1. And in order to get from A to C1, we need to get to its midpoint C2, ad infinitum . The strategy is always the same: to find a way of taking something finite (in this case, the racetrack) and dividing it into infinitely many parts; then arguing that a related task (here, running to the finish line) that looked to be finite really involves an actual infinite number of subtasks (here, reaching all the midpoints); and then concluding that, because one cannot complete an infinite number of tasks, the...

In answer to the question, whether a fair coin's flips might approach 75% heads and 25% tails, I'd like to say No. For the trivial reason that approaching 50% heads/50% tails is just what we mean by "fair coin."

Easier asked than answered. Philosophers are people who do philosophy, and so your question really amounts to what philosophy is. The problem is that what philosophy is is itself a philosophical question. Many grand disputes in the history of philosophy can be viewed as conflicts over how to proceed in philosophy, over what the rules of the game are, over what philosophy's method and central problems ought to be. So, I suppose one "definition" of philosophy might simply be the discipline which takes as its subject its own nature . Of course, that has a circular whiff about that as well. Philosophy is a little like pornography: hard to define, but one knows it when one sees it. But really, most concepts are like pornography in that respect: there's pretty widespread agreement about how they apply, but little hope of working out independently intelligible definitions. That fact itself is something of great philosophical interest. But don't ask me to define precisely what I mean...

That's a good question and a really good answer would have to involve getting into the details of Wittgenstein's thought. But perhaps one thing that might be said at the outset is that saying something clearly and saying something that's easy to understand are not the same thing. A good textbook in advanced mathematics contains many clear statements – but that doesn't mean it's easy for anyone to understand them. Whether something's easy to understand depends in part on the clarity of the thought's expression, but also on the subtlety or complexity of the thought being expressed.

First, there's a difference between showing that an argument for permitting homosexuality is bad and showing that homosexuality shouldn't be permitted. To show the latter, you need an argument to that very conclusion; it won't do to show that some argument for permitting homosexuality is actually a bad argument. Refuting an argument in support of X is different from giving an argument for not-X. Second, I'm not sure you succeed in even showing that the argument you consider for permitting homosexuality is a bad one. You want to say that if the argument were correct then it would also permit incest; since the latter shouldn't be permitted, something must be wrong with the argument. But I'm not sure I agree that the argument you consider about homosexuality really would also apply to incest. That's because I think that incestuous relationships do often lead to psychic harm for one or both of the individuals involved (as opposed to homosexual relationship, which don't lead to such harm – at...

I would say that a well-taught first course in formal logic would presuppose no mathematical knowledge and no mathematical sophistication. The material is technical in nature and freely employs symbols and terms drawn from the vernacular of mathematics -- but all these should be explained by your instructor. From teaching this material many times, I know that some "math phobic" students freeze up at the sight of parentheses and greek letters. But I always tell them that these symbols are their friends! They serve to make our lives easier by allowing us to say just what we want to say very briefly and perspicuously. We could in principle do without them -- but that would in fact make our lives a lot harder.

I'm not so confident that perusing a panelist's publications will helpthe layperson see which questions animate the philosopher. Those publicationsusually begin far into a long conversation and it might be hard to find in themthe simple questions that kicked off the discussion in the first place. So I'll place one on the table. I'd love to have a satisfying answer to this question: "What are we saying when we claim that 5+3=8?" Ilike Thomas' suggestion though. I'll start: I'm not sure I understandwhy many colleagues are as convinced as they are that work in empiricalpsychology is relevant (if not central) to many long-standingquintessentially philosophical questions. It seems to me that the 20th Century saw a number of subtle criticisms of this conviction that have been more ignored than answered. (I'm not assuming its irrelevance. I'm just puzzled at times by the confident, or at least untroubled, assumption of relevance, especially in the face of powerful dissents.)

When a set theorist says that such and such collection is "too big" to be a set, what he typically means is that if that collection were taken to be a set a contradiction would arise. The collection of all sets is such a collection. If we assume it's a set then, applying the argument that generates Russell's Paradox, we arrive at a contradiction. And so we conclude that we were wrong to assume that the collection of all sets is a set. As set theorists put it, it's a proper class , not a set. Is there any way of telling whether a collection is "too big", i.e., a proper class, or whether it's a set, besides seeing whether the assumption that it's a set leads to a contradiction? No. So really, "too big" is just a colorful way of saying "leads to a contradiction if assumed to be a set".

For a relevant entry, see Question 264 .