Just a footnote to Mark Collier's helpful post. I actually said that irrationally formed beliefs are not likely to lead to actions which get us what we want (rather than cannot get us what we want). And that claim is enough to explain why we should in general care a lot about forming our beliefs in a rational way . Which in turn is enough to counter the original questioner's worry that philosophy "uses as its main tool a mechanism [rational thought] that is the opposite of what is most important to us": in general , rational belief-formation matters for getting whatever is important to us. Even if pockets of irrationality, episodes of self-deception, etc. can -- by good fortune -- happen to promote our welfare.

Well, mathematicians all the time talk about infinitary structures. To start with the very simplest examples, they talk about the set of all natural numbers, {0, 1, 2, 3, ...}: and they also talk about the set of all subsets of the natural numbers. And they introduce "infinite cardinal numbers" which indicate the size of such infinite sets. There's a beautiful theorem by Cantor which shows that, on a very natural understanding of "number of members", the set of all numbers has a smaller number of members than the set of all subsets of the set of all numbers. So there are different infinite cardinal numbers, which we can order in size. Indeed, again on natural assumptions, there's an infinity of them. Is that little reminder about what mathematicians get up to enough to settle the question whether "infinity exists"? Well, perhaps not. There remain a number of questions here. Here's one (so to speak) about the pure mathematics, and one about applied mathematics. First, then, we might say "Sure,...

A logically valid inference is one that is necessarily truth-preserving. That's pretty much a matter of definition. We just wouldn't count an inference as logically valid if it didn't meet that condition. (In other words, necessarily preserving truth is a necessary condition for an inference to count as valid. Perhaps it is sufficient too; or perhaps rather more is needed for a genuinely valid inference: e.g. some relation of relevance between the premisses and conclusion. But we needn't worry about that. For present purposes it is enough that it is agreed on all sides that to count as valid an inference must at least be truth-preserving.) Now, the "laws of logic" are general principles specifying which types of inference are indeed logically valid. So again, it is pretty much a matter of definition that instances of the laws of logic have to be truth-preserving. That's what it takes for a law to be a law of logic. It isn't as if we can first identify a class of principles as laws of logic...

"Often the questions seem to me of the utmost seriousness, but a serious response doesn't seem fashionable." The implication seems to be that serious answers aren't much in evidence here. Which is an extraordinary thing to say. For there is a really remarkable amount of good, patient, serious, philosophy here in the answers from my co-panellists. To be sure, their answers are often enviably zippy, witty, done with a light touch, with memorable examples. But seriousness in philosophy isn't at all the same as solemnity. As to the question: facts aren't the sort of thing that are rational or irrational. It's beliefs, belief-forming policies, methods of argument, desires shaped by our beliefs, the actions they lead to, and the like (and the people who have beliefs, belief-forming policies, etc.) which are rational or irrational. But the facts are the facts, and that's that. Some facts are really difficult to get our heads around intellectually (like the facts of quantum mechanics). Some facts,...

The background issue here is: what's the best way of extending our talk of one set's being "larger" than another from the familiar case of finite sets to the infinite case? Now, in the finite case, we can say that [L] the set A is larger than the set B if and only if there is a 1-1 mapping between B and some proper subset of A (i.e. there is an injection from B into A , which isn't onto). But equally, in the finite case, we can say that [C] the set A is larger than the set B if and only if there is a 1-1 mapping between B and some proper subset of A , but no 1-1 mapping between B and the whole of A. In finite cases, [L] and [C] come to just the same: but in infinite cases they peel apart. [L] says the set of naturals is larger than the set of evens; [C] denies that. So isn't [L] the more intuitive principle to adopt? No! For take the case where A and B are both the very same set, the natural numbers. There is a 1-1 mapping from the...

We seem naturally to be prone to over-interpret our environment and to see natural events as the results of intelligent agents at work. And you can see why our evolutionary history should have led to this cast of mind: it was much better for our ancestors to be too quick to diagnose potential agents around them (predators or other dangerous creatures) than to be too slow! So, we seem to be hardwired to be over-ready to see signs of agency in the world and to be susceptible to crediting supernatural explanations of natural events. And so it doesn't take much dedication on the part of those caught up with stories of the supernatural to keep them propagating. For more, much more, on these lines, see Dan Dennett's very readable Breaking the Spell . Now, Mark Collier and Eric Silverman both gesture towards this sort of answer to the original question, but they also both remark that this doesn't settle what to believe about God. Well, yes, not settle . But still, if you do come to think that the...

Here's a simple argument. (1) There are four prime numbers between 10 and 20. (2) But if there are four prime numbers between 10 and 20, then there certainly are prime numbers. (3) And if there are prime numbers, then there must indeed be numbers. Hence, from (1) to (3) we can conclude that (4) there are numbers. Hence (5) numbers exist. Where could that simple argument be challenged? (2) and (3) look compelling, and the inference from (1), (2) and (3) to (4) is evidently valid. So that leaves two possibilities. We can challenge the argument at the very end, and try to resist the move from (4) to (5), saying that while it is true that there are such things as numbers, it doesn't follow that numbers actually exist . This response, however, supposes that there is a distinction between there being X s and X s existing. But what distinction could this be? Well, someone could use mis-use "exists" to mean something like e.g. "physically exists": and of course it doesn't follow from...

There's no settled usage for the term "naturalism" in philosophy. But I guess that most of those who think of themselves as naturalists would say that we should recognize the sciences as the best way of finding out about the world and "refuse to recognize the authority of the philosopher who claims to know the truth from intuition, from insight into a world of ideas or into the nature of reason or the principles of being, or from whatever super-empirical source. There is no separate entrance to truth for philosophers."* On this view, philosophy's questions may be more sweeping than those of the special sciences or concern the relationships between different special sciences (and perhaps other forms of enquiry), but the methods of good philosophy are continuous with those of science. Now, that's still not very specific, but you can already see that naturalism understood this way involves something a lot more than just unfriendliness towards the supernatural. You could cheerfully endorse the...

Suppose I think (a) that it is normally wrong to kill humans, because so doing deprives them of a future life. But I also think (b) women have a "right to choose", and it is permissible to have at least a reasonably early abortion. Then I seem to be in trouble. For by (a) killing a very young human being in utero should be wrong, as it surely deprives it of the long future life it would otherwise have had, while by (b) killing it is permissible. On the face of it, then, my moral views (a) and (b) aren't consistent with each other, but imply that a certain act is both wrong and not wrong -- which is absurd. And note, I can't just shrug my shoulders and cheerfully say "ok, mymoral views are inconsistent" because inconsistent views don't give me anyguidance about what to do, and my moral views are supposed to help guide me! I want to decide to do in various circumstances, and inconsistent moral injunctions are no use at all for deciding. So I need to revise (a) or revise (b), or at least spell out ...

Eric Silverman's reply, if anything, sounds over-optimistic. Certainly, in the UK, there is a now a very significant over-supply of completing philosophy PhDs. I'm afraid that "average decent" doesn't cut it on the job market. Good perhaps for the future of philosophy, but tough on a large number of people who entered into grad school with hopes of an academic career.