To my mind, the mistake occurs here the moment you start speaking of "the image spoon". There is no "image spoon". There is just a spoon, and it is in a glass, and you see it. (So the real spoon is in the glass, right where you thought it was.) The spoon looks to be bent, certainly, so perhaps it follows that there is an image of a bent spoon in your head. (Some philosophers would deny that, but I don't think we have to deny it, or should deny it, in order to resolve this puzzle.) But the image of a spoon is not a spoon, and it is not bent, any more than, if I draw a picture of a bent spoon, the graphite somehow becomes a spoon or becomes bent. It's just a picture of a bent spoon, made out of graphite on paper. Nor, most philosophers would hold, is the image of the spoon what you see. What you see is the spoon, in the glass. Perhaps you see the spoon in the glass in virtue of the fact that you have an image of a spoon in your head, but that is a different matter. Please don't think I'm saying this...

I lost you right here: "this second relation of self-similarity must be self-similar". What is the "second" relation? I thought it was just the relation of self-similarity, which, as you say, the relation of self-similarity presumably has to itself, since everything is self-similar. It's perhaps worth noting that we can construct an analogous set of questions using identity rather than self-similarity: Everything is identical with itself; so the relation of identity is, as a relation, also identical with itself. But again, the relation identity has to itself is just identity: The relation of identity bears itself to itself. So it doesn't seem to me that this line of argument requires "the Universe [to be] crammed with an infinity of relations of self-similarity", and I'm not sure why it seems so obvious that it isn't. That said, however, the kind of language you use here—talking of a relation standing in itself to itself—can cause problems. Some relations, as we have just seen, stand to...

I'd suggest that this puzzle is largely a linguistic one. Consider the relation being larger than . Can one perceive that relation? There's a temptation to say that one cannot perceive the relation itself , because the relation itself "has no color or visual size or shape", and so on and so forth. And maybe that's so. Ask a metaphysician. (Of course, what answer you get will depend upon which metaphysician you ask!) But the examples with which you began suggest a different question. Can one perceive that one thing is larger than another? Here, it seems to me, the answer is clearly that one can. We perceive that kind of thing all the time. But how can we perceive the relation if we can't perceive the relation itself? The answer, I think, is that this question is just confused. What we perceive is that the objects are so related . Perception, as people sometimes put it, has propositional content, and relations figure in these contents. One might yet wonder how it is that we manage to...

It is crucial, I think, to recognize that the relevant question here is not: Are the lives of humans more valuable than the lives of (other) animals? The objection to killing animals need not presuppose that animals' lives and humans' lives are of equal value. Most defenders of animal rights would not, I think, hold such a view. Their claim, rather, is that animals' lives are of sufficiently great value that they ought not to be killed. Note that saying that animals ought not to be killed does not imply that it is never morally permissible to kill an animal. Humans ought not to be killed, but most people would hold that it is sometimes morally permissible to kill human beings, for example, in self-defense. If (say) cows lives are of less value than are the lives of humans, then there may be circumstances in which it is permissible to kill a cow but in which it would not be permissible to kill a human being. But it does not follow from that fact that it is permissible to kill a cow just...

Regarding your second question, whether the incompleteness theorem limits what we are capable of knowing, people disagree about this question. But the short answer is: There is no decent, short argument from the incompleteness theorem to that conclusion. If it does limit what we are capable of knowing, then it will take a very sophisticated argument to show that it does. One might think it followed from the theorem that we cannot prove that PA is consistent. But we can. I proved it yesterday, in fact, in my class on truth. The incompleteness theorem says only that we cannot prove that PA is consistent in PA , if PA is consistent. (If it's not, then we can prove in PA that PA is consistent! But that won't do us much good, since we can also prove in PA that PA is not consistent, and indeed prove absolutely everything else in PA, e.g., that 2+3 = 127.—This last remark assumes that we have classical logic at our disposal.) So when I proved that PA was consistent, I didn't do so in PA. I...

There are a lot of questions here. First, I think your assumption that "all religions claim they are right and all others wrong" is false. I am a Christian, and neither I nor, I think, anyone else at my church would make any such claim. I believe there is profound truth in Christianity, and it is the right form of faith for me. But that is not to say that there is no profound truth in other faiths nor that they are not the right forms of faith for others. Second, Pascal's wager, as I recall it, wasn't really specific to any one religion. Though if you did suppose that, in order to gain salvation, you had to follow one of the following five faiths, all of which were mutually exclusive, then you would be in a bit of a bind. But perhaps the conclusion should be that you should follow one of them. Which? Flip a five-sided coin. That said, I think the going view is that Pascal's wager had other problems, anyway. See the discussion of it at the Stanford Encyclopedia.

This question concerns, in effect, number-theoretic functions that grow very fast. We can say a lot about them. The operation in play here is called "superexponentiation", and is also known as "tetration". We can define it as follows: superexp(0) = 1 superexp(n+1) = 2^(superexp(n)) So superexp(4), e.g., is: 2^(2^(2^2))), and K is superexp(7). Noting that superexponentiation is just repeated exponentiation, we can now define superduperexponentiation as follows: supdupexp(0) = 1 supdupexp(n+1) = superexp(supdupexp(n)) supdupexp(4) is already an enormous number: It is superexp(2^16) = superexp(65,536) = 2^(2^...), where there are 65,536 2s in the tower. supdupexp(5) is unimaginably huge. It is 2^(2^...), where there are supdupexp(4) 2s in the tower. But there is no need to stop there. Let's rename superexponentiation 2-exponentiation, or exp 2 for short; and let's call superduperexponentiation 3-exponentiation, or exp 3 for short; and let's just write 2^x as exp ...

I'm not sure why that should be possible. Indeed, suppose we accept that it is not possible for an omnipotent being to make some contradiction true. Then—if we assume that anything possibly possible is possible (this is the modal axiom known as "4")—it follows immediately that such a being cannot make it possible for a contradiction to be true, either. If s'he could, then it would be possible that it was possible for a contradiction to be true, in which case it would be possible for a contradiction to be true, which it is not. That said, there are some philosophers who think that some contradictions are true, and they would have an easier time, I take it, with this kind of question.

Things get tricky anyway when you mix modal operators and the quantifiers, so it's best if we just leave it to the propositional case. And I'll add, just by the way, that it is quite controversial whether such "operator" treatments are correct for any of these cases, more so for tense, perhaps, than for the rest. Most semanticists nowadays, I believe, would take tense in natural language to be quantificational. And David Lewis, of course, held the same about "necessarily". There are really two kinds of questions here: Can one write down some plausible logical principles governing (say) a language with two such operators? And then, can one develop a semantics for this language and prove soundness and completeness? As for the former question, the interesting issue is what principles should connect the two kinds of operators. Presumably, for example, we should have "If it is necessary that p, then it is always the case that p", but not conversely; and I suppose some people would have us assume "If...

It seems to me that it is sufficient if there is a description of the act under which both parties consent to it. (I find myself tempted to say: ...and under which they both perform it. That may not be necessary but probably is.) Whether there are other descriptions under which one or another of the two parties has not consented to it seems irrelevant (especially if it is not a description under which they perform it): It will always be possible to find such descriptions. (Note that this is different from saying that there are descriptions under which one of the parties would withhold consent. Whether the existence of such descriptions would be relevant is a more difficult question.) If so, then the fact that X and Y, in the first example, happen to think of the consequences of their encounter in different terms does not seem to undermine their consenting to: having a sexual encounter. (And that, of course, is a description under which both of them perform the act.) There are undoubtedly...