The argument here actually requires two more premises: (iv) that different numbers have different successors and (v) that 1 is not the successor of anything. If (v) failed, 1 could be its own successor and the only number. If (iv) failed, then 2 could be 1's successor and also its own. It's perhaps also worth noting that, although (ii)-(v) do imply that there are infinitely many numbers, it does not follow from them that there are sets that have infinitely many members. This is because (ii)-(v) say nothing about sets, and we cannot simply assume that there is a set containing all the infinitely many numbers. Analogues of (ii)-(v) hold in so-called hereditarily finite set theory, in which there are no infinite sets. (Indeed, one can consistently add the axiom "there are no infinite sets" to this theory.) Finally, the general observation that not everything can be proven does not imply that one can't reasonably ask that (ii)-(v) be proven, nor that one might not worry that, say, (iii) is to close to ...

Even if the conditional isn't material, it's clear that this kind of inference has to fail. Suppose my roof leaks whenever it rains. Then it seems true to say: If my roof is leaking, then the streets are wet. But the streets aren't wet because my roof is leaking. Rather, there is a third cause of both these events. Even if there has to be a "link" between them for the conditional to be true, then, the link needn't be directly causal.

Some people think that all pornography is, in some sense, abusive. Maybe that's even analytic, if one distinguishes "pornography" from "erotica", as many people do. For what it's worth, I doubt there's any very clean way to make that distinction (these look like what Bernard Williams called "thick" concepts, if ever there were any), so let me just talk about "sexually explicit media". I don't myself see any reason to think that sexually explicit media, as such, has to be abusive or degrading or necessarily bad in some other way, even if most sexually explicit media is in fact bad in some way, such as presenting abusive sex as unobjectionable or even normal. That said, some people like their sex rough, at least some of the time, and some people even like to roleplay situations in which one of the partners is abusive towards the other. None of that is very surprising. Human sexuality is varied and complex, just like humans. Rough sex, in that sense, is not abusive. It is (or at least can be)...

I could explain the puzzle, but there are already good explanations at the Stanford Encyclopedia of Philosophy and at the Internet Encyclopedia of Philosophy .

Thanks to everyone for their contributions, and especially to Bette for reminding us of the importance of hearing women's voices on such topics. I'll add one more point, along the same lines. The questioner says that, if a fetus has a right to life, then abortion is immoral and should not be permitted; if not, then it isn't immoral and should. But surely this is wrong. I have a right to free speech, but it does not mean that I have the right to cry "Fire!" in a crowded theater. Other people have rights, too, and their rights can sometimes out-weigh mine. The same is true in the case of abortion. The mere fact that the fetus has a right to life is compatible with a pregnant woman's having other rights that might out-weigh the fetus's right to life in some cases. For example, the woman herself has a right to life, and I for one have a very hard time seeing why that right should not trump the fetus's similar right if the pregnancy is endangering the women's life. Similarly, a woman has a right not...

To answer this question properly, we would need to make some of the terms used in the question more precise. Math only works with precise definitions. But there is a natural way to do this, and it does bring us close to Gödel's work. A diophantine equation is any equation of the form: f(x,y,z...,w) = 0 where f is a polynomial (i.e., something like x 3 + 3x 2 y 2 + 4xy 3 ) and the question is: Is there an integral solution to the equation ? I.e., a way of assigning integers (positive or negative whole numbers, or zero) to x, y, z, ..., and w so that the equation comes out true? One very famous such equation is: x 7 + y 7 = z 7 This is what people call "Fermat's Theorem for 7". We now know that it has, indeed, no integral solutions, and the same goes for any other prime exponent except 2. Diophantine equations crop up all over mathematics. So, in a famous lecture in 1900, the great mathematician David Hilbert posed the question: Is there some general...

Without disagreeing with Stephen's fine response, let me point out one other issue. You say that " logic is governed by a few axioms...and then follows deductively, without any normative claims". But t here is no "following deductively" without logic: logic is about the correct norms of deductive reasoning. So this conception is flatly circular: a point made a long time ago by Quine in his paper "Truth by Convention". I should say that there are philosophers who deny that logic is about reasoning at all. On this view, logic is about a certain relation between propositions, implication, that it aims to characterize. But then the dispute just shifts to whatever one thinks does characterize the norms of reasoning, e.g, decision theory. And, for what it's worth, my own view has always been that these philosophers have too simplistic a conception of what sorts of norms logic articulates. But that is a larger issue.

On (1): Different people have different reasons to be vegetarian. Besides the ones mentioned, there are many others. One important one, nowadays, is an environmental concern. Animal farms emit enormous amounts of greenhouse gases; they produce large amounts of pollution; etc. It's also true that animals raised for slaughter are fed a lot more protein (and other foodstuffs) than they will ever produce. They are, if one wants to think of them this way, very inefficient food factories. Regarding the latter part of (1), obviously this depends upon one's reasons, but most vegetarians I know would never carry an alligator bag. On (2), I'm not sure I understand the question, but perhaps the point is that shrimp, crabs, and insects do not plausibly suffer. If that is the point, I don't disagree, actually. If one's reason not to eat chicken, say, is that chickens are intelligent, sentient creatures, etc, etc, then this reason certainly does not apply to scallops, or shrimp, so far as I can see. There will...

The usual term would be something like "singular proposition", as opposed to a "general proposition". A singular proposition is one that is about some particular object. For example, the proposition that the Dalai Lama is German is a singular proposition. A general proposition would be something like: One and only one person is the spiritual leader of Tibetan Buddhism, and that person is German.

The claim that "logic is universal" is the claim that the norms of correct reasoning are universal. It is not the claim that everyone follows those norms, or that everyone reasons well. In the story as told (apocryphal or otherwise), the tribesmen are failing to make a certain inference. That makes them poor reasoners, but it doesn't threaten the universality of logic.