# Do infinite sets exist? Most mathematicians say yes, but to me it seems like infinite sets can only exist if we use inductive reasoning but not deductive reasoning. For example, in the set {1,2,3,4,...} we can't prove that the ... really means what we want it to. No one has shown that the universe doesn't implode before certain large enough "numbers" are ever glimpsed, so how can we say they exist as part of an "object" like a set. We can only do this by assuming the existence of the rest of the set since that seems logical base on our experience. But that seems like a rather weak argument.

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 ...

# If the sentence "If p then q" is true, must the sentence "q because p" also be true? For example, "if it is raining, then the streets are wet" and the sentence "The streets are wet because it is raining." Are there any counter-examples where "If p then q" could be true while "q because p" could be false?

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.

# Is it possible to make a legally meaningful distinction between porn which is abusive and porn which is very rough? I think this question is very relevant for our time.

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)...

# Could someone explain the Frege's puzzle? Is it directly related to semantic stuff? How?

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

# Many pro-choice advocates maintain that, though abortions should be permissible, they are regrettable nonetheless. For instance, Bill Clinton famously said that he wanted to keep abortions "safe, legal and rare." I don't understand this view. To my mind, whether abortion is immoral turns on the question of whether a fetus is a person with a right to life. But this seems a clear dichotomy--either fetuses have such a right, or they don't. If they do, then abortion is immoral. If they don't, then not only should abortion be permitted, but there is nothing objectionable about them at all. Indeed, it is every bit as innocuous as using condoms. Sometimes I think that what is happening is that people who advocate this position are still captive to some kind of residual pro-life sentiment. They believe that abortions should be permissible, but they can't shake the feeling that they are still, somehow, a bad thing. (And not just because of circumstantial considerations, such as that women who need abortions are...

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...

# Is it possible for a mathematical equation to both be fundamentally unsolvable and also have a correct answer?

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...

# Me and my professor are disagreeing about the nature of logic. He claims that logic is prescribes norms for correct reasoning, and is thus normative. I claim that logic is governed by a few axioms (just like any in any other discipline, i.e. science) that one is asked to accept, and then follows deductively, free of any normative claims. My question is: which side is more sound? Thank you.

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.

# There were some questions about vegetarian diets recently, and I'd like to ask a few follow-up questions if I may? First, what is the philosophy in favor of vegetarian diet? is it mostly that it is healthier, or is it moral objections to using animals for food? if the latter, how come so many vegetarians wear leather shoes and carry alligator bags? are they being poseurs or are they just superficial in their thinking? Second, if people object to the way cattle or chicken are raised to be slaughtered, that's fine if we don't want them to suffer. Eating shrimp, crab, insects, and the like would also give us plenty of protein we need for a healthy diet. Finally, in parts of the US prairie, protectect ungulate populations (deer, elk) have no natural predators. To prevent overbreeding which would lead to overgrazing which would lead to mass starvation, state Conservation Departments survey their ungulate populations every spring in order to determine how many hunting permits to issue each fall. If the...

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...

# I've heard some philosophers of mind use the term 'singular content'- but what does that mean?

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.

# I read once that an African tribe was asked a simple logical problem paraphrased as follows: "Berlin is a city in Germany. There are absolutely no camels in Germany. Are there camels in Berlin?" The tribe could not provide a definitive answer, instead saying things like "I have never been to Berlin, so I cannot say whether there are camels or not" or "If Berlin is a big city, there must be camels" in other words, completely missing the logical puzzle and instead providing more pragmatic answers. Now this story may be apocryphal, since I cannot find where I read it, but it raises an interesting question. To what extent is logic universal, is it culturally biased/culturally learned, and how do we explain the answers of the tribe?

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.