One further point. Toward the end, you write: These seem to be grave problems for theapplicability and effectiveness of FOPL to natural language arguments.(I am not referring to the “limits” of FOPL where extensions such asmodal, tense, or second-order logic might accommodate the richer partsof natural language, but rather to the apparent inability of anylogic(s) dealing with these problems.) Waiving the issue about vagueness, there isn't any problem dealing with such quantifiers in a second-order context. Both of the quantifiers I mentioned, "Most" and "Eq", can be defined in second-order logic, so the caveat at the end kind of gives the game away. That said, what perhaps is puzzling about these quantifiers is that, as is the case with second-order quantifiers, there is, as I said, no sound and complete set of rules for them, with respect to the intended semantics. In that sense, there is no "formal" logic for these quantifiers. But, again, that is not to say that one cannot write down some...

There are a lot of different questions here, and we need to disentangle some of them. First, some of the questions you are raising about "most", "few", and the like have nothing to do with their vagueness. Consider, for example, a quantifier I'll write "(Most x)(Fx;Gx)". This is what is called a binary quantifier (similar to your "restricted" quantifiers): Unlike the usual way of representing "all" and "some", it forms a formula from two open sentences. Now, define the quantifier, semantically, so that "(Most x)(Fx; Gx)" is true if, and only if, there are more Fs that are G than there are Fs that are non-G. (More generally, we'd have to talk about satisfaction, but waive this complication.) It can be proven that this quantifier cannot be expressed by any formula of FOPL. It can also be shown that there is no sound and complete axiomatization of the logic of this quantifier. That isn't to say you can't write down some sound rules. But you can't write down a complete set of rules: No matter...

No, there's no reason whatsoever that being religious should make someone less successful as a scientist. Whether one is a "person of faith" has nothing to do with whether one is capable of reason and the like. Any suggestion to the contrary is, frankly, not just insulting but ignorant. Moreover, the question contains several other assumptions that are simply false. First, a religious scientist need have no "need to prove his faith true" by scientific means. She may simply think that science and faith don't really intersect all that much, not because she "partitions" or "compartmentalizes", but for much the same reason she might think science and poetry don't intersect all that much. Second, a non-religious scientist may well have some irrational investment in, say, the truth of some hypothesis that she formulated as a graduate student and interpret all her data in terms of it. Being non-religious doesn't insulate one from bias. Third, it is simply a myth that scientists rely upon "reason and...

It's important to avoid a certain confusion here. One might say, about Alex's argument, that if there does not exist at least one true statement, then of course "There is a pen on my desk now" is not itself a true statement; hence the argument is circular. Of course, the first part is true; but the conclusion does not follow. For the argument to be circular, the claim "There is at least one true statement" would need to be used in that argument, but it is not. What is used in the argument are simply (analogues of) the following two premises: (i) snow is white; (ii) if snow is white, then "snow is white" is true. Alex claims to know (i) by observation; it's less obvious how we know (ii), but one who claims to know it seems on pretty firm ground. From (i) and (ii), then, it follows that "snow is white" is true and so, by a simple logical inference, that there is at least one true statement (viz, "snow is white"). To challenge this argument, one must either challenge (i) or (ii) or one must find...

I don't myself see that there is any real distinction between philosophers and "philosophologists". I've never even encountered the latter term before. It's hard to imagine doing philosophy without reading, understanding, and criticizing it, and I don't honestly know how one could read, understand, and criticize philosophy without doing it---unless, I suppose, one was some kind of skeptic about philosophy.

Stewart Shapiro's Thinking About Mathematics . ( On Amazon .) It's a well-written textbook that covers most of the basics.

I'm not sure why it seems to you that the existence of the sub-conscious could never be established. The idea is simply that our behavior is in part caused by mental states of which we are not consciously aware. Perhaps a better example would be anger: I might come to realize at some point that I have long been angry with X for something he did, and that this anger, of which I'd not previously been consciously aware as such, has been causing me to behave badly towards X. That seems pretty unproblematic to me. It also isn't obvious there is any real difference with the case of Pluto. Suppose you think that mental states are, ultimately, physical states, states of the brain. Then if we knew more about the brain, perhaps we could verify the existence of sub-conscious states like that one.

I'm no expert on art, just someone who enjoys it, but I certainly would agree with you that Picasso can be hard to understand. Most of his painting (and sculpture) isn't what one would describe as "beautiful", though there are paintings of his that are beautiful: For example, "Child with a Dove" (see it here ). But what's beautiful about paintings like this one, to my mind, is what they convey emotionally and less anything to do with sheer physical beauty. And one finds a similar sort of emotional intensity in many of Picasso's other works. His portrait of Gertrude Stein ( here ) is just brilliant. Now, to be sure, Picasso's later works can be more challenging. It can be very hard even to see what's happening in paintings like "The Guitar Player" ( here ) or "Afficionado" ( here ). And, in this case, I think it can help a great deal, as one tries to learn how to see these paintings, to learn something about the aesthetic that lies behind them. Picasso did not decide to paint in the ways he...

Most dualists hold that the mind acts through the brain somehow (assuming they hold that the mind "acts"). Hence brain damage would diminish the mind's ability to act, much as damage to other parts of one's body might. Most dualists (but not epiphenomenalists) would also hold that changes in the body (mostly, the brain) can have effects upon the mind. For example, eating a strawberry causes certain conscious sensations. Perhaps damage to the brain might also be held to cause damage to the mind. The issues here thus seem, broadly, to concern mind-body interaction, which is a big issue for dualism anyway.

There have been several attempts to formalize an inductive logic, but none that have been as uniformly successful as formalizations of deductive logic. See the Stanford Encyclopdia entry for more.

Of course there could be all kinds of differences between races, including differences in native intelligence, ability to learn, and so forth. The only significant question is whether there are such differences, and there has never been any decent reason to believe that there are. Part of the problem here is that people often speak as if "race" is a well-defined notion, perhaps even a notion with biological significance. But it is not.