There certainly is a fallacy here, but I don't know what it should be called. In the end, perhaps it is a simple fallacy of equivocation: an equivocation between two senses of the word "definition". Philosophers since Aristotle (at least) have distinguished two types of definitions: definitions of words and definitions of things . Personally, I find the latter notion hard to understand, but the idea is that a definition of a thing tells you what it really is. So gold, for example, might be defined as that element that has atomic number 79. This is very different from saying that the word "gold" is defined this way. The word "gold" could not have been defined that way before the establishment of modern chemistry, but, nonetheless, the true definition of gold—what gold really is—is: the element with atomic number 79. You cannot, therefore, find out what the real definition of a thing is by consulting a dictionary: That will tell you only how the word is defined. What the "real definition"...

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...

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.

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...

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...

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...

I'm not sure there's much "logic" there, frankly. First, the relevant doctrine is that of transubstantiation , not transfiguration. The latter term refers to the events described in e.g. Luke 9, when Jesus appears "transfigured" in the presence of Elijah and Moses. Second, I'm not entirely sure why it is so obvious to you, or to your friend, that the consecration does not transform the elements into the body and blood of Christ. The fact that they do not look much like flesh and blood has nothing to do with it. (The Wikipedia article on the topic is excellent, by the way.) That said, transubstantiation is controversial within Christianity. It is, as was said, a pillar of the Catholic faith, but it is not widely accepted outside Catholicism. I see no reason to suppose that all "dogmatic doctrine[s] canonized by the [Catholic] church" must stand or fall together. One might reason thus: If one of them turns out to be wrong, that diminishes whatever general reason one had to suppose that official...

This is a difficult and somewhat contested question. Obviously, you cannot argue for the rules of argumentation except by arguing, and if there are rules of argumentation that must be followed if an argument is to be compelling, then one had better follow them. So there is, obviously, a kind of circularity in any such argument. But it is an unusual, and somewhat confusing, kind of circularity. The worst kind of circularity is what is called "begging the question". That is when you simply assume, perhaps tacitly, precisely what you want to prove. It's a bad kind of circularity because, if you assume that P, you can hardly help but reach the conclusion that P. That's not what's happening in the case we're considering. Suppose we're trying to prove the logical principle of disjunctive syllogism, which says that, if "A or B" is true and "not A" is true, then B must be true. We might argue for it as follows. Suppose that "A or B" is true and that "not A" is true. Since "A or B" is true, either A is...

On (a), I might add that possible worlds are an extremely useful tool. As Lynne mentioned, there are many different ways to understand what they are supposed to be. For many purposes, however, one can simply regard possible worlds as a certain kind of mathematical construct. The reason they are then so useful is that they allow us to understand, in a precise way, the logic of such expressions as "necessarily" and "possibly". In fact, there are many notions of necessity and possiblity, and their logics can be very different. Perhaps more importantly, though, possible worlds help us to get a grip on so-called "counterfactual conditionals", like "If JFK had not been assassinated, the US would not have become as seriously involved in Vietnam". It might interest you to know that, even before "paraconsistent" logics came on the scene—these are the logics that allow contradictions to be true...and, of course, false—logicians were interested in logics of necesssity and possibility that allow "It is...