I can only think of one thing to say in response to Allen's remarks, and that would be "Amen!"

And one might add that the cells themselves are hardly immune from "renewal" at the molecular level. So the short version is: If identity requires complete coincidence of matter, then essentially nothing but sub-atomic particles survive over any reasonable stretch of time. That does rather suggest, though the contrary view is certainly held, that identity over time simply does not require complete coincidence of matter. What it does require is not very clear, but that is no reason to despair. Of course, the question didn't ask about complete coincidence of matter. But it's unclear why anything less might suffice. And, if it does, then you run into issues about transitivity: A might share much of its matter with B, which shares much of its matter with C; but A and C do not share much of their matter.

Another question worth considering here is whether the "practice" of Chistianity, as you understand it, is really as connected to the beliefs with which you are becoming disillusioned as you suggest. I'll speak at some length about this. What I have to say may not seem very philosophical, and in some ways it won't be. But there are profound questions here about the relationship between faith and belief, and what I will have to say is related to my own views about that relationship. It seems to be quite commonly believed that one cannot "really" be a Christian unless one accepts certain doctrines of faith, for example (and since it is Good Friday), that Jesus rose bodily from the dead on the third day after he was executed by the Romans (the doctrine of the resurrection). That, in doing so, he made himself the supreme sacrifice, "the Lamb of God who takes away the sin of the world", as the Agnus Dei has it (the doctrine of sacrificial atonement). That Jesus is the only way to God (John 14:6). And...

To give a similar but somewhat different answer, one might think the problem with the line of reasoning in the question comes here: "But it seems bizarre to say that Russell's invention of the theory of types entails that Bush is the 43rd president...". We were talking about the statement, "If Russell invented the theory of types, then Bush was the 43d president", and now we're talking about entailment? Why? What do these have to do with each other? The move from talking about the truth of conditionals to talking about entailment is what lies, in many ways, behind the invention of (formal) modal logic, by Lewis and Langford in the 1920s. One of the central ambitions of early modal logic was to formalize the notion of entailment. It was with reference to this that Quine spoke of modal logic's being "conceived in sin, the sin of confusing use and mention"---of confusing "if p then q" with "`p' entails `q'". Now, that said, it is undoubtedly a serious question whether the English indicative...

With all due respect to Professor Green (hi, Mitch!), even that is not the final word. I think perhaps Professor Nahmias was assuming that John knows perfectly well that Mushmush is a green person, Mushmush being his neighbor and all that, and that John has some minimal degree of logical competence. Still in that case, most people would hold that it does not logically follow that John believes that Mushmush should be killed. There are two quite different reasons for this. One involves the fact that we cannot, even in principle, actually deduce all the logical consequences of everything we believe. It seems extremely plausible, in fact, that there are propositions of the form "All F are G" and "x is an F" that I believe, where I do NOT believe the corresponding proposition of the form "x is G", simply because I have never gotten around to inferring it. Note carefully that the claim is not that I believe that x is NOT G, just that I fail to believe that it is. In this kind of case, though, you...

There are a few kinds of support. One is that one can prove certain special cases of the conjecture that seem inherently unrelated, so one thinks that these special cases must really be true because a certain generalization of them is true---and that's what one conjectures. But conjectures are often based upon a dim and hard to express appreciation for "what is going on", so that it just sort of seems as if the thing ought to be true. One can sometimes give reasons to think things ought to work out that way, but they wouldn't be the kinds of reasons that would count as a proof.

The notion of completeness for logics links two notions: A notion of what is provable or deducible in some formal system of logic, and a notion of what is valid , which is itself defined in terms of a notion of interpretation. It's probably best to think of the latter as primary. We have some system of logical notation, and we have a way of interpreting it that gives rise to a class of "valid" formulas. What we'd like to have then is a proof-method that will be complete in the sense that, if a given formula is valid, then it will be provable by that method. More generally, we can think not just of the class of valid formulas but of some notion of implication or entailment that relates formulas: So we say that some bunch of formulas A, B, C, ... entail some other formula Z. Then what we want is a proof-method that will be complete in the sense that, if Z really does follow from A, B, C, ..., then there is a way of deriving Z from A, B, C, ..., by the proof-method. You can't always have...

Yes, and me. I'm not sure what the worry is here. I think it's clear that there are some philosophical views that are plainly wrong. There may be some truth in them somewhere, but research over the years has shown that the view is wrong. (Examples: Plato's theory of forms; Hobbes's theory of government.) So who says they're wrong? Well, the people who have done the research mentioned. This is no different from science. There are scientific theories that are wrong, and the people who say so are the scientists who do the work.

I know several people who believe such things, or at least say they do. One group thinks that there are true contradictions that involve very special cases. The usual example is the so-called liar sentence, "This very sentence is not true". There is a simple argument that the liar sentence is both true and not true, and some people believe just that. Other people, though, think there are contradictions involving much less special cases. An example would be what are called "borderline cases" of vaguepredicates, like "bald". People often want to say that there are somepeople who aren't bald and aren't not bald either. But the so-called DeMorgan equivalences entail that this is equivalent to saying that theperson is both bald and not-bald (or, strictly, both not-bald andnot-not-bald). People who hold such views are known as "dialetheists". See this article for more.

And, to add to all the confusion, one can say: Obama is identical to a good speaker; and also: Hitler is identical to a good speaker. But it certainly doesn't follow that Obama is Hitler. The reason, in this case, is because what stands on the right-hand side of the identity here is not a name, but what philosophers and linguists call an "indefinite". Exactly how indefinites work is a matter of some controversy, but one (older) way to resolve this puzzle is to treat "Obama is identical to a good speaker" as meaning: There is a good speaker with whom Obama is identical. Or, in logical symbolism: (∃x)(good-speaker(x) & x = Obama) Now the fallacy should be clear. The crucial point is that the only really well-defined notion of validity in logic is one that applies only to formal, logical representations. To apply the notion of formal validity to arguments in ordinary language, one has to "translate" the ordinary arguments into logical notation, and it is not always clear how this is to be...