There's a nice article on intuitionist logic in the Stanford encyclopedia.The differences between it and classical logic become more profound inconnection with quantifiers such as "all" and "some". For example, inintuitionistic logic, it can be true both that not everything has someproperty and that there is nothing that does not have it! Andin the intuitionistic theory of the real numbers, we can actually findsuch a property and prove such a statement: We can prove that not everynumber is either positive, negative, or equal to zero; we can alsoprove that there is no number that is neither positive, nor negative,nor equal to zero. Think about that for a while. Of course, we can only "prove" it in the sense that it follows fromthe principles of intuitionistic analysis. Whether it is true dependsupon whether those principles are true. Since you mentioned internal consistency, perhaps I'll mention something even stranger, so-called paraconsistent logics . These are systems in which...

On Dan's comment. The distinction between so-called weak counterexamples and strong ones is, of course, important. But it really is possible to prove, in intuitionistic analysis, the negation of the claim that every real is either negative, zero, or positive. The argument uses the so-called continuity principles for choice sequences. I don't have my copy of Dummett's Elements of Intuitionism here at home, but the argument can be found there. A short form of the argument, appealing to the uniform continuity theorem—which says that every total function on [0,1] is uniformly continuous—can be found in the Stanford Encyclopedia note on strong counterexamples . There is an important point here about the principle of bivalence, which says that every statement is either true or false. It's sometimes said that intuitionists do not, and cannot, deny the principle of bivalence but can only hold that we have no reason to affirm it. What's behind this claim is the fact that we can prove that we will not...

Philosophers tend not to speak of "a philosophy" the way that phrase is used in ordinary language. You will see people talk, for example, about Russell's philosophy of mind, but that just means Russell's theory about the mind. In so far as people speak of Russell's philosophy, they just mean Russell's work or, again, theories. There's no significant relation between "philosophies" of this kind and religious belief. In the ordinary sense, I suppose "a philosophy" is a set of values or principles. A particular religion might then have a "philosophy" associated with it, but particular "philosophies" will not necessarily have religious elements.

I'm not sure what you mean by "unconcious". If something braodly Freudian, then I'm not in a position to answer this. But there is another notion of "unconscious" that figures heavily in comtemporary empirical psychology: It is the idea of processing or information that is inaccessible to conscious reflection. For example, the standard view in linguistics nowadays is that our ability to speak and understand our native languages depends upon all sorts of unconcious processing. The evidence for this view is the explanatory success of linguitic theory. It is possible, for example, to state an extremely general principle governing when a pronoun can be "bound by" an antecedent (that is, "refer back" to it) which will account for why the first but not the second of these can mean that John saw John in the mirror: John saw him in the mirror. John saw himself in the mirror. There are all kinds of similar contrasts that the principles explain. Compare, for example: Bill thinks it would be...

Identity is an important notion in mathematics. There certainly areexamples of geometrical theorems that demonstrate identities, some ofthem very important. Consider, for example, the (Euclidean) theoremthat the three lines from the vertices of a triangle bisecting theopposite sides meet in a single point. Any time one uses a word like"single", identity is in play. Frege gives a similar example in section8 of Begriffsschrift to illustrate the same point:Mathematical identities can have substantial content. Elementaryarithmetic may be a better illustration of how important identitiesare, though, since basic arithmetical facts are all equalities, thatis, identities. That said, the question arises how identity is to be characterized.There are various ways to proceed. But any characterization is going tohave to deliver four basic properties of identity: reflexivity (thatis, a=a), symmetry (that is, a=b → b=a), transitivity (a=b & b=c →a=c), and substitutivity, which says (roughly) that if you have …a...

The sort of remark made in the second paragraph is one I see and hear a lot. But, frankly, I just don't get it. In particular, why is it supposed to follow that I can't use language to speak of a reality that is independent of language? I can use language to speak of all kinds of things that seem to have nothing very much to do with language: Flowers, rocks, supernovas, non-recursive sets, and so on and so forth. Obviously, everything that can be said has to be said inlanguage. But that is so mind-numbingly obvious that I can't see howanything of consequence could follow from it: It's "analytic" in pretty much the "bachelors are unmarried men" sense. And even if one assumes, more strongly, that anything that could be thought at all could be said, nothing in this vicinity follows.

I don't see how the conclusion follows. Coercion is an obvious counter-example. If someone holds a gun to your head and so coerces you, say, to make crank phone calls, I don't see that you should be held "ultimately accountable" for upsetting the recipients. And coercion hardly has to be "life or death" or even physical. But once you've allowed that much, then it's easy enough to see how external influences might have an effect. In very simple terms, what coercion does is change the costs and benefits of different actions. It seems obvious that external influence, such as addiction, can do the same thing. (I'm really not sure what you have in mind vis-a-vis battered wives. But how can we toss that in when we're supposed to be waiving physical coercion?)

My understanding is that they're not really comparing it to anything. The idea is that the structure of DNA is, in itself, so complex that it could not have been produced by the kinds of processes postulated in the theory of evolution. There are ways of measuring complexity in such cases, or at least there are ways of trying to do so, but it is extremely difficult to provide a good account of this kind of complexity. A large part of the reason is that DNA is finite, and most of the mathematics relevant to the study of complexity counts everything finite as supremely simple. Still, there are ways one can go here (using, for exmaple, the resources of information theory). But part of the criticism of many arguments by proponents of intelligent design is that they operate with inadequate accounts of complexity. The more fundamental criticism of these arguments, though, or so I take it, is that there simply isn't any remotely plausible argument that the structure of DNA is too complex to be produced by...

The answer to question 50 is also an answer to this question, I think: The universe doesn't have an end.

The question what the relation is between thought and language is, to my mind, one of the most fundamental issues in contemporary philosophy. That is to say, what one's view is about this matter will profoundly shape one's views on many other topics. What one's impression is of the current state of play will, however, depend upon what one has read. There are, as you say, many philosophers would suppose that thought is somehow dependent upon language. One famous example is Donald Davidson, who argues explicitly for this conclusion in "Thought and Talk". On the current scene, John McDowell is perhaps the most visible proponent of the view. Hilary Putnam has held a version of this view in the last several years, and it can be found as well in the writings of Michael Dummett. I could easily continue. On the other hand, however, there are plenty of philosophers who reject this view and hold, as you suggest, that the ability to think does not depend upon the possession of language skills. Jerry Fodor, for...

That question can be taken in many different ways. One way is historical. I'm in no position to answer that question. Another would be sociological. I'm not in a position to answer that question, either. Part of the problem here is that I'm not an historian or sociologist. Another is that there's substantial vagueness in the phrase "religion as we know it". Here's a different question I can answer: Does relgious faith have to be bound up with or somehow a consequence of one's fear of death? The answer to that question seems to me obviously to be "No". I suppose there are particular forms of religious faith that are so bound up with a fear of death, and perhaps these are the most visible in American public life or the most salient to non-believers, but there are plenty of forms of religious faith that are not.