I'm not sure studying model theory is important to philosophy, broadly speaking. For some areas, such as the philosophy of set theory, it surely is important that one know at least something about the basic results and techniques, since they are used regularly in set theory itself. But I doubt one really needs to know very much model theory to work in philosophy of mathematics generally. (I don't, and I do.) A general understanding of how what models are and how they are used in studying logical consequence is probably adequate, which means, roughly: What you'd get in any good sequence of logic courses. Of course, if you want to work in logic itself, then how much model theory you need to know will depend upon which branch of logic interests you.

I don't work on this sort of thing, so I won't comment upon the question how one might actually decide whether something has intrinsic value. But I will comment on the overall orientation of the question. It seems to be assumed that, if someone could deny that the object has any value, then that would someone call into question whether it had intrinsic value. But why? People can be wrong about all kinds of things. The fact that something is of intrinsic value does not imply that any particular person will recognize that value. It does not even imply that anyone at all will recognize that value. We might all be completely ignorant of it. So let me ask a question back: What's the significance of the restriction to "intrinsic or innate" value here? Does this worry seem more pressing when those words are included than if they are not? If so, why?

There aren't a whole lot of textbooks on this sort of thing. A more current text is John Burgess's Philosophical Logic . And, depending upon your interests, you might have a look at something like Graham Priest's Introduction to Non-classical Logic . Working through a serious textbook on modal logic would also be worth doing. The two classics are by Chellas and by Hughes and Cresswell. A quite different route would be to look into linguistic semantics. Many forms of philosophical logic—tense logic, modal logic, epistemic logic—originated as attempts to deal with some of the features of natural language that are omitted by quantification theory. But the relation between the logical treatments and natural language were always pretty obscure, and around 1960 people started to get much more serious about dealing with natural language in its own terms. Formally, much of linguistic semantics looks like philosophical logic (especially in certain traditions), but it is targeted at an empirical...

Classical logic (at least, understood from the perspective of classical semantics) rests in part upon the so-called "law of bivalence". This is usually formulated as: Every formula is either true or false. To put it slightly differently, we can begin with the idea (which emerges from Frege) that the "semantic value" of a formula is its truth-value. Then classical semantics involves two claims: (i) every formula has exactly one truth-value; (ii) there are only two truth-values, Truth and Falsity. This formulation does not involve any reference to "not", so the contrast between "not: A is true" and "A is not true" is avoided. There are several sorts of alternatives to this. The one you mention can be presented in several different ways. One involves retaining (ii) but dropping (i) and, indeed, not replacing (i) with anything. So, on this view, a formula can have zero, one, or two truth-values. Logics can be built on this idea, and they "work" in the sense that one can rigorously present them, study...

I would have thought that the obvious theistic response would be that it is the existence of the eternal multiverse that is at issue. I.e., why are there any universes rather than none? From what I've read of Hawking's response to this, it does not seem to me to be very impressive. As usual with these things, it fails to take the motivations of its opponent at all seriously. None of that is of course to say that the theistic argument referenced is any good.

There was a nice post about this sort of argument on the Sojourner's blog recently: http://blog.sojo.net/2010/10/01/what-glenn-beck-and-alan-grayson-have-in-common/ .

No, I do not think the incompleteness theorem has any such consequence. First of all, although the incompleteness theorem does apply to some formal systems some people would classify as systems of logic (e.g., second-order logic), its primary application, as usually understood, is to formal systems of arithmetic or, more generally, or mathematics. What people usually mean by "logic" is first-order classical logic, and the incompleteness theorems do not tell us anything about its consistency or validity. (There is a nice proof of the undecidability of first-order logic that rests upon the completeness theorem.) There are two versions of the incompleteness theorem. The first shows that no (sufficiently strong) formal system is ever complete (if consistent), in the sense that you can either prove or disprove every statement formulable in that system. The second shows that you cannot prove such a system to be consistent without relying upon assumptions that are, in a very specific sense,...

One of the things usually taken to be distinctive of mathematical and logical truth is that such truths are in some very strong sense necessary , i.e., they could not have been false. Assuming that it is in fact true that 2 + 2 = 4, how could that have failed to be true? (Or, to take a logical example: How could it fail to be true that, if Goldbach's conjecture is true and the twin prime conjuecture is also true, then Goldbach's conjecture is true?) Presumably, the answer to this question depends upon what, precisely, one thinks "2 + 2 = 4" means, but it is hard to see how one could accept the statement that 2 + 2 = 4 as both meaningful and true and think that it might not have been true. It's important to be clear that this statement does not say anything about how actual objects behave, e.g., that if you put two oranges on a table with two apples and no other pieces of fruit, then you'll have four pieces of fruit. Weird things might happen in some worlds, but that would not make it false in...

Quine's views on this matter vary over the years. Early (meaning in "Two Dogmas" and related works of that period), he was prepared to deny that there are any analytic statements. Later, especially in Philosophy of Logic , Quine's view mellows a bit, and he is prepared to recognize a very limited class of such statements, namely, truths of sentential logic, such as "It is raining or it is not raining" and the like. That's still a pretty limited set, as Quine seems unprepared to regard even what one would normally regard as truths of predicate logic as analytic (e.g., "If someone loves everyone, then everyone is loved by someone"). But mostly this is because Quine thinks there's no clear sense in which that sentence is properly analyzed as a truth of predicate logic. This is connected with the doctrine of ontological relativity. In so far as it is properly so analyzed, I think Quine would regard it as analytic. So mathematics, for Quine, is quite definitely out as analytic. There are going to...

Just a minor correction, or perhaps elaboration. The (most?) famous argument of Geach's against emotivism (in "Ascriptivism", Phil Review , 1960), concerns embeddings in the antecedents of conditionals, such as: If sodomy is wrong, then it ought to be against the law. The contrast here is with something like, "If OUCH, then I should go to the hospital". That just makes no sense. Geach was not necessarily assuming a truth-functional analysis of conditionals, but the point is easiest to see from that perspective: The conditional is supposed to be true so long as its antecedent is false or its consequent is true; but the emotivist view is that "sodomy is wrong" does not have a truth-value, because it is not truth-evaluable. The response one tends to see from anti-realists turns on a different sort of understanding of conditionals, as so-called "inference tickets". But on this too, the jury has not yet reported.