Gödel's first incompleteness theorem says that, for any suitable formal theory which is consistent and includes enough arithmetic, there will be an arithmetical sentence -- a "Gödel sentence" -- which that theory can neither prove nor disprove. This theorem is a bit of mathematics: its proof is undoubtedly a sound mathematical proof. (That's why those obsessives who plague internet discussion groups with purported refutations of Gödel are so very annoying! -- they are refusing to follow, or are incapable of following, a relatively straightforward bit of purely mathematical reasoning.) The question of the significance of Gödel's theorem, however, is quite another matter. To investigate that , we need to engage in philosophical reflection. Some have held, for instance, that Gödel's theorem can be used to show that minds are not machines. Let's not worry here about why that's been said: the simple point we need now is just that, to decide the merits of this interpretative view, we need...

I am entirely certain that two and two is four, that the earth is bigger than a peanut, and that Hitler didn't win World War II. Aren't you?

You need to be careful to distinguish things and ideas here. Is the question about square circles or about the idea of a square circle ? Compare: there are no such things as unicorns . It would plainly be wrong to say that they "neither exist nor don't exist": unicorns definitely don't exist! But the idea of a unicorn exists and seems coherent enough. Indeed, we are tempted to suppose that there could have been things that fitted the idea. Likewise, there are no such things as round squares . Like unicorns, round squares definitely do not exist. But this time, though we can frame the idea of a round square -- we grasp that something counts as a round square if it is round and a square! -- the idea is a self-contradictory one in the sense that nothing can possibly count as fitting our idea here.

Read some good introductory books. Here's a list that I really must get around to updating.

A while back, I wrote a piece for our grad students about getting published that others working to publish a paper might find useful.

A philosopher of mathematics is interested in questions like: what are numbers? what kind of necessity to arithmetical truths have? how do we know the basic laws of arithmetic are true? what about sets -- do they really exist over an abover their members? is there a universe of sets? there are various set theories, how can we decide which is true of the universe of sets? And so on. You don't have to be a working mathematian to think about these matters (though obviously you have know a little about the relevant mathematics you want to philosophize about). Nor do you have to be a logic-expert. Most mathematicians aren't interested in the philosophy of mathematics (just as most scientists aren't interested in the philosophy of science, and most lawyers aren't interested in the philosophy of law). They just go about doing their maths. And among mathematicians, a serious interest in logic is a rarity: you can certainly be a mathematician without being a logician (e.g. by being a fluid-dynamicist or an...

The given argument is indeed valid on the classical definition of validity which you give (there is no possible way in which the premisses could be true and the conclusion false). On the other hand, we are tempted at first blush to suppose that the conclusion of good argument ought to be connected by some relation of relevance to the premisses. So either we have to reject the classical definition as unsatisfactory, or we have to revise our first thoughts about relevance as a requirement for being a good argument. Different logicians jump different ways. Defenders of some variety of "relevance logic" insist on building in the relevance requirement for validity, and so need to revise the classical definition. But it turns out to be not at all easy to do this while preserving other (equally compelling) intuitions. And defenders of the relevance requirement fight among themselves as the best way of implementing it. Which is why the majority of logicians think it better to isolate the neat...

Suppose that you have a conceptual problem about e.g. your notion of moral responsibility (or justice, or freedom, or causation, or whatever). How could doing your philosophical thinking in Sanskrit terms possibly help? Either the concepts available in Sanskrit are the same as yours -- in which case, they will raise the same problems, and the move gains you nothing. Or they are different concepts -- in which case, thinking about them won't resolve the problems you started off with, which were problems to do with your concepts, and again the move gains you nothing (except additional problems).

Consider the schema: "For every p , given Op , it follows that p : but it is not the case that for every p, given p , it follows that Op ". For what fillings for O does this come out true? Certainly if we put Op = it is necessary that p , we get a truth. But equally Op = Jack knows that p works too. And if we put Op = p and q (for some fixed contingent q ), the result will again be true. So just requiring the schema to hold isn't enough to fix it that Op = it is necessary that p as opposed to the alternatives. Hence requiring the schema to hold is certainly not enough to define the notion of necessity. [A little wrinkle. The scope of the negation in the schema is important here. For take the variant schema "For every p , given Op , it follows that p , but given p , it doesn't follow that Op ". This variant isn't satisfied by Op = it is necessary that p. For suppose p is a proposition of the form ...

There is no possible way that you can be over six foot tall yet not over five foot tall. So the argument "I am more than six feet tall. Therefore I am over five feet tall." is certainly a valid one [on a classical account of validity: see below]. Standardly these days, an argument is said to be sound if it (a) is valid, and (b) has a true premiss/true premisses. So whether the given argument is sound depends on whether the premiss is true or not (which we are not told!). If you are over six foot tall, the argument is a sound one (since it is valid): if not, then not. Let's say that an argument is formally circular if its conclusion is identical to one of its premisses. The given argument is plainly not formally circular in this sense. As to whether it counts as "circular" in some other, looser sense, that will depend on how this looser sense is explicated. Standardly these days, we use talk of tautologies/tautologousness in a narrow sense -- not for logical truth/logical validity...

Let's call the argument about Allen's plastic jug "argument (A)". Then consider the following claims: (1) Kripke's doctrine : It is impossible that something should contain water without containing H 2 O. (2) The classical account of validity : An inference is valid if and only if it is impossible that the premiss(es) be true and conclusion false. (3) Non-ambiguity : The sense of "impossible" as it occurs in Kripke's doctrine is the same as the sense of "impossible" as it occurs in the classical account of validity. (4) The intuition : Argument (A) is not a valid argument. Then, as Allen points out, we can't hold all these together. If we accept Kripke's doctrine, then it is impossible for the premiss of (A) to be true and the conclusion false. If we accept non-ambiguity and the classical account of validity, then it immediately follows that argument (A) is valid. And that clashes with the intuition. What to do? Let's set aside the option of disputing (1). As it happens I...