Note, desiring something isn't needing it. I may desire a villa in Tuscany, but I don't need one. And, whatever is the case with needs, plainly it isn't the case that for every human desire there is a way of fulfilling it (especially given other people's desires). Maybe lots of us would love a villa in Tuscany; but we can't all get one. And in fact we often desire flatly impossible things. Lots of humans would love totime-travel: but of course that desire doesn't make itpossible. And maybe lots of us would love to live for ever: butthere's no reason to suppose that merely having the desire makes eternal life possibleeither.

I'm really glad you enjoyed the Gödel book! Suppose that S is Goldbach's conjecture. And suppose theory T is your favourite arithmetic (which includes Robinson Arithmetic). Then Theorem 9.3 applies to S . So if not- S is not logically deducible from T , then S must be true. So if we had a proof that S is a "naturally" arising Gödel sentence -- i.e. a demonstration that T proves neither S nor not- S -- we'd ipso facto have a proof that S is true. That means that establishing that that S is a "naturally" arising Gödel sentence for T -- if that's what it is -- is at least as hard as proving Goldbach's Conjecture itself. Which, the evidence suggests, is very hard! As to the "odds": my hunch is that GC is true, and can be proved in PA -- but I wouldn't bet even a decent meal out on it!!

Those who believe that there are contradictions which are true don't think that all contradictions are true. They don't think that "ordinary" contradictions like "the book is fully open and the book is fully closed" can be true. It is only special cases, like Liar propositions and other paradoxical propositions, and perhaps some others, that are claimed to be true and false at the same time. [' Fully open' vs ' fully closed' here, for the reasons that Richard gives in his next posting!]

No, Gödel does not show that there are true claims S that are not provable. He shows rather that, given a consistent formal theory T which contains enough arithmetic, then there will be a true arithmetical "Gödel sentence" G which is not provable in T. But that Gödel sentence G , though it can't be proved in T , can and will be provable in other formal theories (for example, G is provable in the theory that you get by adding to the axioms of T a new axiom Con(T) that encodes the claim that T is consistent). So if we reflect on the axioms of T and accept them as true, and so have good reason to think that T is consistent, we'll have good reason to think T 's Gödel sentence G , which is provable in T + Con(T), is true. (And there's nothing especially mysterious about the notion of truth here: it is the common-or-garden notion of arithmetic truth that is invoked when we say of even the simplest sentence of formal arithmetic, as it might be "1 + 0 ...

There is in fact a tradition of 'constructivist' mathematics which does not endorse the law of excluded middle. Very roughly, suppose you think that mathematical truth consists in the possibility of a constructing a proof. Then there is no reason to suppose that, inevitably, either P or not-P -- because there is no reason to suppose that (for each and every P) there is a possibility of proving P or is a possibility of disproving P. To find out more about this, you can read (at least the opening couple of sections of) this article . Note, however, that refusing to endorse the law of excluded middle for mathematical propositions is not to deny the law, nor is it to assert that there is a "third state" between truth and falsehood. Still, for certain other purposes, it can be useful to explore three-valued logical systems (even multi-valued logics), which allow more "states" that the classical true/false pair. From the same source, here's another article -- note the section on ...

How do you show of any bad pattern of reasoning that it is indeed unreliable (whether or not that kind of reasoning is called a fallacy or is dignified with a Latin name)? By coming up with some example arguments that rely on that kind of reasoning yet are evidently and uncontroversially terrible arguments. Such counterexamples reveal that kind of reasoning to be hopelessly unreliable. Suppose that, when the wraps are off, someone's argument relies on the pattern i f A, then B; B; therefore A. That's plainly an unreliable pattern -- just think of e.g. the instance "If Jo is a woman, Jo is human; Jo is human; therefore Jo is a woman"! So the original argument is fallacious. Or suppose that, when the wraps are off, someone's argument moves from the premisses that Xs are not A and a Y is constituted of nothing but Xs to the conclusion that a Y is not A. Then that again is plainly an unreliable pattern -- just think of e.g. theinstance "H2O molecules aren't wet; a...

Take, as an extreme case, the argument "The earth is round; hence the earth is round". The inferential move is trivially deductively valid (there is no possible way the premiss can be true and the conclusion false); and the premiss is true. So the argument is sound . But of course, the argument would be quite useless in an exchange with a latter-day flat-earther! He could rightly complain that, given his views, that argument just begs the question at issue between you. So, in that context, the argument would be sound but question-begging.

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...

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...

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...