A nice question. Yes, if the predicate "F" and the predicate "G" are co-extensive (i.e., are true of exactly the same things), it would be wrong to conclude that the property corresponding to "F" is the same as the property corresponding to "G". (Nick gives some good examples of this in his response.) You seem to think that the set example conflicts with this observation, but it doesn't. If we establish that x is an element of S if and only if x is an element of T, we can indeed infer that S equals T. But that's different from inferring that the property of being an element of S is the same as the property of being an element of T. And it's that inference that would conflict with our observation. Perhaps you think that this last claim can nevertheless be inferred because you think that if S is identical to T, then the property of being an element of S is identical to the property of being an element of T. But that isn't right. Washington, D.C. is identical to the capital of the...

Again, this awaits a Descartes scholar, but what Descartes says in his "First Meditation" is this: Andbesides, as I sometimes imagine that others deceive themselves in thethings which they think they know best, how do I know that I am notdeceived every time that I add two and three, or count the sides of asquare, or judge of things yet simpler, if anything simpler can beimagined? So it seems Descartes argues that, as far as we can tell by the end of the "First Meditation",we could be wrong about (and so can't be said to know) basiccomputational facts, wrong about (what philosophers often call) analytictruths (like "A square has four sides"), and even wrong about "yet simpler"matters (like logical laws, perhaps).

To prove Y from X is to show that if X is true then Y must be. We could say that to prove Y, period, is to prove Y from assumptions that are true. (Should those assumptions turn out to be false, we might say "We thought we had proved Y, but it turns out we were wrong: we had proved Y from assumptions we now know to be false.") So, can we prove anything? Sure. Why not? We may well have already. If in fact we've correctly derived propositions from true assumptions, then we've proved them. Perhaps you're worried by the fact that the bare derivation of Y from X doesn't tell us whether we should accept X and, therefore, Y -- or whether we should reject Y and, therefore, reject X. That's true: the bare derivation doesn't tell us whether we've proved Y (in the above sense). The derivation only gives us grounds for accepting Y if its premises are true. In judging that Y is true, we express our confidence that the truth of X is far more likely than the truth of not-Y. Perhaps your worry...

I suppose one could tell a story about the evolutionary advantage ofappreciating that "If a tiger is nearby, it's best to run" and "A tigeris nearby" together imply "It's best to run". Someone who doesn't graspthis inference is less likely to get down to the business ofprocreation than someone who does. (This isn't a justification of thisinference. Rather, I assumed the inference was correct and then offeredan evolutionary explanation based on that assumption for why humansmight find the inference compelling.) But logical inferences aren'thardwired in the sense that we have no choice at all about the kind oflogic we can adopt. See some of the responses to Question 168 fordiscussions of alternative logics.

The negation of "Everything is a goat" is not "Nothing is a goat". Asentence and its negation must have opposite truth values; that is, ifone is true, the other is false. A sentence and its negation cannotboth be true and they cannot both be false. But, as I think yourealize, "Everything is a goat" and "Nothing is a goat" can both be false: if there are some things that aren't goats and somethings that are, then the two claims will be false. So this shows that"Nothing is a goat" is not the negation of "Everything is a goat".Might the negation of "Everything is a goat" be "Something is a goat"?No, for both these claims could be true: imagine that there is at least onegoat and furthermore that everything is a goat. (All these errors are facilitated by the false assumption that nouns like "nothing" and "everyone" function like "Harry" or "the animal in the shed" do. For more on this error, see Question 49 .) What then isthe negation of "Everything is a goat"? It's "Something is not a goat....

The kind of logic that most mathematicians assume in their work is known as classical logic . Classical logic accepts the Law of the Excluded Middle, which states that for every statement P, "P or not-P" is true. Some logicians and mathematicians (though not many) work within systems of reasoning that do not assert this Law. Most constructive logics fall into this category, in particular intuitionism does. For a little more on intuitionism see Question 139 . These days, there isn't much of a debate within the mathematical community about which is "the correct logic". There has been considerable debate about this within philosophy. Especially pertinent here are the writings of the Oxford philosopher Michael Dummett.

A reductio ad absurdum argument has the following form:we assume that X is true, deduce some absurdity from it, and thenconclude that not-X must be true after all. You can view reductio as arule of inference that allows us to infer not-X from our derivation ofan absurdity from X. Why does this work? Becausededuction is a process that leads from true assumptions to trueconclusions. (See also here .) But an absurdity cannot be true.Therefore, our assumption X is not true. But either X is true or not-Xis true. Consequently, it must be not-X that's true. I agree withAmy that the final upshot of a reductio argument needn't be surprising,so I'll interpret the question differently. Perhaps you're asking thefollowing: when we deduce our absurdity, how do we know whether it's soabsurd that we have no choice but to consider it false and thus toconclude, by reductio, not-X — or whether instead to conclude that,contrary to what one might have thought, the alleged absurdity is true after all! Youmight say...

When someone says "I expect the unexpected" we might hear that alongthe lines of "I fathered someone fatherless". That is, we mightinterpret him as meaning that he expects some event which he also doesnot expect. That does seem like a contradiction. But isn't that tomisunderstand what he's trying to say? What he expects is not someevent (which he also doesn't expect); rather, what he expects is thathe doesn't expect some event. His expectation applies not to some eventitself but rather to his non-expectation of some event. What he expectsis that there will be some event that he does not expect. Thisexpectation is a second-order expectation: it applies to his first-order non-expectation of some event. (My expectation that a credit card billwill soon arrive is a first-order expectation. My expectation that Melanie will expect me to pay for dinner is a second-order expectation. See here for a similar distinction.) That's why, as Peter Liptonsays, "Even if you expect the unexpected, you may...

(This evening, shortly after reading this, I had dinner at arestaurant in NYC — and there was Mayor Bloomberg at the next table. I heard someone say, "Nothing's impossible after all.") I'm not sure what an infinitely small probability would be. Perhapsjust a probability of 0? But that sounds like an impossible event. Soperhaps you're asking whether all events have some finite non-zeroprobability of occurring — and whether the events we call "impossible"really just have a very small finite probability. Philosophershave spent a lot of time trying to figure out what we're actuallysaying when we assign a probability to an event. Are we making someclaim about the world? Or are we making a claim about our degree ofconfidence in some judgment about the world? I won't go into that hereand instead will say a few words about impossibility. Philosophersoften distinguish between different kinds of impossibilities. Somesituations would conflict with the laws of logic: for instance, thestate of...