On April 10, 2014, in response to a question, Stephen Maitzen wrote: "I can't see how there could be any law more fundamental than the law of non-contradiction (LNC)." I thought that there were entire logical systems developed in which the law of non-contradiction was assumed not to be valid, and it also seems like "real life" suggests that the law of non-contradiction does not necessarily apply to physical systems. Perhaps I am not understanding the law correctly? Is it that at most one of these statements is true? Either "P is true" or "P is not true"? or is it that at most one of theses statements is true? Either "P is true" or "~P is true"? In physics, if you take filters that polarize light, and place two at right angles to each other, no light gets through. Yet if you take a third filter at a 45 degree angle to the first two, and insert it between the two existing filters, then some light gets through. Based on this experiment, it seems like the law of non-contradiction cannot be true in...

I won't address the issue about physics, but yes: There are plenty of logical systems that allow for the possibility of true contradictions. For the most part, these are motivated by various sorts of paradoxes, such as the liar paradox (which has to do with truth) or the Sorities paradox (vagueness) or Russell's paradox (set theory). But there can be, and have been, deeper motivations, connected with questions about the limitations of human thought, and even Buddhist notions about the nature of ultimate reality. If you're interested in that sort of issue, have a look at Graham Priest's book In Contradiction or his more recent book Towards Non-Being , which is on a slightly different but related topic. I'll add that my own view is that contradictions cannot be true and that, even if they could, that would not help us solve the sorts of paradoxes I mentioned. But that doesn't mean such views aren't worth taking seriously. I could be wrong!!!

Just for clarity, and not that Prof Rapaport needs me to tell him this, but it is important to distinguish the question whether contradictions can be true from the question whether one can get oneself into a situation in which one was believed . I rather suspect that we most or even all of us have contradictory beliefs of one sort or another, and that might motivate the view that classical logic is not a good theory of how we ought always to reason . But as Gilbert Harman famously pointed out, it isn't obvious that logic should be in the business of formulating norms of reasoning. Maybe what it does is simply study the notion of truth-preservation. So classical logic might be a good theory of validity, but not a good theory of how to reason, and maybe paraconsistent or relevance logics (or probabilistic analogues thereof) are better theories of the latter. For what it's worth, my own view is that Harman's point, though fundamentally correct, needs very careful handling and that, even in the...

Suppose I have never played a game of chess. If I now make the claim that I've won all the games of chess I've ever played, is that claim true, false, or undefined? A group of friends had an argument over this, and I figured that philosophers are deeply logical thinkers that can give us the answer and also to get a proper understanding of why the answer is what it is.

The claim is true. There is no game of chess that you have ever played and lost. That said, if you say that every game of chess you have ever played you have won, then you have said something very misleading . But that is different from saying something false. H.P. Grice started the development of a theory that would explain that difference.

Is it possible for two tautologies to not be logically equivalent?

The term "tautology" has no established technical usage. Indeed, most logicians would avoid it nowadays, at least in technical writing. But when the term is used informally, it usually means: sentence (or formula) that is valid in virtue of its sentential (as opposed to predicate, or modal) structure. I.e., the term tends to be restricted to sentential (or propositional) logic. It is clear that Rapaport is assuming the sort of usage just mentioned: "a tautology is a 'molecular' sentence...that, when evaluated by truth tables , comes out true no matter what truth values are assigned to its 'atomic' constituents". Hence, on this definition, "Every man is a man" would not be a "tautology". Which is fine. It's logically valid, but not because of sentential structure. It is all but trivial to prove, as Rapaport does, that all tautologies are logically equivalent. In fact, however, it is easy to see that Rapaport's proof does not depend upon the restriction to sentential logic. One can prove ...

I have a question about "solved" games, and the significance of games to artificial intelligence. I take it games provide one way to assess artificial intelligence: if a computer is able to win at a certain game, such as chess, this provides evidence that the computer is intelligent. Suppose that in the future scientists manage to solve chess, and write an algorithm to play chess according to this solution. By hypothesis, then, a computer running this algorithm wins every game whenever possible. Would we conclude on this basis that the computer is intelligent? I have an intuition that intelligence cannot be reduced to any such algorithm, however complex. But that seems quite strange in a way, because it suggests that imperfect play might somehow demonstrate greater intelligence or creativity than perfect play. [If the notion of "solving" chess is problematic, another approach is to consider a computer which plays by exhaustively computing every possible sequence of moves. This is unfeasible with...

This is a very good question. It is reminiscent of the debate over the so-called "Turing Test", in particular, of an objection to the Turing Test made by Ned Block: his "Blockhead". See the SEP article on the Turing Test for more on this. In the case of chess, it is generally believed that chess is solvable in principle. There are only finitely many possible moves at any stage, etc. So, in principle, a computer could check through all the possibilities and determine the optimum move at each stage. Practically, this is impossible at present, as there are too many moves. But if chess had been solved, and if a computer were simply programmed to make the best move at each stage, then it seems quite clear that no "intelligence" would be involved. Of course, this does not by itself show that "intelligence cannot be reduced to any...algorithm", and the question whether it could be is hotly disputed. There are some famous (or infamous) arguments due to Lucas and Penrose that attempt to establish...

In writing mathematical proofs, I've been struck that direct proofs often seem to offer a kind of explanation for the theorem in question; an answer the question, "Why is this true?", as it were. By contrast, proofs by contradiction or indirect proofs often seem to lack this explanatory element, even if they they work just as well to prove the theorem. The thing is, I'm not sure it really makes sense to talk of mathematical "explanations." In science, explanations usually seem to involve finding some kind of mechanism behind a particular phenomenon or observation. But it isn't clear that anything similar happens in math. To take the opposing view, it seems plausible to suppose that all we can really talk about in math is logical entailment. And so, if both a direct and an indirect proof entail the theorem in question, it's a mistake to think that the former is giving us something that the latter is not. Do the panelists have any insight into this?

I probably should have noted before that, in the case of the different proofs of the first incompleteness theorem in Boolos, Burgess, and Jeffrey, the first proof they give is indirect or, as it is sometimes put, non-constructive: The proof shows us that, in any given consistent theory of sufficient strength, there is an "undecidable" sentence, one that is neither provable nor refutable by that theory; but the proof does not actually provide us with an example of an undecidable sentence. The second proof, which is closer to Gödel's own, is direct and constructive: It does give us such a sentence, the so-called Gödel sentence for the theory. By doing so, it gives us more information than the first proof. It shows us, in particular, the there will always be an "undecidable sentence" of a very particular form (a so-called Π 1 sentence). This is a good example of why constructive proofs are often better than non-constructive proofs: They often give us more information. But it does not directly...

Anyone with any mathematical training will be familiar with the fact that proofs in mathematics do much more than just show that the statement proved is true. One way this manifests itself is that we often value different proofs of the same theorem. Thus, as Jamie Tappenden once pointed out, Herstein's Topics in Algebra , which was the standard algebra text when I was a student, contains three different proofs of the Stone Representation Theorem . Boolos, Burgess, and Jeffrey's Computability and Logic , one standard text for an intermediate logic course, similarly contains multiple proofs of several of the key results, including Church's Theorem on the undecidability of first-order logic and Goedel's First Incompleteness Theorem. And, oddly enough, I myself have just re-proven an existing result in a way that, I think, is clearly better. But not because the original proof wasn't convincing! It's an interesting question, though, why we value different proofs. Somehow, they seem to throw...

Is it wrong to fantasize about sex with children? If a pedophile never acts on their fantasies are they still guilty of having evil thoughts, assuming that their abstinence comes out of a genuine desire not to do harm?

So far as I can see, there's nothing wrong with fantasizing about sex with children. There's nothing wrong with fantasizing about anything you like. If that seems crazy, then it's probably because you are thinking that someone who fantasizes about something must actually wish to do that thing. But that is just not true. As Nancy Friday makes very clear in My Secret Garden , her classic and groundbreaking study of female sexual fantasy, fantasy is not "suppressed wish fulfillment". The point runs throughout the book, which you can find on archive.org , but maybe the best statement is on pp. 27-8, though see also the poignant story that opens the book (pp. 5-7). I'd post an excerpt, but the language maybe isn't appropriate for this forum! As Friday's studies reveal, people fantasize about all kinds of things. Some women fantasize about being raped. It's a very common fantasy, in fact. That does not mean these women actually want to be raped, on any level. As Friday remarks, "The message...

Do philosophers avoid figures of speech in peer reviewed philosphy journals? What about in everyday life; is there a lower standard of conduct when talking to non-philosophers?

By "figures of speech", I'll assume you mean something like metaphor. And, if so, then, no, philosophers do not avoid metaphor, at least not entirely. Here is one of my favorite philosophical metaphors, from W. V. O. Quine: "The lore of our fathers is a fabric of sentences. ...It is a pale gray lore, black with fact and white with convention. But I have found no substantial reasons for concluding that there are any quite black threads in it, or any white ones." Quine would later describe that lore as a "web", which has proven very fruitful. What is true is that philosophers (at least the philosophers I know) try not to settle for such metaphors. One tries to "unpack" the metaphor, and make the underlying point as explicit as possible. But it is, I think, pretty widely appreciated that there is a limit to how far one can go in that direction. Really good metaphors are, as people who work on metaphor say, "inexhaustible", in some sense. There's always more you can dig out of them. That's maybe not...

I recently heard someone make an argument, something like- "if you accept that there is morality in sex, for example that a father having sex with his daughter is wrong, you can't say gay sex isn't immoral because people should be able to do whatever they want because it causes no harm to others" Is this argument or proof begging the question? Philosophically, what is wrong with this argument.

The main thing wrong with the argument is that it is terrible. Don't we think it's wrong for parents to have sex with their children precisely because we think that it is harmful to the children? One might also think that children have no genuine capacity to consent to sex, an issue that also arises in other settings, such as between a boss and an employee. In such a setting, there are always issues about coercion, even if such coercion is not explicit. Presumably the thought is supposed to be that there are forms of sex that are morally suspect, even though they do not cause any sort of harm. But then one wants to know what those are supposed to be. Then we could consider whether and why they are morally suspect. The example given, as I said, is a very bad one.

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