## On April 10, 2014, in response to a question, Stephen Maitzen wrote: "I can't

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...

### Because the present

Because the present questioner refers to my reply to Question 5536, I'll chime in here to clarify what I said there.

My point was about the fundamentality of LNC. I wrote, "I can't see how there could be any law more fundamental than the law of non-contradiction (LNC)." I gave the following reason: "Let F be any such law. If the claim 'F is more fundamental than LNC' is meaningful (and it may not be), then it conflicts with the claim 'F isn't more fundamental than LNC' -- but that reasoning, of course, depends on LNC." So that's why no law could be more fundamental than LNC, because LNC would need to be true before (in the sense of logical priority) the claim that some other law is more fundamental would even make sense.

If someone can make sense of the claim that some law is more fundamental than LNC, I'm all ears.

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

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.

It would, of course, be equally true that you've lost every game of chess you've ever played. Bad news to go with the good.

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.

## So I'm reading The Power of Logic, 4th edition. While on a section describing

So I'm reading The Power of Logic, 4th edition. While on a section describing Modus Tollen it says that, Not A; If A, then B; So, Not B is an example of Modus Tollen. My question is how can that be if the conclusion of Modus Tollens is suppose to deny the consequent? Am i reading it wrong or just missing something? Keep in mine im still not beyond chapter 1.

There is either a typo in the book you are reading, or else you reported its contents wrong. Modus tollens is: If A, then B; not-B; so not-A. The version you reported is fallacious. It's a version of the fallacy of asserting the consequent.

There is either a typo in the book you are reading, or else you reported its contents wrong. Modus tollens is: If A, then B; not-B; so not-A. The version you reported is fallacious. It's a version of the fallacy of asserting the consequent.

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

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 (as he of course knows) by exactly the same argument that all "valid" sentences are logically equivalent. (All logic texts prove this. It is, for example, general law (12) on p. 64 of Warren Goldfarb's Deductive Logic, which is the text we use at Brown.) The argument goes through so long as "valid" means "true in every interpretation" and "equivalent" means "have the same truth-value in every interpretation". In particular, it doesn't matter what is meant by an interpretation (at least as long as the interpretations are classical).

A couple other remarks on the discussion.

First, when Rapaport says that "two [interpreted] sentences...are logically equivalent if and only if they have the same truth values (no matter what truth values their atomic constituents, if any, have)", he is talking not about logical equivalence in general but about "truth-functional" equivalence, since only sentential constituents can have truth-values. And "Snow is white" and "Obama was born in Hawaii" are not, of course, truth-functionally equivalent. The sole atomic constituent of each sentence is that sentence itself. (Constituency is so defined as to make it a reflexive relation, trivially.) And an interpretation can perfectly well assign these two sentences different truth-values. So this is just a case of P not being equivalent to Q.

Second, as Rapaport notes, there is some controversy whether the basic notion here should apply to sentences (or formulas) or to propositions. But if it applies to syntactic items (as I would prefer), then the sentences and formulas have to be regarded as "interpreted", i.e., as having fixed meanings, or at least the "logical constants" have to be so regarded. Otherwise, indeed, no sentence will be "always true". But if the meanings are held fixed, then, for many purposes, the difference makes no difference, since the notions are inter-definable: A sentence is (logically) true if the proposition it expresses is (logically) true; a proposition is (logically) true if there is a (logically) true sentence that expresses it. The remaining philosophical issue is which notion is more fundamental.

Finally: For lots of interesting material on the word "tautology" and the history of its use in logic, see Burton Dreben and Juliet Floyd, "Tautology: How Not To Use a Word".

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 ...

## This is a follow-up to Miriam Solomon's statement describing philosophy:

This is a follow-up to Miriam Solomon's statement describing philosophy: "Philosophy involves more than deductive logic--it involves the exercise of "good judgment" which in fact we do not understand very well." (june 5, 2014) Can someone tell me more about what this "good judgment" is, please? I studied philosophy in college and I can't recall any of my professors ever suggesting that there was some elusive guiding principle in philosophy beyond what could be articulated...Instead, I was taught that it was about starting with premises and then executing deductive reasoning. Are you now saying that there's something mystical in there that philosophers can't articulate but which guides their work? That seems counter the way I learned philosophy, where the professors seemed particularly intent on articulating things clearly.

I'll just add that, for similar reasons, "good judgement" is equally important in mathematics, and nothing is more deductive than mathematics.

I'll just add that, for similar reasons, "good judgement" is equally important in mathematics, and nothing is more deductive than mathematics.

## I have a question about "solved" games, and the significance of games to

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...

Update: An interesting article about one of my computer science colleagues on the subject of cheating in chess and touching on the nature of "intelligence" in chess just appeared in Chess Life magazine; the link is here

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

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 Π1 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 address the issue about explanation.

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

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?

I'm sympathetic to most of what Professor Heck says, if we consider things from a deontological or even a consequentialist point of view, where the relevant consequences are external to the agent. Fantasy does not violate anyone's rights, and fantasy that never motivates action will not result in actions that harm anyone. But I think there is a plausible way of looking at things that would still find fault with fantasizing about having sex with children, and that would come from the aretaic (or virtue-theoretic) way of thinking, according to which the primary bearer of value is to be found in characteristics of agents. One who indulges in fantasies about sex with children is doing something that both reflects--and also perhaps perpetuates and sustains--a certain trait of character that we might think is not entirely wholesome or admirable. To the extent that we can regard one who indulges in such fantasies as having a trait of character that is improvable, we might also think that some attempt to eliminate or at least diminish the inclination to indulge in such fantasies would result in that person having some improvement in character. It may be that habituation can only go so far, and that virtue theorists (such as Aristotle) overrate the extent to which one can habituate better character traits, but it certainly does seem that a virtue theorist could find the character of someone who tends to indulge in such fantasies at least improvable, and this way of looking at things does, I think, put a different face on this kind of case than what Professor Heck has indicated.

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?

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 true of the metaphor from Quine just quoted, but there are tons of quite striking examples in Donald Davidson's paper "What Metaphors Mean". (There's also a lot of irony there, since Davidson's view is precisely that metaphors have no meaning beyond their literal meaning.)

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

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.

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.