## 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 ...

## Are there any philosophers who deny that the principle of explosion is a valid

Are there any philosophers who deny that the principle of explosion is a valid principle while at the same time both being not accepting of a paraconsistent logic and being accepting of the Law of Non Contradiction?
According to the article on paraconsistent logic at the Stanford Encyclopedia of Philosophy : "A logical consequence relation, ⊨, is said to be paraconsistent if it is not explosive." So denying explosion just is accepting a paraconsistent logic.

## How does one determine which side in an argument must shoulder the burden of

How does one determine which side in an argument must shoulder the burden of proof?

The other guy has the burden of proof. And yes, I'm serious. It's that bad.

But, to elaborate a little bit, I despise burden of proof type arguments. I do not know of any reasonable way of telling who "ought" to have the burden of proof, and I'm not sure I understand what is supposed to follow from someone's having it. People often end arguments saying something like, "Since they have the burden of proof and haven't met it, it is reasonable for us to believe my view". But this seems to me an odd way of thinking about philosophy.

I mean, I do hope that some of the philosophical views I hold will have some influence and help us understand certain sorts of things better than we do. But whether any of my views might actually be true I very much doubt. And the fact that the other guy hasn't been able to knock my view down doesn't seem like good reason to believe it, even if my view is more common-sensical than his (a common test). Philosophy seems to me to be much more a hunt for understanding than it is one for truth, and I'm not sure "burden of proof" has much to do with understanding.

The other guy has the burden of proof. And yes, I'm serious. It's that bad. But, to elaborate a little bit, I despise burden of proof type arguments. I do not know of any reasonable way of telling who "ought" to have the burden of proof, and I'm not sure I understand what is supposed to follow from someone's having it. People often end arguments saying something like, "Since they have the burden of proof and haven't met it, it is reasonable for us to believe my view". But this seems to me an odd way of thinking about philosophy. I mean, I do hope that some of the philosophical views I hold will have some influence and help us understand certain sorts of things better than we do. But whether any of my views might actually be true I very much doubt. And the fact that the other guy hasn't been able to knock my view down doesn't seem like good reason to believe it, even if my view is more common-sensical than his (a common test). Philosophy seems to me to be much more a hunt for understanding than...

## If the sentence "If p then q" is true, must the sentence "q because p" also be

If the sentence "If p then q" is true, must the sentence "q because p" also be true? For example, "if it is raining, then the streets are wet" and the sentence "The streets are wet because it is raining." Are there any counter-examples where "If p then q" could be true while "q because p" could be false?

Even if the conditional isn't material, it's clear that this kind of inference has to fail. Suppose my roof leaks whenever it rains. Then it seems true to say: If my roof is leaking, then the streets are wet. But the streets aren't wet because my roof is leaking. Rather, there is a third cause of both these events. Even if there has to be a "link" between them for the conditional to be true, then, the link needn't be directly causal.

Even if the conditional isn't material, it's clear that this kind of inference has to fail. Suppose my roof leaks whenever it rains. Then it seems true to say: If my roof is leaking, then the streets are wet. But the streets aren't wet because my roof is leaking. Rather, there is a third cause of both these events. Even if there has to be a "link" between them for the conditional to be true, then, the link needn't be directly causal.

## Me and my professor are disagreeing about the nature of logic. He claims that

Me and my professor are disagreeing about the nature of logic. He claims that logic is prescribes norms for correct reasoning, and is thus normative. I claim that logic is governed by a few axioms (just like any in any other discipline, i.e. science) that one is asked to accept, and then follows deductively, free of any normative claims. My question is: which side is more sound? Thank you.

Without disagreeing with Stephen's fine response, let me point out one other issue. You say that "logic is governed by a few axioms...and then follows deductively, without any normative claims". But there is no "following deductively" without logic: logic is about the correct norms of deductive reasoning. So this conception is flatly circular: a point made a long time ago by Quine in his paper "Truth by Convention".

I should say that there are philosophers who deny that logic is about reasoning at all. On this view, logic is about a certain relation between propositions, implication, that it aims to characterize. But then the dispute just shifts to whatever one thinks does characterize the norms of reasoning, e.g, decision theory. And, for what it's worth, my own view has always been that these philosophers have too simplistic a conception of what sorts of norms logic articulates. But that is a larger issue.

Without disagreeing with Stephen's fine response, let me point out one other issue. You say that " logic is governed by a few axioms...and then follows deductively, without any normative claims". But t here is no "following deductively" without logic: logic is about the correct norms of deductive reasoning. So this conception is flatly circular: a point made a long time ago by Quine in his paper "Truth by Convention". I should say that there are philosophers who deny that logic is about reasoning at all. On this view, logic is about a certain relation between propositions, implication, that it aims to characterize. But then the dispute just shifts to whatever one thinks does characterize the norms of reasoning, e.g, decision theory. And, for what it's worth, my own view has always been that these philosophers have too simplistic a conception of what sorts of norms logic articulates. But that is a larger issue.

## I read once that an African tribe was asked a simple logical problem paraphrased

I read once that an African tribe was asked a simple logical problem paraphrased as follows: "Berlin is a city in Germany. There are absolutely no camels in Germany. Are there camels in Berlin?" The tribe could not provide a definitive answer, instead saying things like "I have never been to Berlin, so I cannot say whether there are camels or not" or "If Berlin is a big city, there must be camels" in other words, completely missing the logical puzzle and instead providing more pragmatic answers. Now this story may be apocryphal, since I cannot find where I read it, but it raises an interesting question. To what extent is logic universal, is it culturally biased/culturally learned, and how do we explain the answers of the tribe?

The claim that "logic is universal" is the claim that the norms of correct reasoning are universal. It is not the claim that everyone follows those norms, or that everyone reasons well.

In the story as told (apocryphal or otherwise), the tribesmen are failing to make a certain inference. That makes them poor reasoners, but it doesn't threaten the universality of logic.

The claim that "logic is universal" is the claim that the norms of correct reasoning are universal. It is not the claim that everyone follows those norms, or that everyone reasons well. In the story as told (apocryphal or otherwise), the tribesmen are failing to make a certain inference. That makes them poor reasoners, but it doesn't threaten the universality of logic.

## Why does inconsistency entail validity?

Why does inconsistency entail validity?

Without disagreeing with anything Alex has said, let me just add one more thing: There are logicians who sympathize with this sort of question, and so who would deny that an argument with inconsistent premises is always valid. There are logics, that is to say, that do NOT validate all inferences of the form: A & ~A, therefore B, for arbitrary B. Such logics are called "paraconsistent, and if you'd like to read about them I'd recommend the Stanford Encyclopedia article as a start.

Without disagreeing with anything Alex has said, let me just add one more thing: There are logicians who sympathize with this sort of question, and so who would deny that an argument with inconsistent premises is always valid. There are logics, that is to say, that do NOT validate all inferences of the form: A & ~A, therefore B, for arbitrary B. Such logics are called "paraconsistent, and if you'd like to read about them I'd recommend the Stanford Encyclopedia article as a start.