Tuesday, August 30, 2005

All of None

Some might deny that "all ravens are black" means the same thing as "there are no non-black ravens". The latter claim is true even if there are no ravens at all. But it seems odd to say that the former would be true in such a case. After all, it would be just as true to say "all ravens are entirely non-black", since there are no black ravens to contradict it. From these we can infer that "all ravens are both black and entirely non-black", which seems very odd indeed. Logicians mutter that it's just another way of saying that no ravens exist at all. But it clashes sorely with common linguistic usage. We might instead hold that universal statements have existential import: to say "all ravens are black" implies that there are some ravens.

This would provide a simple resolution of the raven paradox - for while randomly sampled red herrings might confirm that there are no coloured ravens, they do nothing at all to confirm the existence of black ravens, so if the universal claim is committed to such existence, then the red herrings do not confirm that "all ravens are black" after all. Our intuitions are saved! (Not really: read on.)

It's worth noting just what the randomly sampled red herrings really show. They don't specifically confirm that there are no coloured ravens. This implication is very much implicit -- red herrings don't say anything direct about ravens. Rather, the fact that our random sample turned up a red herring indicates that red herrings are probably quite common. (If you pick an object at random, chances are you're going to pick an object of a common rather than rare type, for obvious reasons.) We might say that it directly confirms the claim "everything is red herrings". This in turn then implicitly confirms the logical entailments, "there are no ravens", and particularly, "there are no non-black ravens", which the modern logician takes to be equivalent to "all ravens are black".

But the red herring doesn't just confirm that all ravens are black (without existential import); it equally confirms that all ravens are green, and that no ravens exist at all. It is only from our background knowledge that some black ravens exist that we rule out the other equally "confirmed" hypotheses, and say that the observation supports the specific claim that all ravens are black (rather than, e.g., that no ravens exist at all).

I think this helps explain why the herring/raven confirmation seems paradoxical to people. The observation in itself says nothing specific about ravens, and that's why we suppose that it cannot confirm claims about ravens (e.g. that all ravens are black). But it can confirm that there are no (or relatively few) ravens. And while this alone cannot support "all ravens are black" over "all ravens are green", the latter can be ruled out by our background knowledge.

Through most of this I've been assuming, with modern logicians, that universal statements have no existential import. "All X are Y" is true if no X's exist, no matter what Y might be (it can even be a contradiction, as we saw above). But in fact the herring confirmation of "all ravens are black" goes through regardless once we factor in background knowledge. Given that some black ravens exist, the existential condition is already satisfied, so all we need to confirm in addition is that there are no non-black ravens, which the randomly sampled red herring does help to confirm.

But enough about ravens already. I'm wondering: do universal statements have existential import?

Let's take an example: (M) All mermaids are green.

Is M true? False? Neither? I don't know what to make of it. The problem with stipulating that empty universals are vacuously true was mentioned above: it implies the truth of apparent contradictions such as "all mermaids are not mermaids". But existential import fares no better, for that implies that empty universals are all vacuously false, which is implausible for tautologies like "all mermaids are mermaids" (which is surely true even if no mermaids exist).

Perhaps what we need is for the quantifier to range over non-existent possibilia in such cases. We go to some appropriate possible world where mermaids exist, and assess the truth of the universal in that world.

Are all unicorns white? I think so. Are they all black? Nope! Is it possible to get a coloured unicorn? Maybe in some distant possible world, but that need not contradict our judgment that all unicorns are white in the appropriate world.

Hempel actually discusses this issue in his original raven paradox paper ('Studies in the Logic of Confirmation', pp.16-17). One reason he offers, to think that universal statements don't have existential import, is that some just obviously don't. An astronomer's universal hypothesis about what happens in extreme conditions "need not imply that such extreme conditions ever were or will be realized".

But in fact this supports my view, not his. We want to say the astronomer's hypothesis is true only if his prediction would happen were the extreme conditions to occur. In other words, the hypothesis must hold true in the nearest possible worlds where the extreme conditions obtain. Otherwise we would hold it to be false. Logicians, by contrast, would hold the universal statement to be vacuously true if the conditions never actually obtained. They would say the opposite prediction would be just as (vacuously) true. But this is clearly wrong. His prediction is not a vacuous one, and it can be either true or false, no matter whether the conditions actually obtain.

(One way to take my point here is to say that such claims are not properly analyzed in terms of universal quantifiers and material conditionals. They are better understood as counterfactual claims, if they refer to nothing actual.)

1. Logicians just are not practical.
If you measure them against that standard and compare it to yourself and your relitively intuitive standards they are doomed to fail.
They are, however, likely to fair well in terms of internal consistancy and so forth.

2. You only need to post your comments once. (Try refreshing your browser if it doesn't seem to be appearing.) I deleted the four extra copies for you :P

3. Would you extend the same approach to counterfactuals in purely propositional logic? e.g.

If the moon is made of green cheese then I am a hamster.

Is this true or false?

4. It depends what you mean by "if-then". It's true by the definition of material conditionals, of course. But these are notorious for failing to capture our common-sense understanding of conditional statements. (It certainly sounds false to my ear.) Actually, I talked about this stuff a bit last year. There are definite connections between these issues, which I hadn't noticed before you mentioned it. Thanks for the reminder.

5. Medieval logicians regarded all categorical affirmative statements as having existential import, and their logical work was quite as sophisticated as some of the most sophisticated twentieth-century work. (There was a sort of logical dark ages between the fourteenth and the nineteenth centuries, in which knowledge of logic regressed; many of the logical discoveries of nineteenth and twentieth centuries were really re-discovering things that were already discovered by medieval scholastics. The primary disadvantage of the medievals is that they had no literal calculus, so had to write everything out longhand.)

There has been a lot of good work in this area by scholars of medieval logic, particularly Gyula Klima. The following article is particularly relevant:

Existence and Reference in Medieval Logic (PDF)

6. Reminds me of this quote from Peirce:

"Every phoenix, in rising from its ashes sings 'Yankee Doodle,'" will be, we may be confident, not in conflict with any experience. If so, it is perfectly true."

7. Please, this has been going on and on. If propositions were fully defined as they should be then paradoxes would not appear .. eg.. ' all existing ravens are black' is complete.
Or you could define it even further to one's taste.

Maybe it should be considered as an exercise in student ability to detect that they are being led up a garden path by their own assumptions.
Following ever more remote logical processes to explain something that inevitably ( due to lack of definition) can not be clear in itself.

8. Anon, nobody is forcing you to discuss these issues if you don't want to. But you are in no position to criticize the rest of us for doing so. Two points:

1) Your claim that the paradoxes under discussion are "inevitably" unclear, is quite an assumption in itself! (And demonstrated false by my previous posts which have gone a long way to clear up the raven paradox, to my mind.)

2) Your example sentence means exactly the same thing as "all ravens are black", so it faces all the same questions. For one: supposing that there are no existing ravens, then what is the truth value of your sentence?

Of course, one can clear up the definitions by comparing the following sentences:
(i) If there are any ravens, then all of them are black;
(ii) There are some ravens, and all of them are black;
(iii) In the nearest possible world with ravens, all of them are black.

But my question is, which of those three, or something else entirely, is meant by the commonplace sentence "all ravens are black"?

9. haha thanks richard - I was using an inferior browser unfortunatly

10. Richard, why does the sentence have to mean one of them. Why can't we just say that the sentence is vague until put into a particular assertion. i.e. used pragmatically

11. Okay, yeah, that might be a better idea actually.

12. You point out that "material conditionals" (what exactly you mean by this I don't know), and (I believe) by implication logic in general, "are notorious for failing to capture our common-sense understanding of conditional statements."

If so, doesn't that say more about our common sense than about logic? Genius suggests that if you compare your intuition and logic, you'll find logic comes up short. I suggest the exact opposite: if you measure your intuition against logic and often find any difference at all, you should be seriously questioning your intuition.

13. covaithe,
I think you are misinterpreting me a little.
I agree that logic is better than intuition. I just noted that logicians need not create practical pieces of logic. Just because the logic is true doesnt mean it is applicable. Such a mismatch is likely to create an apparent "paradox".

14. Covaithe, I'd agree that logic takes precedence when it comes to making inferences. But the controversy here is over how to translate a common-sense English statement into the language of logic. It is perfectly legitimate for our intuitions to rebel here and say, "No, that isn't what I meant at all. That's a bad translation."

Material conditionals interpret the statement "if P then Q" as being logically equivalent to "Either P is false or Q is true". This is, in many cases, simply a bad translation of what English speakers mean when they use the words "if-then".

Consider:
(D) "If I die tomorrow then I will sing the next day"

D seems false. So not-D is true. But if translated as a material conditional, D means "Either I won't die tomorrow, or I will sing the next day." And the denial of this, i.e. not-D, is "I will die tomorrow and I will not sing the next day."

So, according to "logic", if D is false we can infer that I will die tomorrow. This is patently absurd. The material conditional translation fails to capture what is meant by the English sentence D.

15. That does make more sense, Genius, and I'm sorry if I misinterpreted. Still, your original statement is, at best, confusing.

Richard, what does D mean, in English? To me it seems just as contradictory, albeit perhaps in a vaguely poetically evocative way, as the material conditional version. Also, remember that via Epimenides and Goedel, there are large classes of logical statements that are neither true nor false. So just because the material conditional versions of D and not-D are both absurd doesn't mean that that mc translation from English is wrong, necessarily. The English statement itself might be absurd.

(Of course I realize, and you probably do too, that in English D is implying that if the speaker dies he/she will go to heaven, where there will be singing and such. As such, the m-c version of D is no more false-seeming than the English statement, to my ears.)

16. The material conditional version of D is not contradictory at all. It is very easily true. It is true if I do not die tomorrow.

The English sentence has rather more vexing truth conditions. I read it as a counterfactual: If I did die tomorrow, then I would sing the next day. More rigorously: In the nearest possible world where I die tomorrow, I sing the next day. But this is clearly false: in the nearest possible world where I die, I stay dead.

So the material conditional is a bad translation of (what I mean by) the English sentence.

17. I feel like this is totally off the subject now, but what the hell.

"In the nearest possible world where I die tomorrow, I sing the next day. But this is clearly false: in the nearest possible world where I die, I stay dead."

This assumes that it's impossible to sing while dead, i.e. that there is no afterlife. Surely in a discussion of the applicability of pure logic, we don't need to make such a controversial assumption. If the speaker of the English sentence D does believe in an afterlife involving singing, then there's nothing wrong with the counterfactual, and it seems to me like its truth conditions are identical to those of the material conditional version.

18. > Still, your original statement is, at best, confusing.

Appologies Covaithe :)

19. How can you say the truth conditions are identical? Here are the truth conditions:

m-c version is true if (1) either I do not die tomorrow, or else I sing the next day.

c-f version is true if (2) were I to die tomorrow, I would sing the next day.

Clearly (1) and (2) are not identical.

And, yeah, we are a bit off-topic. But it's important to point out that any application of logic goes beyond "pure logic". Pure logic presumes you've already made appropriate translations, and set the truth values accordingly. But these are extra-logical tasks. "Pure logic" has nothing to say about these questions, and the assumptions that logicians make about them could well be mistaken.

20. Let A be "I do not die tomorrow", and B be "I sing the next day". Then the m-c version is "A or B". Truth table:

A | B | A or B
==================
T | T | T
T | F | T
F | T | T
F | F | F

The c-f version, "were I to die tomorrow, I would sing the next day", is not-A ==> B. Yes? Then,

A | B | not-A ==> B
=====================
T | T | T
T | F | T
F | T | T
F | F | F

The truth tables show that 'not-A ==> B' is true exactly when 'A or B' is true. This is what I mean by saying that the truth conditions are identical for the m-c and the c-f version. Have I made a mistake somewhere?

21. Yup, your "==>" operator is the material conditional. What you've shown is that the material conditional (not-A ==> B) is logically equivalent to (A or B), like I said at the start. Counterfactuals are completely different.

You can't write a truth table for counterfactuals, because they depend on more than actual-world truth values. They also depend on truth values at other possible worlds.

To illustrate: suppose that there is no afterlife, so if I died, I would not sing the next day. Further suppose that I will not die.

Then (D) is false on the c-f version, but true on the m-c version.

Those are the different truth conditions. m-c depends only on actual truth values: in particular, the truth of A suffices for the truth of (not-A ==> B). But clearly the truth of A does not suffice for the truth of D! The mere fact that I will survive does not logically suffice to establish that if I were to die I would sing the next day. It is not so easy to prove there is an afterlife ;)

22. Clearly I am not familiar with the kind of counterfactuals you are using, and/or I'm failing to understand your explanation of the distinction between them and normal statements of implication, where "==>" is simply shorthand for "implies".

You say that counterfactuals "depend on more than actual-world truth values. They also depend on truth values at other possible worlds." But this is exactly what truth tables capture! In any given world, no possible statement A can be both false and true, so when the truth table captures all possible truth values of A, it does so in all possible worlds. This is the great advantage of formal logic; that it applies to all possible worlds.

You say "suppose there is no afterlife, so if I died, I would not sing the next day", or, C ==> not-(A or B), where C is "there is no afterlife", and A and B are as before. And, "Further, suppose that I will not die [tomorrow]." So, A. But then you conclude that "(D) is false on the c-f version, but true on the m-c version." Huh? I guess I'm just totally failing to understand under what conditions a couterfactual is considered "true". It certainly doesn't seem to have any relationship to logical statements as I know them.

23. Ack, I hadn't quite finished reading your last paragraph when I wrote the above. I'm not sure whether it changes my mind; need to think.

24. I think we may be mischaracterizing the meaning of the truth tables. As you observe, A being true does not prove (not-A ==> B) is true, obviously. But I don't think that the truth table claims this. The point of truth tables is to show that two or more statements are equivalent. It's not a proof that either of the statements is "true". Perhaps using T and F in the columns is misleading; it's traditional, but that's not necessarily a good reason to continue the practice.

To show that an implication X ==> Y is true, you have to show that it's impossible for Y to be false without X also being false (not-Y ==> not-X, the contrapositive). In other words, to show that the conditions in the row of the truth table where there's an F for X==>Y never occur, in any possible world.

Considering D again, if we write it in the m-c version as (A or B), then we look at the truth tables and identify the rows with an F, we see that the only such row is when A is false and B is false. Then we ask, does this ever happen? Is there any possible world in which both "I die tomorrow" and "I do not sing the next day" are true? Yes! It's possible in any world where there is no afterlife. So D, interpreted as a material conditional statement, is false. But we might instead consider E: "If there is an afterlife, and I die tomorrow, then I will sing the next day." Is *this* true? Going through the same process, one might say no, since in worlds with an afterlife that prohibited singing it would fail. But G: "If I die tomorrow and there is an afterlife where everyone sings all the time, then I will sing the next day." That seems to work, because there are no possible worlds where I die tomorrow and that have a singing-all-the-time afterlife in which I do not sing the next day.

Where does the counterfactual fit into this? I'm still not sure. :-P

25. We should distinguish entailment from material implication. What you call "implication" above is really entailment. There's no truth table for entailment, because truth tables only involve a single world at a time, whereas entailment depends upon all possible worlds.

"A" being true really is sufficient for the truth of (not-A ==> B). The truth table shows this. "A implies B" just means that either A is false or B is true. The only truth values that matter are the actual world. You don't need to appeal to other possible worlds at all. (Though of course you could choose to apply the truth table to any world. You could say that at world w, "A implies B" is true at w iff "B or not-A" is true at w. But again, this formula doesn't appeal to any other worlds besides w, the world where we assess the truth of the statement.)

Entailment, on the other hand, holds necessarily. To show that X entails Y, you must show that it is impossible for X to be true and yet Y false. "X entails Y" is like saying "X implies Y in all possible worlds". That's a much stronger claim than plain old "X implies Y", that is "Y or not-X", which only makes claims about the actual world.

Anyway, we're getting rather far afield here. The crucial points are:

1) The truth of a material conditional "A implies B" depends only upon the actual-world truth values of A and B. It is a simple truth-functional connective like "AND" or "OR". "A implies B" is true if and only if either A is false or B is true. You've misanalyzed it above. Other possible worlds have nothing to do with it.

2) The truth of counterfactual statements (and most English uses of "if-then") depend upon more than just actual world truth values. This is no simple truth-functional connective. Knowing the truth values of A and B will not tell you whether the counterfactual "If A had been false then B would have happened" is true. This depends upon more than the actual truth of A and B. It depends on what WOULD have happened if A HAD been false. This is a modal claim. Much more complicated.

26. Simple version:

Material conditionals only depend upon what IS (actually) the case. If you know what IS true, then you can know the truth of all implication claims.

Counterfactual claims instead depend upon what MIGHT HAVE BEEN. Knowing everything that IS true (at the actual world) does not tell you what MIGHT or WOULD have been (at other possible worlds).

27. You say that a truth table only describes a single world. How can this be, since it includes rows where A is true and A is false? A row of a truth table, say the row for A is true and B is true, describes, not a single world, but collection of all worlds where A and B are both true. Together, the rows of the truth table describe all worlds.

To put it another way, a truth table isn't a claim about the correctness of any assertion, it's just a very explicit way of writing that assertion. By writing a truth table for "X ==> Y", we aren't claiming anything about whether "X ==> Y" is true or not, we're just explaining in very explicit detail exactly what we mean by "X ==> Y", exactly what relationship it describes between X and Y.

Except for tautologies (e.g. "A ==> A"; more generally, relationships where the truth table contains only T's), the truth of any logical statement is always qualified by which worlds it holds true in. When we prove theorems in mathematics, we are only proving that they are necessary given our axioms, i.e. that they hold true in all the worlds where our axioms are correct. So I think your "entailment" is not what I'm trying to describe, or what I mean by "==>". But I certainly don't that the truth of "not-A ==> B" is restricted to a single world. Clearly there's a large class of worlds where it is true, and another class of worlds where it is not true.

I kind of suspect we're getting close to the point in the discussion where we're saying the same things, just in different words.

28. I mean that the various columns of a truth table are all talking about the same world. The actual truth of (A or B) only depends upon the actual truth of A and of B. It doesn't depend upon what happens at any different possible worlds.

Now, I'm still not convinced whether you're clear on the difference between material conditionals and counterfactuals.

"Considering D again, if we write it in the m-c version as (A or B), then we look at the truth tables and identify the rows with an F, we see that the only such row is when A is false and B is false. Then we ask, does this ever happen? Is there any possible world in which both "I die tomorrow" and "I do not sing the next day" are true? Yes! It's possible in any world where there is no afterlife. So D, interpreted as a material conditional statement, is false."

This reasoning is completely wrongheaded. To find out whether (A or B) is true, you do NOT ask whether there is "any possible world" in which it is false. The question is not whether the formula is necessarily true. We're merely asking whether it's true, simpliciter. That is, true at the actual world. Its truth is simply determined by the truth values of A and B at the actual world.

Here's how truth tables work: You find out the truth value of A, and the truth value of B. You then find the row with the appropriate values. Now you look across to the column of our other statement, (not-A ==> B) or whatever, and the table tells you the truth value of that. No other possible worlds are involved.

If you think the truth of "A implies B" depends upon any worlds apart from the actual one, then you are simply mistaken.

Here are the two crucial issues:

1) Do you understand the difference between material implication and counterfactuals?

2) Do you see why material implications are a poor translation of many English "if-then" statements? That is, do you agree that "if not-A then B", in English, often means something VERY DIFFERENT from "either A or B"?

If the answer to any of these questions is 'no', then please re-read my previous comments carefully. I'm afraid I'm feeling too frustrated to continue the discussion.

Perhaps I should offer a concrete example before finishing up. Consider: "If today were Friday then pigs would fly". As a material conditional, this is equivalent to "Either today is not Friday, or pigs fly", and thus is true -- simply because today is not Friday.

But of course the counterfactual is NOT true. If today had been Friday, pigs wouldn't suddenly grow wings. It's absurd. Moreover, the translation is OBVIOUSLY a bad one. "If today were Friday then pigs would fly" does not mean the same thing as "Either today is not Friday, or pigs fly". Anyone who claims otherwise simply doesn't understand English!

It's just an obviously bad translation. And that's the only point I was making all those comments ago.

29. If you're not enjoying the discussion, then perhaps we should let it go. I admit I've been enjoying it, it in a kind of exasperated way, perhaps, but enjoying nonetheless. I'm twisted like that. :)

Your point about English and poor translations to logic is well taken. However, logic does have a (spoken) language, and it does (unfortunately, perhaps) overlap with English. And as always, whenever there are two ways of interpreting some statement, different people are going to interpret it different ways. If I said to you, "if today were Friday then pigs would fly", it would be correct to assume that I meant "Either today is not Friday, or pigs fly". I probably wouldn't make such a statement, but if I did, that's what I would mean. The way I use the "if...then..." language construct, the statements are identical. Too many years of studying formal math proofs and computer programming have completely changed the way my personal brain interprets and uses the words "if" and "then". You might argue that I'm not speaking English, I'm speaking "logic", and I probably wouldn't disagree. I don't think I'm alone, though.

30. when I saw this
"If today were Friday then pigs would fly"
I was a bit confused because my brain immediatly skipped to pigs flying every friday. After more thinking - it still sounds funny. having said htat I never pretend to be normal !

You know, covaith - we should spend more time butting heads we will probably go a long time before getting too frustrated !!

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