August 31, 2003

The Architecture of Teaching

It's a horrible room, Evans 70: underground, clockless, windowless, with old blackboards and cheap free-standing metal chairs. Fortunately for me (and them), it is not also horribly overcrowded.

I follow Ken Wachter, who is teaching demography. I precede an upper-division mathematical logic class.

I soon learned when I took mathematical logic that p -> q was supposed to be the same thing as q v (~p). But I never believed it. It didn't make sense that p -> q should automatically be true whenever p was false. Consider the sentence: "If Socrates were a parrot, he would be 200 feet tall." p = "Socrates is a parrot" q = "Socrates is 200 feet tall". Since p is false, p -> q is supposed to be true. But it clearly isn't. "Implication" in mathematical logic is not what I mean by "If... then... "

Posted by DeLong at August 31, 2003 11:43 AM | TrackBack

Comments

My understanding of p->q has always been that it means "if it is the case that p is true, then it is also the case that q is true," and that it has nothing to say about what happens if p is false. Is that wrong?

Posted by: David J. Balan on August 31, 2003 12:00 PM

There's quite a lively debate within philosophy and linguistics about just what the relation between p->q and ~p v q is. Most philosophers (and I think most linguists) think there are two classes of conditionals, and some think that p->q is true just in case ~p v q is for one of those classes.

The boundary between the two classes is not sharply defined, but they are meant to be illustrated by the following pair

(I) If Oswald didn't shoot Kennedy, someone else did.
(S) If Oswald hadn't shot Kennedy, someone else would have.

Since a more-or-less full-convinced Warrenite believes (S) but doesn't believe (I), these say different things. But they are both of the form ~Oswald shot Kennedy -> Someone else shot Kennedy. So we conclude there are (at least) two conditionals.

For reasons not altogether clear, conditionals like (I) are called indicatives and conditionals like (S) called subjunctives. No one thinks that conditionals like (S) are true whenever p is false. But many people think that conditionals like (I) are true whenever p is false. (I'm not one of them, but they are many.) Here's the best reason to think that. The following argument looks valid.

Either the butler did it or the gardener did.
So, if the butler didn't do it, the gardener did.

But all it takes for the premise to be true is that the butler did it. (Unless we have a very strange theory of what 'or' means.) So that's enough for the conditional in the conclusion to be true. But that means "If the butler didn't do it, the gardener did." is true if what comes between the if and the comma is false. If this argument can be generalised to all conditionals like (I), it means any conditional p->q that is grammatically like (I) is true if p is false. This isn't a knock-down argument, but it does seem like a serious consideration in favour of the way the mathematicians treat ->.

Posted by: Brian Weatherson on August 31, 2003 12:24 PM

The "implication" in which p > q is equivalent to ~p v q is called "material implication". It is not the same as logical implication, where q is deducible from p.

Material implication is a artificial invention of the logicians, but it has the advantage that it makes possible a two-valued logic, where every sentence is either true or false. It also has the advantage of making all the logical connectives truth functional, which means that the truth value of a compound sentence is entirely a function of the truth values of its constituent sentences.

The man in the street, if asked whether "If Socrates is a parrot, then he is two hundred feet tall" is true of false, would be hard put to give an answer. Most likely he would say "neither". By using the concept of material implication, we treat the sentence as true, and close any truth value gaps. It's not intuitive, but it works. Ask your local computer programmer if you don't believe it.

The sentence "If Socrates were a parrot, he would be two hundred feet tall" is not subject to treatment as a material conditional. Instead it is what is called a "counterfactual conditional". Counterfactuals are statements having to do with the causal connections among things, and they are not truth functional. The uses of the subjunctive "were" is a sign of a counterfactual conditional. "If Socrates were a parrot, he would be two hundred feet tall" is *false*. If Socrates were a parrot, he would be about ten inches tall.

Posted by: Joe Willingham on August 31, 2003 12:24 PM

Perhaps there's something to be learned from the fact that a relation that's tough for a even a smart guy to follow is at the very center of scientific method, i.e. absence of disproof is not a proof. To use our example, the disproof of the thesis would be to show that Socrates is a parrot and yet not 200 feet tall. However is Socrates is not a parrot, we cannot say anything about the relation.

One must be careful to seperate the varacity of a relation from the veracity of a clonclusion based on that relation. The relation is positive, the conclusion is a negation. "Yes, we have no bananas."

Posted by: Matthew Ernest on August 31, 2003 12:28 PM

Suppose you want to prove 'q'. Then, the assertions ('p' is true) and (p -> q) will yield the assertion ('q' is true). On the other hand, the assertions ('p' is false) and (p -> q) can yield -either- the assertion ('q' is true) -or- the assertion ('q' is false). Any other behavior would be unsatisfactory.

Matt

Posted by: Matt on August 31, 2003 12:48 PM

Matt, substitute the word "neither" for the word "either" and "nor" for "or" in the next to last sentence in your post above and you're right.

Posted by: Joe Willingham on August 31, 2003 12:58 PM

This is called the paradox of material implication.
See http://www.earlham.edu/~peters/courses/log/mat-imp.htm

Apparently the ancient Greeks (as always) were already discussing it.
See
http://mathforum.org/epigone/historia_matematica/speismoiwoo/01e301c36ae8$5deee4c0$6101020a@dm.uniba.it

Posted by: Andrew Boucher on August 31, 2003 01:04 PM

What I'm saying is that if 'p' is false, then 'p -> q' has no bearing on the truth value of 'q', i.e., 'q' may be either true or false.

Matt

Posted by: Matt on August 31, 2003 01:06 PM

Joe Willingham

Well done. Fun stuff, I always thought.

Lise

Posted by: lise on August 31, 2003 04:22 PM

"If Socrates were a parrot, he would be about ten inches tall."

Lovely. We have a parrot named Oliver, but that will do. Say "cookie...." We have several small parrots. "Wanna come? Wanna come"

Posted by: lise on August 31, 2003 04:34 PM

Lise,

A friend of mine has a parrot who is quite nice, except that he likes to chomp on the furniture. I hope yours don't do that!

Posted by: Joe Willingham on August 31, 2003 06:06 PM

I think that there is a parallel in mathematics, namely dividing by zero. We still read of some great proof finally offered, and a few weeks later ..oops... line 3754 is a dividing by zero. Why it compares of course that that if you divide by zero you can make a proof, I understand of almost anything. Likewise if you start from false premises, is there anywhere you cannot end up?

Posted by: secular clergyman on August 31, 2003 06:18 PM

Logic can tell you only whether you have reasoned correctly from the premises to the conclusion. It is up to other sciences to tell you whether the premises are true or not.

For example logic says that the following syllogism is correct:

All swans are white.
Harvey is a swan.
Therefore Harvey is white.

But it is from the zoologists that we must learn the truth or falsity of the premises.

Posted by: Joe Willingham on August 31, 2003 07:03 PM

Mr. Clergyman:
The answer to your question:

if you start from false premises, is there anywhere you cannot end up?

is no. A false premise implies anything. We're talking about the logical expression q v (~p). This is an "or" and it is sufficient condition for the expression to be true that either part of it is true. Given that p is known to be false, ~p is true. Therefore, the expression is true no matter what the value of q.

Posted by: Jonathan Goldberg on September 1, 2003 12:12 AM

Brad: There are other logics that try to capture a notion of implication closer to human intuition, such as intuitionistic logic and relevance logic.

These other notions of implication come at a price, though. Intuitionistic logic surrenders the notion that statements are true or false: some statements remain in an in-between state. Relevance logic goes considerably further -- to make sense of the truth of a statement, you have to consider different possible worlds, including possible worlds where logically impossible statements are true.

Implication in classical logic is what you get when you try to satisfy these three requirements:

1) Statements are either true or false.

2) The truth of "p implies q" can be determined simply by knowing the truth or falsity of p and q.

3) "implies" captures modens ponens, i.e. it's the logical connective that fills in the blank in statements like "if p is true and p ___ q is true then q is true".

Any other notion of implication is in practice considerably more complicated.

Posted by: Walt Pohl on September 1, 2003 01:35 AM

Uh, oh... Both p->q and (q or not p) are false here! What you are saying is that "if it's a parrot then it's 200 feet tall" sounds worse than "it's 200 feet tall or it's not a parrot", I beg to disagree!

We should (and intuitively do) expect both p and q to be statements not only about a 6ft tall man but rather about all objects that have lengths or might or might not be parrots and could be called Socrates! It would otherwise be meaningless to use these rather general expressions just for a specific person. Try it:

p = "Socrates is a parrot"
q = "Socrates is 200ft tall"
Socrates is any known object with length that might or might not be a parrot

Choosing "Socrates = a 6ft tall man" satisfies both p->q and (q or not p), even though it makes p->q look weird. Now, choosing "Socrates = a 200 ft building" *falsifies* both p->q and (q or not p)! Either the statments are ridiculously narrow ones about a 6ft tall man (or something else that fail both p and q), or they are both false - no wonder the example makes perfectly good logic look weird!

Drawing a Venn diagram should make it clear that p->q and (q or not p) are identical. As it should when considering sensible statements that are generally true. And "if it's a car then it's a vehicle" works for some reason much better than "it's a vehicle or it's not a car"!

Posted by: Mats on September 1, 2003 03:09 AM

Parrots and Parrots

Many species of parrots polish and sharpen and possibly exercise beaks by chewing on wood and leather and cloth. That means "toys." We have small chewy parrots, and so make lots of parrot toys. They are disciplined happy chewers. Happy, because from the sounds they make we are convinced they often simply play. Simon the parrotlet chews on toes and stops when we say no biting.

Posted by: lise on September 1, 2003 03:55 AM

"All swans are white.
Harvey is a swan.
Therefore Harvey is white."

- But it is from the zoologists that we must learn the truth or falsity of the premises. -

There is where the interest resides. Some swans are black and white, some are black, signets are sometimes grey, and whites can be off-white. [San Francisco has wonderful black swans roaming the zoo.]

This summer we are worried about a lack of swans in a nearby small lake that has had nesting swans in summer as long as anyone can recall. There is the interest.

Posted by: lise on September 1, 2003 04:44 AM

Swans. This splendid passage was written by an Irish or Scottish writer. I found the passage in a "first chapter" of a novel reviewed recently in the NYTimes, but I can not find the author. I save the reviews and will come on it again.


"Several swans were sailing on the lake amid dark clutches of wildfowl. The occasional lone heron flew between the island and the bog. Nothing was sharp. The lanes of watery light that pierced the low cloud from time to time seemed to illuminate nothing but mist and cloud and water. The sedge of Gloria Bog and the little birches had no color. The mountains were hidden."

Posted by: lise on September 1, 2003 05:24 AM

I don't know why I write this, no one ever seems to pay attention to what I write *sigh*, but I actually have the answer to your question, Brad.

``Since p is false, p -> q is supposed to be true. But it clearly isn't. "Implication" in mathematical logic is not what I mean by "If... then...''

Your intuitions are pointing you to counterfactual reasoning rather than implication. Actually, the difference between counterfactuals and implication is much greater than the difference between material and logical implication.

Basically, given a statement "IF a THEN b", and we know that a is false, counterfactuals are all about creating an "alternative universe" or "mental space" in which a *is* true, and deciding if, in such a universe, b would also be true (or, measure the degree of beleif we would have that b is true).

In the linguistics universe, one of the best treatment of counterfactuals (in my opinion) is by UCSD's Gilles Fauconnier with his Mental Spaces theory. In mathematics/computer science, one of the better known authors is Judea Pearl of UCLA.

Counterfactual reasoning, besides being an interesting research topic, also has a direct impact on policy decision making (I think, for obvious reasons). In fact, I beleive that Pearl has published some papers about counterfactual reasoning using some common econometric models.

As other posters have mentioned, I don't think implication is "wrong" (and I don't think this is your suggestion). Logic is fairly well-behaved and well understood with implication working the way it does. On the other hand, counterfactual reasoning is still very much an open research topic...

Posted by: Amit Dubey on September 1, 2003 07:18 AM

Whoops,

Re-reading the comments, I noticed that Joe Willingham already mentioned counterfactuals. One thing worth mentioning, though: while many formal semanticists really really want there to be a strong link between logic and language, it usually is not so strong as advertised... The mapping from language to logic is messy (not including if..then constructions, the "or" is different between logic and language, capturing negation is nontrivial, language has more quantifiers, and even the quantifiers which overlap are used differently, and set membership in human language has a "fuzzy" cognitive, rather than formal, basis).

Logic, basically, is a formalization of human language. And while that formalization is rigorous, you lose all the things all those dead logicians didn't want to model (either intentionally or by ignorance). In some cases, this leads to non-obvious meanings for logical operators.

Also, if you're interested, the link to the Pearl paper is here (actually, I goofed, Alexander Balke is the first author):

ftp://ftp.cs.ucla.edu/pub/stat_ser/R232-U.ps

But the paper many not be so interesting unless you like building Bayes' nets in your spare time.

Posted by: Amit Dubey on September 1, 2003 08:00 AM

Amit Dubey -

"Intuitions are pointing you to counterfactual reasoning rather than implication. Actually, the difference between counterfactuals and implication is much greater than the difference between material and logical implication."

Well, I pay attention. Another interesting post though no parrots or swans found.

Deduction takes us only so far. There was and is always inference to begin and end with. Inference has to ground and allow proper flexibility to our thinking. Imagine Darwin and Einstein and know what inference is.

Actually, I often try to use birds as a frame of reference.

Posted by: lise on September 1, 2003 08:07 AM

This is "material implication". It was an early attempt to formalize implication, and my belief that it is taught to beginners mostly because the formalization has all been worked out, and is thus cut and dried and easy to teach. No one takes it seriously any more, I don't think.

When far-fetched implications are considered as present-time counterfactual conditionals, they might seem just barely acceptable as sort of a glitch in the system. But suppose that in a historical situation I say (in 1914) that "If Germany wins this war (p), the NY Yankees will dominate the American league (q). For that implication to be false, we need p.~q. But we don't have p, so the implication is good (whether or not the Yankees win).

It would seem that a system of implication that can't handle any actual events isn't a very good one. Or perhaps I should just be glad that logicians have been driven from history.

To me, the conclusion is that material implication is, as I said, purely formal and not much use, except as an opening step, if thought of as a way of formalizing logic. Because, after all, we do use logic to describe actual events.

In fact, material implication DOES have an intuitive translation, but it isn't very exciting or useful one. (~p v q) means that "It is not impossible that p--> q." It excludes the only case when it is impossible that p--> q: i.e., (p.~q), when you have the p without the consequent q. So we have "It is not impossible that a German victory in WWI implies Yankee domination of the American League". I.E, that reasoning was bad, but not impossible.

There's another little quirk. I was taught (Copi) that the --> sign (a single sign) is used as shorthand, since p --> q is shorter than ~p v q. However, this is not true if you work out ~(p-->q), which is longer than p.~q. To me, slipping in the new --> sign is a sort of a wishful attempt to make it seem that ~p v q means more than it really does.

Posted by: zizka on September 1, 2003 10:53 AM

Brad wrote: "It didn't make sense that p -> q should automatically be true whenever p was false". Didn't it!? Now p -> q means exactly ~q->~p. "If it's a car then it's a vehicle" clearly implies "If it's not a vehicle then it's not a car". Hence (p->q) -> (~q->~p). Applying this the other way round gives that (~q->~p) -> (~~p->~~q) so (~q->~p) -> (p->q) and we now have that (~q->~p) and (p->q) is the same thing.

Therefore, saying that p->q is true when p is false is the same as saying that ~q->~p is true when ~p is true, or that r->s is true when s is. Accepting p->q as true when q is – you thus have to accept that p->q is true when p is false.

Why do we fail to see that the parrot and 200ft tallness statement is true then? I tried an explanation in my earlier post. Here is a (near) analogy:

A number times itself equals the number added to itself, right? You may protest here and say that multiplication and addition is different etc. Yet, this just means 'a number is equal to two', which most people would accept right away.

Posted by: Mats on September 1, 2003 11:16 AM

I think that this argument is on two levels. One accepts material implication as a formal system and interprets statements within its definitions.

The other asks, "Is material implication much like we think of as implication?", getting an answer of "NO!"

As I said, all you really can get from (~p v q) is "It's not impossible that p --> q". As if p --> q can be only "impossible" or "valid".

Over the years I've become slightly acquainted with a number of formal systems: game theory, set theory, number theory, systems theory, information theory, Boolean algebra, etc. I'm not good at that kind of stuff, but in all these cases I can easily see what the value of the theory is.

With material implication I just can't. It's as if its authors though that formalization has some kind of magic power regardless of whether there's a real-world interpretation / application or not. (Hao Wang, an associate of Goedel's, has written some amusing stuff about this).

I did like George Spencer-Brown's solution to Russell's Paradox using an imaginary truth-function, but I'm not sure that anyone took that seriously.

Posted by: zizka on September 1, 2003 01:11 PM

I think that this argument is on two levels. One accepts material implication as a formal system and interprets statements within its definitions.

The other asks, "Is material implication much like we think of as implication?", getting an answer of "NO!"

As I said, all you really can get from (~p v q) is "It's not impossible that p --> q". As if p --> q can be only "impossible" or "valid".

Over the years I've become slightly acquainted with a number of formal systems: game theory, set theory, number theory, systems theory, information theory, Boolean algebra, etc. I'm not good at that kind of stuff, but in all these cases I can easily see what the value of the theory is.

With material implication I just can't. It's as if its authors though that formalization has some kind of magic power regardless of whether there's a real-world interpretation / application or not. (Hao Wang, an associate of Goedel's, has written some amusing stuff about this).

I did like George Spencer-Brown's solution to Russell's Paradox using an imaginary truth-function, but I'm not sure that anyone took that seriously.

Posted by: zizka on September 1, 2003 01:13 PM

There's not really a debate. Material implication works for the purposes of physical science, mathematics, and computer progamming, but doesn't reflect ordinary English usage. Ordinary English is not two-valued: rather than every statement being true or false, many statements are neither.

Posted by: Joe Willingham on September 1, 2003 03:42 PM

Zizka: Material implication is _the_ implication of mathematics. Mats' Venn diagram example is exactly why it's the definition mathematicians use.

Posted by: Walt Pohl on September 1, 2003 03:44 PM

zizka: Implication is spectacularly useful for computer programming, since there's a one-to-one mapping between terms in intuitionistic logic and the types of expressions in functional programming languages (this is called the Curry-Howard isomorphism). Implication corresponds to the types of functions, and in fact the type of a function that takes (say) integers to strings is written "int -> string". Programming language types generally drop an explicit negation from the system, though, since we want our type systems to be decidable.

Also, Russell's paradox is not the worst paradox that naive set theory paradox permits. You can imagine fixing up set theory by redefining negation, but unfortunately there's another paradox, Curry's Paradox that doesn't use negation at all! Figuring out how to weaken logic to the point where self-reference doesn't create paradoxes would be a tremendous win, since we could then create a type system for self-modifying programs. (This is important, because there are plenty of programs that can be updated while they are running, like this web browser. :)

Curry's Paradox:

http://plato.stanford.edu/archives/spr2001/entries/curry-paradox/

Posted by: Neel Krishnaswami on September 1, 2003 04:00 PM

There is no single implication for all of mathematics. You can make up whatever logic you like, and as long as every theorem isn't true in it people will be interested in it. Even inconsistent logics are interesting, as long as they aren't explosive (ie, A and not-A implies X, for any X).

Posted by: Neel Krishnaswami on September 1, 2003 04:15 PM

Here is an amsuing statement of Haskell's paradox as a proof that penguins rule the universe.

http://www-personal.usyd.edu.au/~jasong/penguins.html

Posted by: Joe Willingham on September 1, 2003 04:53 PM

I mean of course Curry's Paradox.

Posted by: Joe Willingham on September 1, 2003 04:55 PM

It would seem that, if material implication is mostly usable in programming, math, and physical science, it should not be taught with real-world (historical) examples of any kind.

Programming and math aside, I still suspect that in the physical sciences, even strictly deterministic ones, a form of implication would be pretty useless which requires this to be a valid statement: "If water is flammable, it boils at 0 degrees centigrade". The usefulness of this form of reasoning to a scientist escapes me.

My favorite set-theoretical idea is Hjalmos's "Nothing contains everything; or to put it more spectacularly, there is no universe". In "The Tao is Silent" Raymond Smullyan plays around with this kind of thing.

Or to put it differently, the two values of material implication should be described as "invalid" (p.~q) and "not invalid" (rather than "valid")-- rather like "guilty" and "not guilty" in law (there is no "innocent" in law, as I understand).

Posted by: zizka on September 1, 2003 07:37 PM

P.S. I remember now my argument that the "-->" sign cannot be called an abbreviation. I had it wrong earlier.

While p --> q is shorter than ~p v q, this is not true if you work out p --> ~q, ~p --> q, and ~p --> ~q, and then add up the total number of signs. They come out the same, as I remember, so you've added a new different sign "-->" which really does nothing within the system.

EXCEPT that it tells you that "~p v q" is intended to represent implication. In other words, the new sign "-->" only tells you how the formal system you are working with is supposed to relate to the world outside itself -- its "application".

EXCEPT, as I've argued, that it doesn't really work that way.

Posted by: zizka on September 1, 2003 07:44 PM

All the logical connectives can be reduced to one, NAND, where p NAND q is true if and only if p is false, q is false, or both both p and q is false.

So NOT, If . . . then, AND, and OR are all dispensable.

If don't like the horseshoe (material implication) just rephrase everything using the Scheffer stroke (NAND).

For ~p we have p NAND p
For p & q we have (p NAND q)) NAND (p NAND q)
For p v q we have (p NAND p) NAND (q NAND q))
And p > q is equivalent to p NAND (q NAND q)

Posted by: Joe Willingham on September 1, 2003 09:23 PM

In terms of telling you what's going on, no more and no less, the "v . =" notation seems best to me. The NAND notation is ingenious but saeems like a stunt, though I suppose to a programmer that way of writing might have its point. Anyway, "(p NAND p) NAND (q NAND q))" seems like the hard way to say "p v q". But the horseshoe ("-->" here) really seems like false advertising.

In a formal system "~p v q" interpreted as "p is not seen without q" might be more meaningful than in an empirical or historical system. For example, in arithmetic, "It is possible that 2 + 2 = 4" might be identical to "It is necessary that 2 + 2 = 4".

I think that the counter-factual / subjunctive interpretation is a bit of an evasion, because the same problems arise in less-hypothetical, less-peculiar present statements about the future. IE "If the Axis wins the war, the Holy Roman Empire will survive" is a reasonable implication, whereas "If Germany wins the war, the Yankees will lose (or win) the pennant" is a ridiculous one, but in material implication they are both good.

As I understand, the makers of formal logic (Russell, Wittgenstein, the Logical Positivists) originally had the idea, or some of them, that they would be able to improve public discourse by defining valid implication. Brad's objections and mine pertain to that hope, which seems to have been vain. Seemingly formalized logic is not usable for induction in concrete historical situations.

Posted by: zizka on September 2, 2003 09:40 AM

"Seemingly formalized logic is not usable for induction in concrete historical situations." - Ah, stop it! All that you have is a false statement F and a true statement T:

F: p(x)->q(x) for all x belonging to a reasonably wide set.

T: p(x)->q(x) for x beeing a man named Socrates.

Natural language works fine too: p(x)->q(x) is the same as "If it's a parrot then it is 200 ft tall" with "x" the same as "it". Sadly though, you are careless enough to confuse the qualifications: the english if...then construct is most often used for statements that are expected to be of general validity. Therefore, you confuse the true (but pointless) statement T: "If Socrates were a parrot, then he would be 200 ft tall" for the false but similar (meaningful) statement "It might be any object that is or is not a parrot and has a known length. And, if it's a parrot then it is 200 ft tall".

That's all right, natural language invites you to be careless about the context. But don't blame natural language, and don't blame logic for beeing faulty just because you are too lazy to do the logic.

(I was too lazy to in my previous post which should otherwise have led to Joe Willinghams argument above. Thanks Joe!)

Posted by: Mats on September 2, 2003 12:09 PM

The horseshoe is not false advertising. It makes no claim to capture ordinary usage. The interesting thing is to see what can and what can't be said in a purely truth-functional language. That's the kind of issue that fascinates philosophers.

Of course formal logic alone is not adequate for induction. Nobody ever said it was. It's about deduction, not induction.

Even many mathematicians laughed at Russell and Whitehead and call their attempt to formalize mathematics useless pedantry. The joke was on the critics. The formalization led to the idea of mechanized reasoning, which played a key role in the development of the electronic computer. It has been a frequent irony that the most seemingly useless and recondite regions of science have ended up having the most important practical implications.

Not only that, mathematical logic is an amazing and profound subject. The fact that it doesn't capture the way we talk in many contexts is beside the point.

Induction has its own paradoxes, for example the raven paradox.

Consider the generalization "All ravens are black". We might gather a large number of observations and by induction advance the hypothesis that all ravens are black.

But consider. "All ravens are black" is logically equivalent to "All non-black things are non ravens. ("If it ain't black, it ain't a raven".) My car is not black, and it's not a raven. So my car is scientific evidence that all ravens are black.

Posted by: Joe Willingham on September 2, 2003 12:23 PM

Stop it, stop it! At the one hand there is those limits to all system, like Haskell's paradox and other "Gödel-style" things that occurs when a system refers to itself, like "the set containing all sets". At the other hand we have Brad's trivial but widespread misunderstanding about p->q. Brad writes:

"If Socrates were a parrot, he would be 200 feet tall." [...]is supposed to be true. But it clearly isn't."

So it is false then? But it would be equally false even it Socrates really were a parrot! So you admit that Socrates might be a parrot after all!

No way! it is true simply because a parrot is not 200 feet. Rephrase it slightly "If Socrates were a parrot, parrots could be 200 ft tall". If Socrates were a parrot, the statement would be false. Accepting it as true gives us a guarantee that Socrates is not a parrot!

And we have EXACTLY this in EVERYDAY language:

"if you're good at football I would be mr Beckham"

(meant to be emphatically true, implying in a rather sarcastical way that "you're NOT good at football")

Posted by: Mats on September 2, 2003 01:06 PM

let me just... I promise, just one more:

The statement "If Socrates were a parrot, he would be 200 feet tall." equals "If Socrates were a parrot, the parrot Socrates would be 200 feet tall." Now as "no parrot is 200 feet tall" the statement would be false if Socrates were a parrot.

It is also equal to "If he were not 200ft, Socrates wouldn't be a parrot". Now "Socrates isn't 200ft" so if the statement is true Socrates is not a parrot.

Once we establish that "no parrot is 200 feet tall" and "Socrates isn't 200ft", there is no other variable to make the statement true or false other than the parrotness of Socrates. Above we have first that "Socrates is parrot" makes statement false, and "statement is true" makes Socrates a non-parrot.

If the statement is false, as Brad claims it to be, Socrates hence has to be a parrot.

Posted by: Mats on September 2, 2003 01:57 PM

Several posters have already pointed out that contrary-to-fact conditionals are not stateable with the horseshoe. There seems to be no controversy on this point.

"If Socrates were a parrot, then he would be about ten inches tall is true". "If Socrates were a parrot he would be six feet tall" is false. The logic of counterfactuals is not truth-functional. There is a considerable philosophical literature on this puzzling topic.

"If you were a carpenter, and I were a lady, I would marry you anyway and have your baby".
This statement may be true, but its truth doesn't depend on the truth of the constituent clauses. "If you were a carpenter" isn't a statement at all, so it is neither true nor false. "I would marry you anyway . . . " may be true, but somehow one has one's doubts!

Posted by: Joe Willingham on September 2, 2003 02:15 PM

"Several posters have already pointed out that contrary-to-fact conditionals are not stateable with the horseshoe." aha!

Posted by: Mats on September 2, 2003 02:28 PM

Or is the jargon of "several posters" above just hiding from me that there are two different way of reading natural language. One computer-like, taking everything literally, just transforming sentences the way they can be transformed until some "parsimoneous" statement is achieved:

{"If Socrates were a parrot, he would be 200 feet tall.", "no parrot is 200 feet tall", "Socrates isn't 200ft"} equals {Socrates isn't a parrot;"no parrot is 200 feet tall", "Socrates isn't 200ft"}

Another, more human-like, searching for valuable information and trying to add information were it seem to have been lost in the transmission:

"If Socrates were a parrot, he would be 200 feet tall." is processed like: "hmmm, Socrates, ...don't think he is a parrot, anyway this seem to be about parrots, hmm... but if it is a parrot it's certainly not 200 feet tall. Now it clearly seems like nonsense, lets go find better information somewhere else!

Posted by: Mats on September 2, 2003 03:00 PM

I deduce that Socrates is going to Burning Man and has high standards for the outrageousness of his costumes.

To put more usefully-to-non-programmers what Neel said so thoroughly: I use "p=>q" thinking when representing something that has no time or causation. (Usually, in any but a trivial program, this is a tiny piece of the real problem, which I have represented by a more complicated kind of logic.) It's nowhere to end up, but it's a good solid place to start.

zizka - ""If water is flammable, it boils at 0 degrees centigrade". The usefulness of this form of reasoning to a scientist escapes me."

It's useful to have a form of reasoning that lets you ignore so many statements about the world. Thrifty.

Posted by: clew on September 2, 2003 05:07 PM

Check out this web site on the logical paradoxes.
http://www.wikipedia.org/wiki/Paradox

They are truly wonderful and amazing. The Curry Paradox mentioned by Neel Krishnaswami is an example.

Another paradox (not really a paradox but an impossibility theorem) is the discovery by Nobel Prize winning economist Kenneth Arrow that there is no system of voting that is logically consistent and at the same time accords with our intuitive criteria of fairness. Democracy is logically impossible!
http://www.wikipedia.org/wiki/Arrow%27s_paradox

Posted by: Joe Willingham on September 2, 2003 05:49 PM
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