bonanova

It was talent night at Morty's and the theme was amateur magicians. Not surprisingly, Alex and his buddies filled the first row. A week of drinks says I can guess how the first trick is done, me lads, boasted Alex. I'm somewhat of a prestidigitator, myself, ya know. It was a few years ago, mind you, but the mind is still sharp, and it's an exceptional trickster that will outsmart me. Anyone care to differ? Poor Davey. Even after stroking his beard he couldn't resist the challenge. You're on, he said. Then, with a deafening trumpet fanfare The Great Waldo appeared on the stage. He shuffled an ordinary deck of cards and then handed the deck to Alex! Pick any five cards and hand them to Amber, my beautiful assistant, he commanded. As Amber leaned forward to receive the five cards that Alex selected, it became apparent that what she lacked in beauty she more than made up for in cleavage. Thank you, she smiled, and Ian swore later that he saw Alex actually blush at her words. Amber looked over the cards, stuffed one of them into the front of what might be called a dress, and handed the others to Waldo. The rattle of a cheap set of snare drums pierced the silence as Waldo pored over the cards. Finally Waldo spoke. Ladies and gentlemen, he intoned, the card now resting safely within the bosom of my beautiful assistant is .... the Seven of Diamonds! To prove that I'm correct, I'll ask that gentleman [pointing at Jamie] to come and inspect the card. In two leaps Jamie was on the stage and, none too quickly, he retrieved the card. Please show it to the audience, ordered Waldo. Jamie held up the card. It's ... the Seven of Diamonds! Two minutes later, Waldo was gone, and Alex was deep in thought. Finally Alex spoke. Yes, he said, I think I have it! Was Alex correct? Is there a legitimate way to perform this trick?

CR has found it.

Hi HB, Can you share with us the difference between technical and theoretical correctness? Thanks.

foth, Interesting development. To discuss the connection of the red dots, once the sausage is drawn, what if all the red points were [say] in the lower left corner? Is there a procedure for connecting them to the blues? Starting with them all in one place [arbitrarily close, not coincident] might help develop a general procedure. I have a thought on this, but I'm interested in yours. Edit: Ah, I see you are already thinking along that line.

Itachi-San. Here is someone you can talk to.

You of course can say that All but 2 implies an initial set of 3 or more. Anyone is free to define rules, and publish them, about the meaning of statements. Under any set of rules [a sufficiently complete set] there can be a meaningful discussion. In this case, you and Aristotle could talk. Oh, wait. That poses a different problem. In geometry it was once believed that from other axioms it could be proved that two parallel lines do not cross. But attempts at that proof failed. So they added it as an axiom. Geometries with that axiom are called Euclidean geometries. Then someone said, well, what if we don't have that axiom? Omitting that axiom defines other [equally valid, non-Euclidean] geometries [e.g. hyperbolic.] In logic, ascribing existential import to All [or not] connotes Aristotelian or [non-Aristotelian] Boolean logic. You don't have to "buy" either of them. You choose one, and use it. Neither of them is invalidated by the presence of the other. The French language is not invalidated by the presence of the Russian language. To have a meaningful discussion participants need little more than definition of terms [vocabulary with an agreed upon dictionary] agreed upon rules of grammar and syntax. If participants agree on these, there can be a meaningful exchange. I think that in my very first post in this thread I said that both answers to the OP question are acceptable. One carries the Aristotelian interpretation; the other, Boolean. Some of the discussions here are rather like Americans, who insist that "bum" means "hobo", arguing with Brits, who are adamant that it means something quite different!

In my 100 car analogy, I made sure that his partner had purchased the mercedes and the rolls before the the man made the statement All but two of my cars are Fords. The words "but two" eliminates two of his cars - the mercedes and the rolls - from consideration of being fords. "All the others" then refers to a set of cars, numbering 0-98 - depending on his partner's buying activity to that point.

Hi Itachi-san, I agree that if you have 0 submarines "all but 5 of them" does not make sense. If you have 7 submarines, all but 5 would refer to a set of 2. If you have 6, all but 5 would refer to one of them. If you have 5, all but 5 would be the empty set. If you have 4, all but 5 would not make sense. Did you Google existential import?

Earlier post that turns on the question of existential import [which you can Google] A set may be empty and still be discussed. We can talk about the set of humans living on Mars [empty, so far as we know]. Some have called empty sets non-existent sets. Probably hoping to imply that they can't be talked about. It's more accurate to say the members of an empty set don't exist. Assuming no humans now live on Mars, the ordinary English meaning of this statement is not clear: All humans living on Mars are male. Does the statement assert human life on Mars? Does the statement assert only that if there are humans on Mars, they are male. If the second interpretation is taken, the statement is equivalent to saying No humans living on Mars are non-male. The first is the Aristotelian meaning; the second is the Boolean meaning. Aristotelian logic holds that the Universal Quantifier [fancy word for "All"] has Existential Import [fancy term meaning the set is not empty - it has at least one member.] Boolean logic does not hold that All has EI. Note that if Some is used: Some humans living on Mars are male, then we assert existence of humans living on Mars. Both logical systems call Some the Existential Quantifier. For everyone, Some has existential import [EI]. In Aristotelian logic, All has EI; in Boolean logic it does not. Some have argued that in "ordinary English" All uniformly has EI. If you talk about a class of things, then at least one member of that class exists. Or, ordinary English is always Aristotelian. To see this is not the case consider a tract of private land posted with the sign: All trespassers will be prosecuted. We do not usually take this sign to have asserted the existence of trespassers. It makes the attempt, rather, to ensure there are no trespassers. It means, logically, no one who trespasses will go unprosecuted, or No trespassers will be unprosecuted. In this case, ordinary English takes the Boolean meaning. So, in discussing the statement All my cars but two are Fords, to say that ordinary English demands there must be at least one Ford overstates a point. That's an assertion. Let me attempt a proof, closer to the language of the OP. Assume the posted land has two owners: Jim and John. Anyone on that land, other than Jim or John, would then be trespassing. The sign could then be worded: All persons found on this land other than Jim and John will be prosecuted. The meaning of the sign is clear. The sign does not imply the presence of a person other than Jim or John. The sign makes sense by the rules of ordinary English. Now lets talk about collections of cars. A man tells his partner to go out and buy him 100 cars. His instructions are: Buy me a Mercedes, then Buy me a Rolls Royce, then Make the rest of them Fords. A friend then asks him about his cars. How many cars do you have? - I don't know, he replies, my partner is buying them now. What kind are they? Well, I don't know that either, but he called to say he'd bought a Mercedes and a Rolls Royce. A Mercedes and a Rolls? Isn't that a bit extravagant? Not really, except for those two, all my cars are Fords. At the time the man made that statement, it's possible no Fords had been bought.

Slim chance indeed. But how big is the payoff?[spoiler=CR has it. ]The OP asks about expectation - the result when averaged over all possible outcomes. If you bet $1 and there were 10 equally likely outcomes of 0, 0, 0, 0, 0, 0, 0, 0, 0 and $20 then your expectation would be to double your money. If you made 10N bets, where N is a large number, you'd expect to get a $20N return. The best strategy is to guess a red-black distribution and bet it all on every card. There are 52C26 red card distributions, and one of them pays off $252. The ratio [like the $20/10 = $2 above] is $9.0813.... When Davey gives Alex his stake, he's left with CR's answer.

Let me ask you: How can it be a false statement? If A then B is equivalent to {not A} OR B. If A then B implies either A is false or B is true. In this case, A is false; so the implication is true. Not to be demeaning, I mean this simply factually: Have you ever taken a course in logic? I fear your reply will be that the question is irrelevant - that you're just communicating in English. But you're not; you asked a question whose answer cannot avoid logic. How is "If there were cars they would be Fords" a true statement?

Please read what I said. You claim that something is or is not true. Truth is a logical entity. Your claim is a logical statement. Without employing rules of logic, there is no way to determine whether or not your claim is a true statement - whether you are right or wrong. I can write the following statement in English: two plus two equals five. Now I claim my statement is true and I prohibit you from introducing the laws of arithmetic by claiming that this is not an arithmetic statement; it was written in English and it is correct. So ... two plus two equals five. And please do not talk about rules of arithmetic to comment on whether the statement is correct - I wrote it in English, not in arithmetic.

No it's not considered cheating. Everyone gets a shot at saving his own neck by guessing [loudly enough that everyone else hears] a color. The prisoners may employ any strategy that depends only on what each prisoner, in turn, says for a color. But no other communication is permitted. Just saying the color of the hat in front of you is perfectly permissible. But it's a lousy strategy for saving lives.

The statement is a categorical statement and it has logical import. Logic is a disciple within which meaning may be deduced from categorical statements. In one system of logic, the statement "All A are B" is equivalent to "No non-A are non-B". Within that system no members of A are implied. In another system of logic, "All A are B" implies there is at least one member of A. So ... Sure it matters what system of logic you use. If you don't define your terms and rules, you get endless and meaningless Brainden discussions.

all but two of my cars are fords. <=> Remove two of my cars. All the remainder are fords. In boolean logic, all the remainder may be discussed even though remainder is an empty set. None of the remainder of my cars are non-fords.

You can format text horizontally using a code box: (codebox)|1 2 there is a space before this 2 3 there are two spaces before this 3(/codebox) which gives 1 2 3[/codebox]or specifying a monospace font: (except for that you do need a nonblank character in the first position) (font="Courier New")]|1 | 2 there is a space before this 2 | 3 there are two spaces before this 3(/font) which gives [font=Courier New]|1 | 2 | 3 -- oops - no. leading and multiple blanks are ignored outside of code boxes[/font]

Yeah. What he said. I vote boolean.

If the universal quantifier all implies at least one, then the book, and Martini, are correct. In some forms of logic - I believe the two that differ on this point are Aristotelian and boolean - is does not. Thus, "All four-legged humans with three brains and two left hands are male" makes sense in - and is true in - boolean logic. It is equivalent to the statement "No four-legged ... etc. are not male [female or androgenous]," which is more intuitively seen to be true. In boolean logic, you can discuss all the members of empty sets. But not in Aristotelian logic, where all implies at least one. So the book probably took the Aristotelian view.

No clues for a while. Several are still working. Care to put your worst case strategy into a spoiler?

a squared can be coded as a(sup)2(/sup). but use [] brackets instead of () to get a2 Others: a(sub)2(/sub) gives a2 (s)delete this(/s) gives delete this Colors: (color=#FF0000")red(/color) comes out red Lists: (list=1)(*)one(*)two(*)three(/list) gives ordered lists: one two three (list)(*)bullet(*)bullet(*)bullet(/list) gives unordered lists bullet bullet bullet Spoilers: (spoiler="here's a hint to be hidden")hide this text(/spoiler) does this: Hope that helps.

No. You can assume these are mathematical lines and dots, so that such mundane, real-life, engineering problems don't limit you.

Yes, that's a fair restatement of the problem. If it's possible for any set of N pairs [and below] but impossible for at least one set of N+1 pairs, what's N? Equivalently, what's N+1?