Conditional Reasoning, Skills in Deduction & Forming Proper Conclusions in this JFK Thing of Ours

[Glenn Nall]

right - Jon, the limit of the number of cards, and therefore the scope of the proposition, was established immediately by the introduction, "Four cards are laid out as below" - i'm not sure what the reason would be to assume that that statement was just fodder, without point...? It's the same as "given these four cards" would introduce a test... "this is the test: four cards..." - "you will be asked to decide whether, within these four cards,..." - "Consider these four cards..."

"You will travel to a universe far, far away - a land where nothing exists, a land where Rod Serling plays a short version of solitaire - a Land of Four, and Only Four, Cards..."

you are about to enter...

ok i got carried away. I'm back...

[Thomas Graves]

Glenn's problem boils down to this testable proposition: If A, then B.

One can test the truth of this proposition by examining samples that present either A or B.

If a sample presents A, we know the proposition is true if the sample also presents B.

If the sample presents X, we know the proposition is false if the sample also presents B.

What we don't know explicitly from Glenn is the sample size. Glenn has presented four cards. The E card might have a 4 on the other side, and the proposition would be true so far. The 7 card might have a G or might have a U. A G means the proposition is correct so far. A U means the proposition is incorrect so far.

Glenn hasn't told us how to test the truth of his proposition. If truth is established by one instance out of four, that's one thing. If truth requires consistency among among all four instances, that's another thing. If truth requires consistency among the infinite number of possibilities, that's quite another thing.

Why an infinite number of possibilities? There is a finite number of vowels: a, e, i, o , u. As for even numbers, there are infinitely many.

Dear John,

We can reasonably assume that the question applied to only the four cards mentioned in the exercise itself.

How could / would he have worded the exercise if he'd wanted us to consider all possible cards in the universe that had a rational number on one side and an upper-case letter from the Modern English alphabet on the other? Does the fact that he didn't do this suggest that he wanted us to infer that that was what he had intended?

"Which of the infinite number of cards in the universe would you have to turn over...." LOL

Or then again, we can unreasonably quibble, split hairs, and apply "if, then" "then, if" logic like so many of us do in our JFK assassination "research" and "analysis."

A perfect microcosm (or is it macrocosm?).

Solution for an infinite number of cards: Turn over all of the cards showing a vowel, and turn over all of the cards showing an odd number. If any of the former have an odd number on the other side, or if any of the latter have a vowel on the other side, then God is either messing with you or has made an honest mistake.

--Tommy

edited and bumped

Solution for an infinite number of cards (OR the four-cards corollary): Turn over all of the cards showing a vowel, and turn over all of the cards showing an odd number. If any of the former have an odd number on the other side, or if any of the latter have a vowel on the other side, then God is either messing with you or has made an honest mistake. --T. Graves

For an infinite number of cards, the options change. the falseness of the antecedent can be proved, but its accuracy can never be. This is why the solution is stated to be the "least" number of turns that need to be made in order to prove true or false. with an infinite number of cards, as soon as a vowel is turned with no even #, or as soon as an even # is turned with no vowel, falseness occurs. and to the contrary, the trueness of the problem can never be proved. --G. Nall

Of course you're right, Glenn.

I was just messing with you.

--Tommy

How about if you turn over both cards (one showing vowel, the other showing odd number) at exactly the same time, forever?

Edited August 14, 2015 by Thomas Graves

[Mark Knight]

Glenn,

The reason I only mentioned the E is because, even if we assume a larger deck of cards is available, the MAXIMUM number of different letters in the English language is 26, and the maximum number of vowels is 5.

But the maximum number of even numbers is infinite.

And just because we know that a vowel will ALWAYS have an even number on the opposite side, there was NOTHING in the problem that stated that the converse was true, that an even number would ALWAYS have ONLY a vowel on the opposite side. That would be an ASSUMPTION, since it was not stated anywhere in the parameters of the problem.

THAT is my only "problem" with the stated solution to the problem: that one has to make an unsupported ASSUMPTION that ALL even numbers ONLY have vowels on the opposite side, simply because all vowels have an even number on the opposite side.

[simply put: unless it is expressly stated that the converse of the original "given" is true, we can not decide that it IS true without making an ASSUMPTION that it is true.]

That, to me, is the single flaw in the "correct" answer to the test. [in a court of law, that would be grounds for an objection, "assuming facts not in evidence."]

[Thomas Graves]

Glenn,

The reason I only mentioned the E is because, even if we assume a larger deck of cards is available, the MAXIMUM number of different letters in the English language is 26, and the maximum number of vowels is 5.

But the maximum number of even numbers is infinite.

And just because we know that a vowel will ALWAYS have an even number on the opposite side, there was NOTHING in the problem that stated that the converse was true, that an even number would ALWAYS have ONLY a vowel on the opposite side. That would be an ASSUMPTION, since it was not stated anywhere in the parameters of the problem.

THAT is my only "problem" with the stated solution to the problem: that one has to make an unsupported ASSUMPTION that ALL even numbers ONLY have vowels on the opposite side, simply because all vowels have an even number on the opposite side.

[simply put: unless it is expressly stated that the converse of the original "given" is true, we can not decide that it IS true without making an ASSUMPTION that it is true.]

That, to me, is the single flaw in the "correct" answer to the test. [in a court of law, that would be grounds for an objection, "assuming facts not in evidence."]

Mark,

The mind numbing complexity can be eliminated by remembering that the exercise was intended to be limited to the four cards given.

Once we can accept that, there's a lesson to be learned by thinking about this problem.

Here's a hypothetical analogy: "Although everyone involved was a dirty rotten rat, not every member of the CIA was involved." LOL

--Tommy

Edited August 14, 2015 by Thomas Graves

[Robert Prudhomme]

I agree with you, Mark. The rule only stated that a card with a vowel showing would have an even number on the opposite side; it did not state that cards with even numbers had to have a vowel on the other side. While a certain number of even numbered cards are obligated to have a vowel on the reverse, the other even numbered cards could have anything on the reverse.

Even if there were only the four cards in the test.

Turning over the four can neither prove nor disprove our rule.

Edited August 14, 2015 by Robert Prudhomme

[John Dolva]

every card is "a card"

every card has two sides.

every side is "a side"

every side has an "other side"

if a card.. etc - refers only to "a card" not "all" or "any" card.

anything else is an assumption.

[Thomas Graves]

every card is "a card"

every card has two sides.

every side is "a side"

every side has an "other side"

"If a card has a vowel on one side.." etc - refers only to "a card" not "all" or "any" card.

anything else is an assumption.

Correct.

In the exercise at hand, it's not necessary (in fact it doesn't help at all) to turn over the "K" card or the "4" card to find out whether or not the rule, "A card with a vowel on one side must have an even number on the other side" was actually followed by the person who wrote the letters and numbers on the four cards, but it is necessary to turn over the "E" card and the "7" card.

--Tommy

Edited August 15, 2015 by Thomas Graves

[Glenn Nall]

fascinating. i did more harm trying to clarify things than i did clarify things.

what's fascinating is the many various ways persons read and understand a particular sentence.

i promised myself to go through the different interpretations later, and I intend to. it pumps me up.

i've encountered hospitals and new homes and other assorted variables; but i'll get back to this.

the one thing that jumps out at me is that ONE person got the answer correct immediately - SO - it's not an insurmountable exercise, except that one might wish it to be...

[Glenn Nall]

Glenn's problem boils down to this testable proposition: If A, then B.

One can test the truth of this proposition by examining samples that present either A or B.

If a sample presents A, we know the proposition is true if the sample also presents B.

If the sample presents X, we know the proposition is false if the sample also presents B.

What we don't know explicitly from Glenn is the sample size. Glenn has presented four cards. The E card might have a 4 on the other side, and the proposition would be true so far. The 7 card might have a G or might have a U. A G means the proposition is correct so far. A U means the proposition is incorrect so far.

Glenn hasn't told us how to test the truth of his proposition. If truth is established by one instance out of four, that's one thing. If truth requires consistency among among all four instances, that's another thing. If truth requires consistency among the infinite number of possibilities, that's quite another thing.

Why an infinite number of possibilities? There is a finite number of vowels: a, e, i, o , u. As for even numbers, there are infinitely many.

Dear John,

We can reasonably assume that the question applied to only the four cards mentioned in the exercise itself.

How could / would he have worded the exercise if he'd wanted us to consider all possible cards in the universe that had a rational number on one side and an upper-case letter from the Modern English alphabet on the other? Does the fact that he didn't do this suggest that he wanted us to infer that that was what he had intended?

"Which of the infinite number of cards in the universe would you have to turn over...." LOL

Or then again, we can unreasonably quibble, split hairs, and apply "if, then" "then, if" logic like so many of us do in our JFK assassination "research" and "analysis."

A perfect microcosm (or is it macrocosm?).

Solution for an infinite number of cards: Turn over all of the cards showing a vowel, and turn over all of the cards showing an odd number. If any of the former have an odd number on the other side, or if any of the latter have a vowel on the other side, then God is either messing with you or has made an honest mistake.

--Tommy

edited and bumped

Solution for an infinite number of cards (OR the four-cards corollary): Turn over all of the cards showing a vowel, and turn over all of the cards showing an odd number. If any of the former have an odd number on the other side, or if any of the latter have a vowel on the other side, then God is either messing with you or has made an honest mistake. --T. Graves

For an infinite number of cards, the options change. the falseness of the antecedent can be proved, but its accuracy can never be. This is why the solution is stated to be the "least" number of turns that need to be made in order to prove true or false. with an infinite number of cards, as soon as a vowel is turned with no even #, or as soon as an even # is turned with no vowel, falseness occurs. and to the contrary, the trueness of the problem can never be proved. --G. Nall

Of course you're right, Glenn.

I was just messing with you.

--Tommy

How about if you turn over both cards (one showing vowel, the other showing odd number) at exactly the same time, forever?

[Glenn Nall]

every card is "a card"

every card has two sides.

every side is "a side"

every side has an "other side"

"If a card has a vowel on one side.." etc - refers only to "a card" not "all" or "any" card.

anything else is an assumption.

Correct.

In the exercise at hand, it's not necessary (in fact it doesn't help at all) to turn over the "K" card or the "4" card to find out whether or not the rule, "A card with a vowel on one side must have an even number on the other side" was actually followed by the person who wrote the letters and numbers on the four cards, but it is necessary to turn over the "E" card and the "7" card.

--Tommy

bingo.