A guy goes into a Jewel-store owned by a logician who never lies...

Michael Ossipoff

A guy goes into a jewelry-store.

Working at the counter, as sales-clerk, is a logician who never lies.

The customer goes up to the counter/display-case. In that case is an immense diamond, so large that it’s surely worth 20 million dollars. A sign next to that diamond says the following:

“If, at any particular time, you have given $5000 to the sales-clerk (under no circumstances will it be returned), then, within 60 seconds after your giving him that money, he will give you this diamond, and it will at that time become yours.”

Customer: “That’s a valuable diamond. Is that sign true??

Clerk: “Yes”.

Customer: “I don’t know, are you sure about that? Can I believe that?”

Clerk: I said it’s true, didn’t I?

The clerk leans forward, and says: “I don’t like being called a liar.”

The clerk then raises his right arm, elbow at right-angle, hand open, and says, “I swear to you, I swear on my honor, that that sign’s implication-proposition is true.”

Finally convinced, the customer gives $5000 to the clerk.

The clerk accepts the money, says “Thank you”, and puts it in his pocket.

After a while, the customer says, “Well?”

Clerk: “Well what?”

Customer: “The diamond. Are you going to give it to me?”

Clerk: “No, I’m not going to give it to you.”

Customer: “What?”

Clerk: “Do you think that anyone would sell a diamond that size for that price?”

Customer: “The sign says that you will.”

Clerk: You gave me $5000. I refuse to give you the diamond. Obviously the sign isn’t true, is it.”

Customer: “But you swore that it was true. You raised your hand and swore on your honor that it was true!”

Clerk:

“It was true when I said it was true. It was true then, because you hadn’t yet given me the money. Every implication-proposition is true if its premise isn’t true. The premise of the implication-proposition was that you’ve given me the money. You hadn’t given me the money. The premise wasn’t true. Therefore the implication proposition was true then. Didn’t you know that every implication proposition is true if its premise isn’t true? Such a proposition can only be false if its premise is true and its conclusion is false, as is the case now.

Michael Ossipoff

James R Meyer

This is completely wrong. In classical logic that "every implication-proposition is true if its premise isn’t true" is based on the fact that there is never any change in the truth value of any proposition.

The premise in the implication-proposition in the sign is "If, at any particular time, you have given $5000 to the sales-clerk". In the scenario described, this does not have a fixed truth value, hence the application of a classical fixed truth value analysis is absurd - hence the absurd result, a result that would not hold in any court of law - for obvious reasons.

The same invalid application of classical fixed value logic to various real life scenarios that do not have fixed truth values is a common method of producing conundrums.

Michael

↪James R Meyer

I believe he's referring to this truth table:

p	q	p → q
T	T	  T
T	F	  F
F	T	  T
F	F	  T

p → q is true if p is false.

Harry Hindu

The customer asked if the sign was true, not if the implication-proposition was true. The sign includes the premise and the implication-proposition. The clerk said that the sign was true. But only part of it was true and the other part was false. Therefore the clerk lied.

If at any given moment part of the sign is false and part is true, then the clerk can't say that the sign is true. The sign is both true and false, which just makes the sign illogical. The clerk also lied that he was a logician.

The sign is an IF-THEN statement. How are IF-THEN statements not true? They aren't unless we are missing information to put into the logical system. The implication always follows the premise assuming you have all the right information going in. It is only when you don't have all the information can it seem like the conclusion is false when the hypothesis is true. The missing information is what the clerk didn't tell the customer. If the customer had been given that information then they wouldn't have been tricked. So this isn't an example of how logic fails. It is an example of how one can use logic to get the wrong answers when they don't have all the information needed to get the right answer. In order for logic to work, you have to put in all the relevant information.

Michael

↪Harry Hindu

As the above truth table shows, "if p then q" is true if "p" is false.

p ≔ You have given $5,000 to the sales-clerk
q ≔ He will give you the diamond

So, "if you have given $5,000 to the sales-clerk then he will give you the diamond" is true if "you have given $5,000 to the sales-clerk" is false.

Michael Ossipoff

This is completely wrong. In classical logic that "every implication-proposition is true if its premise isn’t true" is based on the fact that there is never any change in the truth value of any proposition — James R Meyer

I'd checked various articles on the subject, put up by various universities. Their definitions didn't include a stipulation about truth values never changing.

Of course It isn't a matter that affects my metaphysical proposal. It was only intended to show that the usual 2-valued truth-functional implication truth-table that I'd read about could lead to an undesirable conclusion.

Michael Ossipoff

Michael

It was only intended to show that the usual 2-valued truth-functional implication truth-table that I'd read about could lead to an undesirable conclusion. — Michael Ossipoff

The paradoxes of material implication.

Michael Ossipoff

The customer asked if the sign was true, not if the implication-proposition was true. The sign includes the premise and the implication-proposition. — Harry Hindu

Of course the proposition's premise is mentioned, but only as part of the implication proposition.

You can't say A => B without mentioning A.

So no, the sign tells nothing but the implication-proposition.

The clerk said that the sign was true. But only part of it was true and the other part was false. Therefore the clerk lied.

See above.

If at any given moment part of the sign is false and part is true, then the clerk can't say that the sign is true.

The sign doesn't have a part other than its statement of the implication-proposition.

The sign is both true and false, which just makes the sign illogical.

...based on your claim that the sign said more than the implication-proposition. See above.

The sign is an IF-THEN statement. How are IF-THEN statements not true?

By that standard truth-table that i referred to, they're false only if the premise is true and the conclusion is false.

They aren't unless we are missing information to put into the logical system. The implication always follows the premise assuming you have all the right information going in. It is only when you don't have all the information can it seem like the conclusion is false when the hypothesis is true. The missing information is what the clerk didn't tell the customer.

What didn't the clerk tell the customer? The sign spoke for itself, when it told the implication-proposition.

If the customer had been given that information then they wouldn't have been tricked.

I didn't say the clerk was honest or not a crook. I merely said that he didn't lie. The whole truth would have had to include, "After you give me the money, I'll keep the diamond." Of course the clerk didn't volunteer the whole truth (which he wasn't asked about).

If the customer had asked, "If I give you $5000, will you really give me the diamond?", and the Clerk had answered "Yes", then, having been paid, the clerk would have to give the diamond or be a liar.

But the only thing stated or asked about was the truth of the sign's implication-proposition.

So this isn't an example of how logic fails. It is an example of how one can use logic to get the wrong answers when they don't have all the information needed to get the right answer. In order for logic to work, you have to put in all the relevant information.

All the information stated on the sign was the implication-proposition. All the information asked about and answered about was about the truth of that implication-proposition.

Yes, the clerk withheld the whole truth, information that he hadn't been asked for.\

The customer was misled and defrauded.

The clerk was a crook. The sign, by not being honored (true) after the money was given, amounted to fraud. The clerk didn't lie, but his sign's implication-proposition did, by being false after the payment was made. The clerk (who also owned the store) of course committed fraud, and of course that's illegal.

So "Don't try this at home".

Michael Ossipoff

Michael Ossipoff

Let me clarify this more:

The sign includes the premise and the implication-proposition. — Harry Hindu

The sign asserts the implication-proposition. It doesn't assert that proposition's premise, which is only in an "if" clause (as is the nature of an implication's premise)..

Michael Ossipoff

andrewk

I believe the statement was not true when the person first inquired, because the words 'at any particular time' are not constrained to cover only the past, so they cover the future too. It is a universal quantifier: for all t.

If the person is inquiring at time t1, the quantified part of the statement is true for values of t less than t1, by virtue of the above truth tables Null implication) but it is not true for values of t more than or equal to t1. Hence the statement is not true at time t1 because it is universally quantified and it is not true for all values of t.

One might think that removing the universal quantifier will thereby render it true. But then the 't' in the statement (implied in '60 seconds after you have given him your money') becomes a free variable, and in FOPL, a formula containing a free variable entails the version of the formula in which that variable is universally quantified (rule of universal quantification), so we are back where we started.

So I don't think it works. The store clerk lied.

Harry Hindu

So, "if you have given $5,000 to the sales-clerk then he will give you the diamond" is true if "you have given $5,000 to the sales-clerk" is false. — Michael

That is a contradiction, and therefore can't be logical. That's like saying A x B = 1 if A=0

↪Michael Ossipoff

So, all you've done is create an impossible scenario where someone actually receives the diamond? Is it really is no different than a sign saying, “If, at any particular time, you have given $5000 to the sales-clerk (under no circumstances will it be returned), then, within 60 seconds after your giving him that money, a unicorn will appear, and it will at that time become your best friend.”?

TheMadFool

↪Michael Ossipoff

One of those times when time is important in logic.

The sign was true before the man gave the clerk money and then it became false.

However, the IF-THEN logical form isn't defined in temporal terms. If it is then I've never heard of it. From the logic books I've read, the IF-THEN logical form is timeless i.e. we can't change its truth value over time or space. Otherwise, we'd be equivocating all the time, right?

So, the logician has lied.

Michael

That is a contradiction, and therefore can't be logical. That's like saying A x B = 1 if A=0 — Harry Hindu

It's not a contradiction. It's the standard truth table for the material conditional:

p	q	p → q
T	T	  T
T	F	  F
F	T	  T
F	F	  T

Where p is false, p → q is true. In our case, p is "you have given $5,000 to the sales-clerk" and q is "he will give you the diamond".

Michael

Another way to look at it:

p → q is logically equivalent to ¬p ∨ q, so "if you have given $5,000 to the sales-clerk then he will give you the diamond" is logically equivalent to "you have not given $5,000 to the sales-clerk or he will give you the diamond".

If you have not given $5,000 to the sales-clerk then "you have not given $5,000 to the sales-clerk or he will give you the diamond" is true. Therefore, if you have not given $5,000 to the sales-clerk then "if you have given $5,000 to the sales-clerk then he will give you the diamond" is true.

Also, p → q is logically equivalent to ¬q → ¬p, so "if you have given $5,000 to the sales-clerk then he will give you the diamond" is logically equivalent to "if he will not give you the diamond then you have not given $5,000 to the sales-clerk". Do you find this latter conditional problematic?

Harry Hindu

p → q is logically equivalent to ¬p ∨ q, so "if you have given $5,000 to the sales-clerk then he will give you the diamond" is logically equivalent to "you have not given $5,000 to the sales-clerk or he will give you the diamond". — Michael

So then why didn't the clerk give the customer the diamond before the customer gave him the money? The sign would have been true when the customer walked in because the customer had not yet given the clerk the money. Not only that but is the sign true even when no one reads it? If so, then shouldn't everyone who hasn't given the clerk $5000 get the diamond?

If you have not given $5,000 to the sales-clerk then "you have not given $5,000 to the sales-clerk or he will give you the diamond" is true. Therefore, if you have not given $5,000 to the sales-clerk then "if you have given $5,000 to the sales-clerk then he will give you the diamond" is true. — Michael

The word, "or" seems to separate the two statements - making them independent of each other, which means that the conclusion doesn't necessarily follow the premise. All you are saying is "this condition exists or that condition exists". So when the first condition didn't exist, (the customer hadn't given the clerk any money) then the latter condition exists (the clerk should have given the customer the diamond).

Also, p → q is logically equivalent to ¬q → ¬p, so "if you have given $5,000 to the sales-clerk then he will give you the diamond" is logically equivalent to "if he will not give you the diamond then you have not given $5,000 to the sales-clerk". Do you find this latter conditional problematic? — Michael

The latter conditional is saying the same thing as "Give the money to the clerk and he will give you the diamond". The customer gave the money to the clerk, now where is his diamond?

Forget about the "truth" table. Just read the words. They contradict each other, which means that the first statement is never true - ever. In other words, it is a false statement.

Michael

So then why didn't the clerk give the customer the diamond before the customer gave him the money? The sign would have been true when the customer walked in because the customer had not yet given the clerk the money. Not only that but is the sign true even when no one reads it? If so, then shouldn't everyone who hasn't given the clerk $5000 get the diamond?

...

The word, "or" seems to separate the two statements - making them independent of each other, which means that the conclusion doesn't necessarily follow the premise. All you are saying is "this condition exists or that condition exists". So when the first condition didn't exist, (the customer hadn't given the clerk any money) then the latter condition exists (the clerk should have given the customer the diamond). — Harry Hindu

It's:

1) You have not given $5,000 to the sales-clerk, or
2) He will give you the diamond

1) is true, so 2) needn't be.

The latter conditional is saying the same thing as "Give the money to the clerk and he will give you the diamond".

No it isn't. It's saying that if "he will not give you the diamond" is true then "you have not given $5,000 to the sales-clerk" is true.

As both "he will not give you the diamond" and "you have not given $5,000 to the sales-clerk" are true, the material conditional is true.

Forget about the "truth" table. Just read the words. They contradict each other, which means that the first statement is never true - ever. In other words, it is a false statement.

They don't contradict each other, as the truth table shows.

Michael

As I mentioned before, it's worth checking out the paradoxes of material implication.

Harry Hindu

It's:

1) You have not given $5,000 to the sales-clerk, or
2) He will give you the diamond

1) is true, so 2) needn't be. — Michael

Exactly. If 2 isn't logically dependent upon 1 then he will give you the diamond. It would also be incorrect to use the truth table in this instance because now you are saying that 1 isn't necessarily the premise to the conclusion. The two are only related as they are on the same sign. Again, all you are doing is describing two independent conditions, which can be true or false independent of each other.

They don't contradict each other, as the truth table shows. — Michael

I said forget about the "truth" table and just read the words. Actually, maybe you should put both conclusions into the truth table and see if that works. Maybe then you'll see the contradiction.

Michael

Exactly. If 2 isn't logically dependent upon 1 then he will give you the diamond. It would also be incorrect to use the truth table in this instance because now you are saying that 1 isn't necessarily the premise to the conclusion. The two are only related as they are on the same sign. Again, all you are doing is describing two independent conditions, which can be true or false independent of each other. — Harry Hindu

It doesn't need to be logically dependent. The following is a true material conditional:

If my name is Michael then London is the capital city of England.

I said forget about the "truth" table and just read the words.

I have read the words. And I won't forget the truth table, because it's relevant to the topic. The store-clerk is a logician who uses the truth-table of the material conditional to help determine the truth of the sign.

Harry Hindu

An IF-THEN statement:

Therefore, if you have not given $5,000 to the sales-clerk then "if you have given $5,000 to the sales-clerk then he will give you the diamond" is true. — Michael

Make "if you have not given $5,000 to the sales-clerk" = p

and

then "if you have given $5,000 to the sales-clerk then he will give you the diamond" = q

Does that make any sense? The premise contradicts part of the conclusion. As I said, "IF A x B = 1 then A=0" is a contradiction.

Harry Hindu

I have read the words. And I won't forget the truth table, because it's relevant to the topic. The store-clerk is a logician who uses the truth-table of the material conditional to help determine the truth of the sign. — Michael

I'm saying it isn't relevent to the topic. The OP didn't include it. You did later. I'm saying that is you that is off-topic. Just read the sentences. It's more like: if p->q then p is not equal to p.

Michael

↪Harry Hindu

p is "You have given $5,000 to the sales-clerk" and q is "He will give you the diamond".

p → q is equivalent to ¬p ∨ q.

¬p is "You have not given $5,000 to the sales-clerk".

If ¬p is true then ¬p ∨ q is true. So if "You have not given $5,000 to the sales-clerk" is true then "You have not given $5,000 to the sales-clerk or he will give you the diamond" is true.

"You have not given $5,000 to the sales-clerk" is true. Therefore, "You have not given $5,000 to the sales-clerk or he will give you the diamond" is true.

Michael

I'm saying it isn't relevent to the topic. The OP didn't include it. You did later. I'm saying that is you that is off-topic. Just read the sentences. — Harry Hindu

He says that the store-clerk is a logician who talks about implications being true if the premise (actually "antecedent") is false. He's referring to the material implication truth table.

Harry Hindu

p is "You have given $5,000 to the sales-clerk" and q is "He will give you the diamond".

p → q is equivalent to ¬p ∨ q.

¬p is "You have not given $5,000 to the sales-clerk". — Michael

Which is the same as saying that it doesn't matter whether or not p is true or false. q is true regardless of the truth value of p, which means that q is independent of p, which makes p->q false. There is no IF-THEN relationship between p and q.

He says that the store-clerk is a logician who talks about implications being true if the premise (actually "antecedent") is false. He's referring to the material implication truth table. — Michael

Then that is the problem with the OP. He's applying a system that is irrelevant to the circumstances, or to what the words actually mean.

Michael

Which is the same as saying that it doesn't matter whether or not p is true or false. q is true regardless of the truth value of p, which means that q is independent of p, which makes p->q false. — Harry Hindu

That's just wrong. p → q is true if both p and q are true or if p is false. See the truth table.

Then that is the problem with the OP. He's applying a system that is irrelevant to the circumstances, or to what the words actually mean.

So as I have twice brought up, this is an example of the paradoxes of material implication, where "if ... then ..." in classical logic doesn't mean what it does in ordinary language, hence the unintuitive conclusions.

charleton

“If, at any particular time, you have given $5000 to the sales-clerk (under no circumstances will it be returned), then, within 60 seconds after your giving him that money, he will give you this diamond, and it will at that time become yours.” — Michael Ossipoff

This offer is a HIRE offer.
The rest of the scenario is of no consequence and is nothing but sophistry.
"at that time it will be yours" implies a limit.

Michael Ossipoff

I'm saying it isn't relevent to the topic. The OP didn't include it. — Harry Hindu

Though I didn't include it, I quoted from it, in regards to the story's situations.

Michael Ossipoff.

Michael Ossipoff

This offer is a HIRE offer.

It has nothing to do with offering employment.
— charleton

The rest of the scenario is of no consequence and is nothing but sophistry.

It's always easy to make a vague, unsupported statement like that..

"at that time it will be yours" implies a limit.

I didn't say "At that time it will be yours." I said, "At that time it will become yours."

Michael Ossipoff

Michael Ossipoff

He's applying a system that is irrelevant to the circumstances, — Harry Hindu

I portrayed a situation in which a definition of implication that I'd read (articles at various university websites were unanimous about that definition of 2-valued truth-functional implication) gave an undesirable result. So, if you don't like the result, then don't apply it to such situations.

I acknowledged the store's dishonesty, and that the falsity of the implication when the money had been given constitutes fraud.

Michael Ossipoff

Michael Ossipoff

One of those times when time is important in logic.

The sign was true before the man gave the clerk money and then it became false. — TheMadFool

Correct. It became false when its premise was true and is conclusion was false.

However, the IF-THEN logical form isn't defined in temporal terms.

The definitions that I found didn't make any mention of time.

To stipulate that the truth-values never change would be to mention a temporal matter, thereby defining implication in temporal terms..

From the logic books I've read, the IF-THEN logical form is timeless i.e. we can't change its truth value over time or space.

That temporal stipulation contradicts your statement above, that implication isn't defined in temporal terms.

And, I just mention, as a matter-of-fact, that obviously that stipulation limits implication's applicability.

A stipulation that truth-values never change would make logic inapplicable to electronic logic-gates, whose inputs and outputs do change.

...or is it just for implications (but not for AND, OR, NOT or NAND) that truth values never change?

Anyway, Michael mentioned that A -> B is equivalent to (not A) OR (B).

...implying that if you let truth values change for OR, then you're letting them change for implication.

As I've already said, my purpose was to show a consequence of a definition that I'd read about at various logic articles put up by universities.. Those articles unanimously stated the same definition, and it made no mention of time, or any temporal stipulation such as that truth values never change.

I don' speak for sources other than those that I found.

Michael Ossipoff

Michael Ossipoff

Make "if you have not given $5,000 to the sales-clerk" = p — Harry Hindu

Make it whatever you want, Harry. Make it something different from what I said, if you want to, though that's off-topic.

Michael Ossipoff

A guy goes into a Jewel-store owned by a logician who never lies...

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