A guy goes into a Jewel-store owned by a logician who never lies...
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
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
Comments (95)
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.
I believe he's referring to this truth table:
p ? q is true if p is false.
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.
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.
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
The paradoxes of material implication.
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.
See above.
The sign doesn't have a part other than its statement of the implication-proposition.
...based on your claim that the sign said more than the implication-proposition. See above.
By that standard truth-table that i referred to, they're false only if the premise is true and the conclusion is false.
What didn't the clerk tell the customer? The sign spoke for itself, when it told the implication-proposition.
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.
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
Quoting 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
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.
That is a contradiction, and therefore can't be logical. That's like saying A x B = 1 if A=0
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.”?
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.
It's not a contradiction. It's the standard truth table for the material conditional:
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".
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?
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?
Quoting MichaelThe 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).
Quoting MichaelThe 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.
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.
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.
They don't contradict each other, as the truth table shows.
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.
Quoting 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.
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 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.
Quoting 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.
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.
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.
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.
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.
Quoting 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.
That's just wrong. p ? q is true if both p and q are true or if p is false. See the truth table.
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.
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.
Though I didn't include it, I quoted from it, in regards to the story's situations.
Michael Ossipoff.
I didn't say "At that time it will be yours." I said, "At that time it will become yours."
Michael Ossipoff
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
Correct. It became false when its premise was true and is conclusion was false.
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..
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
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
Of course.
Say it how you want. I said "at any particular time".
Save yourself all that elaborate muddle.
At any particular time (be it past, present or future), is the time that the premise is about.
For example, that could be as time in the near future, after you've paid the clerk.
At that time (whatever time that be), "if you've given $5000 to the sales-clerk (as of that time)" is the premise of the implication.
Michael Ossipoff..
No contradiction. It's a universally-agreed part of the truth-table for 2-valued truth-functional implication.
No. The customer didn't receive the diamond. The scenario isn't impossible. Sure, a court would rule that the transaction was fraudulent. But would the customer be able to prove that he gave $5000 to the clerk? Because the customer trusted the clerk, he didn't demand a receipt or bring a witness.
One difference would be that, even the most trusting sucker would be maybe a little less likely to believe that that implication proposition would be true after the payment.
But go for it.
Michael Ossipoff
The sign's implication didn't say anything about the diamond being given without the money being given.
Of course.
As I said, the sign's implication says nothing about a diamond being given to someone who hasn't given $5000 to the sales-clerk.
It more than seems to.
Incorrect. "(not A) OR (B)" and A-> B are equivalent. They mean that B necessarily follows from A.
No. Remember that the first condition of the OR statement is that the customer has NOT given the money.
When the first condition is true (The money hasn't been paid", the second condition needn't be true. That's the nature of OR.
So, the truth of "You haven't given $5000 to the clerk" means that "He'll give you the diamond" needn't be true.
Michael said:
Harry replied:
That's what the customer wanted to know too
Obviously the sign's implication was false after the money was paid: . $5000 was given. The diamond wasn't given. That made the implication false.
That was the clerk's answer. That answer was true, as was the clerk's answer before the money was paid.
Was it fraud? Sure.
Can the customer prove that he paid the clerk? No.
You've been told why the implication was true before the money was paid. Therefore the clerk's assurance at that time was true as well.
MIchael Ossipoff
Eh?
I assume you are an American.
In English English people are not hired, cars are hired.
What I meant was The notice implies that the diamond was for RENTAL.
Are we clear?
No, it isn't at all clear what you're talking about, or where you're getting your ideas.
The sign said "...the clerk will give the diamond to you, and at that time it will become yours"
That isn't a rental offer. It's a sales offer.
Michael Ossipoff
AT THAT TIME. Why is this codicil present?
It's a rental!
I didn't say "At time it will be yours", or "It will be yours only at that time."
I said, "At that time it will BECOME yours.
"...At that time it will become yours" means that, at that time, it will start being yours."
But it won't become yours until it's given to you, which will happen after you pay for it. That's the purpose of saying "at that time".
Michael Ossipoff
t?((p?q)?(¬p?(q?¬q)))
(¬p?(q?¬q))¬?t
I'm talking about the logical implications of the truth table.
Quoting MichaelSo you're admitting that there is more than one logical way to interpret the sign as the customer did.
There is also a classical logical rule that two statements that contradict each other are false.
I'm talking about the implications of the truth table and how those p's and q's get translated into English words. Language is logical and they both need to be consistent with each other.
Of course, which would make the sign (and the sales-clerk) misleading, not false (or lying).
Quoting Harry Hindu
I don't understand the relevance of this.
An objection would have to be a lot more specific than that.
Quoting Harry Hindu
You'd have to specify what's inconsistent.
Something being true at one time and false at a later time needn't be an inconsistency.
"It's raining today" might be true today and false tomorrow.
The truth-table for 2-valued truth-functional implication doesn't contain any contradictions.
The sign's implication-proposition applied to any time. It purported to be a timeless fact.. The customer believed it.
But, when the money was given, the proposition became false then, by virtue of the fact that the clerk refused to give the diamond.
Was the customer misled? Most definitely.
My metaphysics is based on timeless abstract if-then facts. They're true in the sense that if the premise is true, then the conclusion is true...
...and (regardless of whether the premise is true) if the premise were true, the conclusion would be true.
Of course there are if-then propositions, about hypotheticals, for which that latter condition can be demonstrated.
I make no claim about the premises being true.
By the truth-table for 2-valued truth-functional implication, if the conclusion follows from the premise, then of course the implication-proposition will always be true, regardless of whether the premise is true.
So, the 2-valued truth-functional truth-table agrees that those if-then propositions that my metaphysics speaks of are always true..
I expect that the truth-table for truth-functional implication was written as it was, because the case where the premise is false is irrelevant to the the implication's truth, and so, if the implication must always have a truth-value, then it's convenient and reasonable for it to remain true when the premise is false, because a false premise certainly doesn't falsify the implication.
It's perfectly reasonable to say that an implication-proposition is true if it would be true when it counts (when and if its premise is true).
.
...Michael Ossipoff
Quoting andrewk
The proposition's premise becomes true right after the payment is made.
The inquiry is made before the payment. Therefore, the implication-proposition's premise is false, and so the implication-proposition is true, at t1.
At some unknown time after t1, the payment was made, making the implication's premise true. 60 seconds after that, the premise's conclusion is false, and so, at that time, the implication-proposition becomes false.
Quoting andrewk
It doesn't refer to any time other than the time at which that customer has paid $5000. That's a time that's an unknown amount later than t1.
Whatever time you choose as t1 and make the inquiry at that time, the premise becomes true when you have made the payment, an unknown time later than t1. .
At t1, the payment hasn't been made, and so the implication-proposition's premise is false, and so the implication-premise is true.
The premise only refers to one one-sided duration--the time before the payment is made. It doesn't refer to all times. It refers to that one duration.
Giving a name to the time of the inquiry doesn't change that.
Michael Ossipoff
Yes it is. It's true until after you've made the payment, some unknown time after t1.
The implication's premise is false, and the implication is therefore true, until you have made the payment. 60 seconds after you make the payment, the implication becomes false.
The sign doesn't refer to all times. It refers only to time before you make the payment. You choose that time.
Michael Ossipoff
"“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.”"
Call that assertion A1.
The 'at any particular time' is a universal quantifier so, under the rules of FOPL, it can be replaced by a reference to a specific time. Let's say the clerk says to the customer at 10:00am that the sign is true, and that the customer gives the money at 10:02. Under the rules of FOPL, if the sign was true at 10:00 then so was any version of it with the 'at any particular time' replaced by a specific time. So the following statement was true at 10:00*:
"“If, at 10:04, 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.”"
To prove this, call that more specific assertion A2. Under the rules of FOPL we have A1 --> A2, which is a tautology and hence true at any time at all. Call that tautology T1. Adopt a conditional hypothesis that A1 was true at 10:00. Then by Modus Ponens on A1 and T1 we deduce that A2 was also true at 10:00am.
But A2 is false at any time, because money was given at time 10:02 and a diamond was not given by 10:03. So in particular A2 was false at 10:00. By contradiction that entails that our Conditional Hypothesis that A1 was true at 10:00am must be false.
Hence A1 was false at 10:00am when the clerk asserted it was true. So the clerk lied.
IF((I've justified my story by FOPL) AND (Andrew's application of FOPL is otherwise correct))
THEN (Andrew's evaluation of my story via FOPL is on topic and correct)
That implication proposition is true, because it's premise (at least part of its premise) is false.
An argument purporting to use that implication to show that Andrew's evaluation of my story is correct would be a valid argument.
...but it wouldn't be a sound argument, because the implication's premise is false.
Michael Ossipoff
The sign refers to a payment made at any particular time, and then refers to THAT PARTICULAR time in the implication's premise and conclusion.
The implication-proposition is only about two times: The time at which the payment is made, and a time 60 seconds after that.
We need to get that straight: The implication-proposition is only about those two times.
If the customer has chosen 10:00 as his payment-time, then the implication's premise becomes true at 10:00, and was false before 10:00.
...and the implication's conclusion becomes false at 10:01, when the clerk still hasn't given the diamond to the customer.
Had the customer chosen a different payment-time, then the implication would be about that other time instead.
If, by writing a long argument, with letters representing quantities and statements, and using the terminology of FOPL, though the story has only been justified in terms of propositional logic, and then writing a long, elaborate argument in those terms, it's easy to make it too complicated for ourselves, and thereby get ourselves confused about something that needn't have confused us.
Michael Ossipoff
\forall t2\ \forall t1\Bigg(\bigg(Pays(C,5000,t1) \wedge (t2 \ge t1)\bigg)\to GetsDiamond(C,t1+1)\Bigg)
[/math]
where [math]C[/math] is the customer.
Substituting 1002 (10:02am) for both [math]t2[/math], the 'any particular time', and [math]t1[/math], the time the money was paid, gives
[math]
Pays(c,5000,1002) \wedge (1002 \ge 1002)\to GetsDiamond(C,1003)
[/math]
We observe that the money is paid at 10:02.
So both antecedents are true, so the consequent must be true, ie:
[math]
GetsDiamond(C,1003)
[/math]
But observation shows this is false. So the original statement must be false.
Does anybody have a different formalisation to suggest?
Thank you for further exemplifying what I said in this paragraph:
"If, by writing a long argument, with letters representing quantities and statements, and using the terminology of FOPL, though the story has only been justified in terms of propositional logic, and then writing a long, elaborate argument in those terms, it's easy to make it too complicated for ourselves, and thereby get ourselves confused about something that needn't have confused us."
Michael Ossipoff
The above post is much shorter than your statement of the problem in the OP!
If you disagree with it, with which bit do you disagree?
First of all, my justification of my story had nothing to do with FOPL. It was only in terms of propositional logic, using a definition and truth-table that was unanimous among the academic sources that I'd found.
But, instead of evaluating and criticizing my justification, in the terms in which I'd justified it, you change the terms to FOPL, which has nothing to do with my justification of the story...thereby also making the subject too complicated for yourself, and confusing yourself, as described in my paragraph that I quoted.
So, instead of saying what's wrong with my justification, in terms of how I justified it (a propositional logic implication definition about which academic sources were unanimous), you re-write the topic in other terms, and then say that I should say what's wrong with your argument in different terms..
That's an easy and common crackpot technique:
"They won't say what's wrong with my design-proposal for a perpetual-motion machine!" [...maybe because they don't have time to wade through it.]
Michael Ossipoff
For one thing, you said that t2 equals or is greater than t1. But I'd said "...if, at that time, you have given $5000 to the sales-clerk..."
The sign explicitly specified a time after the payment was made.
Then you assign the same time value to t1 and t2.
That's just a first comment, from a look at the beginning of your argument.
For your argument to make enough sense to evaluate it, you'd have to change those parts of it. Only then would there be any point examining the rest of it.
In other words, if you can make those corrections, and still have an argument that seems right to you, then you'd need to do so, in order for your argument to be worthy of further examination.
Michael Ossipoff
I don't agree that those adjustments are necessary but, for the sake of furthering the discussion I'll accept them. Here's a version where [math]t2[/math] strictly exceeds [math]t1[/math]. The money was paid at 10:01:30am.
Do you disagree with it? If so, with which bit?
[math]
\forall t2\ \forall t1\Bigg(\bigg(Pays(C,5000,t1) \wedge (t2 \gt t1)\bigg)\to OwnsDiamond(C,t1+1)\Bigg)
[/math]
where [math]C[/math] is the customer.
Substituting 1002 (10:02am) for [math]t2[/math], the 'any particular time', and 1001.5 for [math]t1[/math], the time the money was paid, gives
[math]
Pays(c,5000,1001.5) \wedge (1002 \gt 1001.5)\to OwnsDiamond(C,1002.5)
[/math]
We observe that the money is paid at 10:01:30 (ie 1001.5).
So both antecedents are true, so the consequent must be true, ie:
[math]
OwnsDiamond(C,1002.5)
[/math]
But observation shows this is false. The customer does not own the diamond at 10:02:30. So the original statement must be false.
Obviously, instead of just saying, "If you've given $5000 to the sales-clerk...", I should say:
"If you've given $5000 to the sales clerk within the most recent 59 seconds..."
Otherwise the implication's conclusion could automatically already be false at the "any particular time".--something that I didn't intend.
I hereby modify the story as described above.
I don't know if that affects your argument.
1. You're saying that the times referred to in the implication can have any value. That's contrary to my story.
Maybe the story would be clearer if I worded the sign like this:
"At any particular time, if you've given $5000 to the sales-clerk within the most recent 59 seconds...."
...thereby getting the "any particular time" out of the "if" clause.
I now hereby make that modification to the story too.
You (or the customer) can choose any time. Given that fixed time that you or he have chosen, the implication is about that fixed time.
The free choice of time is over, as soon as you decide when to make the payment. The implication refers to that fixed time.
2. I'd change "GetsDiamond" to "HasReceivedDiamond".
That would be more consistent with the sign's promise in my story.
I'll post these changes and comments now, and resume soon.
Michael Osspoff
Instead of T2>T1, I'd say:
T1+1 > T2 > T1
Michael Ossipoff
Of course the implication-proposition becomes false at 10:02:30.
The inquiry, and its answer, were made before the payment, which was made at 10:01:30..
At the time of the Clerk's answer to the initial inquiry, the implication proposition was true, because its premise was false, because the payment hadn't yet been made.
The truth of the implication-proposition when the clerk said it was true, is all that's needed for the clerk to not be lying.
The implication-proposition becomes false at 10:02:30, because the clerk hasn't given the diamond within 1 minute after the payment.
So yes, after 10:02:30, the implication-proposition is false.
But it was true when the clerk said it was, before 10:01:30, because its premise was false, because the payment hadn't been made.
Michael Ossipoff
[math]
\forall t2\ \forall t1\Bigg(\bigg(Pays(C,5000,t1) \wedge (t1+1\gt t2 \gt t1)\bigg)\to OwnsDiamond(C,t1+1)\Bigg)
[/math]
where [math]C[/math] is the customer.
Substituting 1002 (10:02am) for [math]t2[/math], the 'any particular time', and 1001.5 for [math]t1[/math], the time the money was paid, gives
[math]
Pays(c,5000,1001.5) \wedge (1002.5 \gt 1002 \gt 1001.5)\to OwnsDiamond(C,1002.5)
[/math]
We observe that the money is paid at 10:01:30 (ie 1001.5).
So both antecedents are true, so the consequent must be true, ie:
[math]
OwnsDiamond(C,1002.5)
[/math]
But observation shows this is false. The customer does not own the diamond at 10:02:30. So the original statement must be false.
It still looks like the clerk was lying.
First the sign. Here's what it should say, and what i'm changing it to:
if you've given $5000 to the clerk, then at any time more than 60 seconds after you gave him that money, he'll have given you this diamond
I hereby change the sign-wording to what the above paragraph says..
The inequality:
I'm getting rid of the inequality, and wording the predicate expression in a different way that says what the sign says:
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So here's how I'd write the predicate expression (though the sign-wording, by itself, is sufficient).
HasPaid(C, $5000) --> HasBeenGivenDiamond(C, at all times at least 1 minute after paying the $5000)
(I don't say he owns it, because he might sell it, or he might be 500 years deceased)
No, I didn't word it algebraically. Does predicate logic format require that?
*************************************************************************************************************
In any case, as I said, the sign-wording is the important thing, because the sign, and not the predicate logic wording, is in the story.
**************************************************************************************************************
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The time of payment is decided by the customer. It can only have one value, the one chosen by the customer. For the purposes of the implication, the time-of-payment isn't universally-quantified. It's a constant that has been chosen by the customer.
There's no need to name or label, the "any time". It isn't, and needn't be, mentioned in the implication-proposition.
Michael Ossipoff
Yes, as i mentioned earlier, of course the implication-proposition becomes false a minute after the money has been paid, because the clerk hasn't given the diamond.
But that doesn't make the implication false before the payment has been made. Before the payment has been made, the implication's premise is false, making the implication true.
Therefore the clerk wasn't lying when he said (before the payment was made) that the implication-proposition was true at that time. It was true at that time.
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But my sign-wording that you've quoted (and my predicate language too) had other problems.
I've fixed those problems in my post before this one.
Michael Ossipoff
Because you're forgetting something important - the interpretation of the customer, which contradicts the clerk's interpretation. Which interpretation is the correct one? Read below.
Then I was right when I said that you used an improper logical system in translating the logical meaning of the sign.
In "If-THEN" statements, the THEN statement is necessarily dependent upon the truth value of the IF statement. This is the way it works in the English language and computer programming (and I would add that a computer is more logical than a logician because a computer doesn't have greed clouding it's interpretation of the symbols on the sign).
If the truth value of the implication-proposition is only dependent upon the truth value of the conclusion, then the truth value of the premise is irrelevant to the truth value of the proposition.
The above is true because it follows both the logic of IF-THEN statements and the logic of the material conditional.
If the material conditional only states that q is true when (but not necessarily only when) p is true, and makes no claim that p causes q, then what exactly is the relationship between p and q? A material conditional is more like simply writing two completely separate statements. Translating to English, it's more like saying,
"Give me $5000."
"I give you the diamond.",
where each part isn't dependent upon each other to be true.
The sign is an IF-THEN statement and that is the logical system that should be used in determining the logical meaning of the sign. The "truth" table produces invalid results precisely because you're using a logical system that doesn't translate to the actual meaning of the sign.
That's a bit silly. Yes it's chosen by the customer but he can choose any time.
I'm not sure what the purpose is of this thread. Is it to show that the truth table for material condition isn't an adequate reflection of how we actually speak?
Presumably which one the clerk intended when he wrote it? Isn't that part of your theory on meaning; the speaker's intention? In this situation, the customer simply misunderstood.
If you have given me $5,000 then I will give you the diamond.
Of course. That's why clerk's scam worked.
Yes the customer was intentionally deceived.
By the definitions that I found in those academic articles, the clerk's interpretation is correct for the 2-valued truth-functional definition and truth-table for implications.
Obviously the clerk's scam would be illegal. But, as I said, the customer has no proof that he paid for the diamond.
Michael Ossipoff
Quoting Benkei
...and when he has done so, by paying, his time of payment becomes a constant. For the purpose of the implication-proposition, the customer's payment-time is a constant.
It isn't a universally-quantified variable, or a variable at all, in the implication-proposition.
Quoting Benkei
It was intended to illustrate how the, otherwise-useful, 2-valued truth-functional definition and truth-table for implication could have a meaning that most people wouldn't expect.
Michael Ossipoff
That makes no difference I'm afraid. If it's random it becomes constant at the time of payment as well.
Quoting Harry Hindu
...but that could depend on the company that's using the computer.
The truth value of the implication-proposition is function of the truth-values of the premise and the conclusion.
But of course, by the standard implication truth-table, if the conclusion is true, the implication-proposition is true regardless of whether or not the premise is true.
The relationship is only that expressed by implication proposition's truth-table. As you said, nothing is said or meant about causation.
The only observation of p and q truth-values that could establish something about whether p always implies q would be an observation of p true and q false.
The metaphysics that I propose is based on abstract if-then facts for which the conclusion demonstrably follows from the premise. Of course there are lots of such abstract if-then facts. A proved mathematical theorem is such a fact.
The standard 2-valued truth-functional implication truth-table was the basis of my story about the diamond sales scam.
I don't know what you mean by that. The sign's implication proposition said that, at any time, if you've given me the $5000, then at any time more than a minute after you gave it to me, I'll have given you the diamond.
Of course. It was.
The truth-table's results are ordinarily useful, but their deceptiveness in the story situation is the point of the story. The customer was intentionally deceived. The truth-table made it possible for the clerk to deceive and scam the customer without lying.
But yes, there are some interpretations, some alternative truth-tables, that say that the truth of the implication-proposition is indeterminate when its premise is false. The 2-valued interpretation doesn't allow that.
And the 2-valued truth-functional truth-table is the more standard one.
But, as a practical matter, in the story, it doesn't matter. The customer can't prove that he paid, and so the scam worked. The clerk (who is also the store owner and a logician) can assure himself that he didn't lie when he scammed the customer, because his truth-table is the standard one.
Personally, speaking for myself, the if-then proposition that seems most relevant to mathematics and metaphysics is one in which the conclusion demonstrably follows from the premise.
Michael Ossipoff
Exactly. If the customer flips a coin to decide when to pay, the time of his payment is still a constant with respect to the implication-proposition.
Michael Ossipoff
Exactly. That makes andrewk right.
Quoting Benkei
If he said that, then he's certainly right about it.
But he has represented some times mentioned in the implication-proposition as universally-quantified variables.
That representation is incorrect, because, as I've been saying, the time-of-payment is a constant with respect to the implication-proposition.
Michael Ossipoff
I don't think that's a reasonable paraphrase of the sign. This version refers only to the present, and whether, at the time the reader is reading the sign, they have already given $5000.
The actual sign emphasises its difference from this paraphrase by use of the words 'at any particular time'. No matter how charitable one is seeking to be, one cannot ignore those words.
What if the words were removed? (goes back to OP to read sign while mentally eliding those words)
Without those words, I think the paraphrase might be considered realistic. However I think almost nobody would pay the money on the basis of such a sign. Those four words are crucial to tricking people into paying the money.
Didn't he get a receipt upon payment of the money? The OP does not mention whether he does, but only a fool would pay such an amount without immediately obtaining a receipt.
But even if the customer had been so unwise, they could immediately call the police and ask them to retrieve the $5000 cash from the clerk's pocket, dust it for fingerprints and ask the clerk to explain how they came to have $5000 cash in their pocket that had been handled by the customer. They'd be hard pressed to come up with a credible excuse, given the sign about the $5000 is sitting there in plain view, and the customer's testimony.
The customer was too trusting.
Yes, the clerk made a mistake when he just put the money in his pocket, instead of in the cash-register. By that mistake, he could be caught.
I didn't know that finger-prints could be gotten from a currency-bill.
I've been short-changed by being given change for 20 when I'd paid with a 50, or being given change for a 10 when I'd paid with a 20.
In the case of the 10 and the 20, I was later reimbursed by the manager.
In the case of the 20 and the 50, I wasn't reimbursed.
Ideally, when paying with a 20, a 50, or a 100, one should have recorded the serial-numbers of the 20s, 50s and 100s that one is carrying.
Then, the clerk would have a hard time explaining how the customer knows the serial number of that 50, if he only paid with a 20.
It might sound like a lot of trouble, but it wouldn't be so laborious to write down the serial numbers of ones 20s, 50s, and any possible one or more 100s that one might (temporarily, I hope) be carrying.
Ii must admit that I've never recorded serial numbers. That's why I lost $30, when I paid with that 50.
Michael Ossipoff
Michael Ossipoff
To escape, he would have to hide the cash somewhere that the police will not find it. The customer should not let him out of his sight, so that he can see where he hides it.
Ok, you're right. It looks as if the clerk's scam would be very difficult, if not.impossible, to succeed with.
I should have demanded a register-count when I was shortchanged when paying with the 50..
Michael Ossipoff
The clerk could have an accomplice, who'd take the money off the premises, to somewhere else, before the police arrive.
Michael Ossipoff
Authorities could ask to look at the security-camera record for the time in question.
Maybe there's a well-concealed security-camera and it's possible to say, "We don't have one yet." (Not necessarily feasible).
Or maybe there's a pre-set-up way that the accomplice could code a signal to an unconcealed security-camera, to delete its record for that day, in a way that successfully mimics a natural failure. That would only work once, before the coincidence became too improbable. And, even once, it might justify a close examination of the camera-system. (So, not necessarily feasible).
So the scam would be problematic.
Michael Ossipoff
Maybe he just prides himself on never lying to a customer.
Michael Ossipoff
The OP never said the clerk wrote the sign. As a matter of fact, the store owner (which isn't a logician) most likely wrote the sign because he is the one that actually owns the diamond.
You're missing the point. The point is that the customer's interpretation of the sign is just as legitimate as the clerk's. The problem is that they both contradict each other, which means that at least one of the interpretations is wrong.They can't both be right at the same time.
Quoting Michael Ossipoff
Of course he does. The diamond and the sign would attract attention. No other customers or clerks saw the customer give the clerk the money? There aren't cameras in the jewlery store? All these other behaviors you tell us the clerk engages to cover up the fact that the customer gave them the money in is dishonest. The clerk is a liar simply by his behavior.
Again you miss the point. It's not about speaking different languages, it's about using the correct terms in ANY langauge to translate to the correct terms of another language. When your logical system ends up being inconsistent with other logical systems, then something is wrong. They should all be integrated into a consistent whole.
You keep claiming that the clerk is a logician. If so, then the clerk would know that there other logical interpretations of the sign and that all logical interpretations should be consistent.
Quoting Michael OssipoffYou obviously don't know much about computer programming. ALL computer languages mean the same thing with IF-THEN statements.
Quoting Michael OssipoffNo. It only depends on the truth value of the conclusion. Just look at the table.
Quoting Michael OssipoffExactly. Now you've just contradicted your statement above. See how illogical this is?
Quoting Michael Ossipoff
You just keep moving the goal posts. This conversation is no longer meaningful.
I said, in the title of the thread, that the store is owned by a logician. I said in my post that the clerk is the owner and logician.
Michael Ossipoff
The clerk's interpretation is correct, by the 2-valued truth-functional truth table and definition of implication that several academic sources were unanimous about.
There are other truth-tables for implication. I wouldn't say that some are more "legitimate" than others.
But the clerk's truth-table is the more widely-quoted one, the more standard one.
I'd said:
You replied:
There was only one clerk in the store. But, as i said, he could have had an accomplice, to remove the money from the store before police arrived. There weren't other customers in the store.
I acknowledged that the security-camera would be a problem, maybe a prohibitive one.
I said, "Don't try this at home."
Of course. He's a liar, because, even though he didn't lie to the customer, he lies to the police about whether the payment was made. So he doesn't really live up to the title of the thread.
And,even though he didn't lie to the customer, of course he defrauded and intentionally deceived the customer. His sign was false (a lie) when the clerk didn't honor it by keeping its promise. So, in that sense, too, the clerk lied (because he'd written and displayed the sign that proved false), even though his earlier assurance was true.
The scam couldn't be repeated. And, after the notoriety of the first time, the store wouldn't get any business. The scam wouldn't be very feasible, if do-able at all. And even if were feasible, it wouldn't be practical.
If he has a 20 million dollar diamond, why does he bother scamming for $5000?
But the clerk didn't lie to the customer when he said the sign's implication-proposition was true, because that statement was correct,when made, by the standard truth-table for 2-valued truth-functional implication.
Michael Ossipoff
I’d said:
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You replied:
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I just keep missing that darn point!
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That was what my objection was about.
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There are different truth-tables for implication.
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He was using the standard truth-table for 2-valued truth-functional implication.
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I’d said:
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You replied:
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I didn’t say that different programming languages mean different things by IF-THEN. You said something about honesty, and that’s what I was replying to. There aren’t dishonest programming languages, but there are dishonest companies. And no doubt phishers and malware-writers use perfectly honest programming languages.
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I’d said:
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You replied:
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Incorrect.
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If the conclusion is false, then the truth of the implication depends on whether or not the premise is true, by the truth-table that I’ve been referring to, the standard 2-valued truth-functional truth-table.
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I’d said:
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You replied:
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No, I didn’t. If the conclusion is false, then the implication proposition is false if its premise is true, and true if its premise is false.
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You continued:
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You got that right.
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I’d said:
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You replied:
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Incorrect. That’s what I’ve been saying from the start.
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It never was.
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Michael Ossipoff
Whether or not the clerk lied isn't what is being argued against. My argument is that he isn't a logician. What I'm saying is that implication-propositions don't translate to logical "IF-THEN" statements that are used by people and computers via their programming. The customer interpreted the sign correctly as a causal relationship between the act of giving the money and the effect of receiving the diamond. If there is no relationship between the premise and the conclusion, then the sign is wrong to be written the way it is.
The clerk's interpretation that an implication-proposition is true if its premise is false is unanimously agreed on by the academic sources i found, for 2- valued truth-functional implication.
Quoting Harry Hindu
As for people:
That was the whole point of the story, ...to illustrate that the standard truth table for such implications can give results that differ from what people ordinarily expect.
And no, i''m not "moving the goalpost". It's something that I've been saying from the start.
As for computer programs:
Of course. So what?
A computer program doesn't interpret an "IF...THEN" statement as a logical proposition that a conclusion follows from a premise.
It takes it as an instruction to do something if a certain proposition t is true.
Loosely said, it often takes it as an instruction to make a variable take a certain value if a certain equality, inequality, or proposition is true. ... when the action called for is the execution of an assignment-statement.
...but it can also just specify an action, such as "IF x = a, THEN PRINT(x)"
In general, it's an instruction, like saying, "If he tries to get in, call the police".
Michael Ossipoff.
As you may know, in logic, "A OR B" means "A", "B" or "A and B". If you just want one of them, you must use exclusive OR, abbreviated xOR.
In human language it's the opposite. If the carnival game operator tells you that you've won a stuffed bear or a parasol, and you take both, you're obviously in the wrong.
If you want inclusive OR, you have to say "A or B or both". Or, more briefly, A &/or B".
No one is claiming that words always mean the same in logic and in human language.
Michael Ossipoff
Quoting Michael Ossipoff
Exactly. The sign is written as an IF-THEN statement. IF you give the clerk $5000, THEN you receive the diamond. IF-THEN-(ELSE) is how we make ANY decision.
You simply need to rewrite the sign so that it actually translates correctly - meaning that you need to remove the IF and the THEN and write it as two indpendent statements.
You still haven't given us the relationship between giving the clerk $5000 and receiving the diamond. Is it a causal relationship, or what? What does the arrow between p and q actually mean?
Quoting Michael OssipoffHuman language is logical.
Must quit for the evening.
Replying tomorrow.
Michael Ossipoff
I’d said:
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You replied:
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Incorrect. As the person who posted my post, I’m the one to say what my point was, in posting it.
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Re-quoting you:
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The truth-table that I (and others) quoted wasn’t complicated. It’s unambiguously expressed in English. No, there was no mis-translation of it in my story.
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I’d said:
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You replied:
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One point that I was making was that a computer language’s IF…THEN statement isn’t the same thing as a logical implication-proposition (…in this case one that’s interpreted in the standard manner of 2-valued truth-functional logic).
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At no time did the clerk tell the customer that the sign was a computer-language statement that was going to be executed after the payment. The clerk merely (correctly) stated the truth-value of the implication-proposition, by the standard 2-valued truth-functional interpretation, at a time before payment was made.
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No mis-translation. No lie.
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As I said above, there was no mis-translation.
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It’s called an “implication-proposition”. Depending on what kind of logic one is referring to, there can be various truth-tables for it. In the standard 2-valued truth-functional interpretation, an implication-proposition is true if its premise is false.
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I’d said:
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You replied:
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:D
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Well, some languages are more logical than others.
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If you like language to be logical, then I recommend Esperanto. It’s more logical than English, and more logical than at least nearly all natural languages.
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Harry, this conversation has, for some time now, consisted only of repetition. It’s time for us to agree to disagree.
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I’m at these forums to discuss metaphysics.
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Michael Ossipoff
In the link that Michael provided on the first page of this thread - and that you agreed with - also explains exactly what it was I've been trying to tell you for several posts now. If you scroll down to the section, "Philosophical problems with material conditional", it explains how the implications aren't completely translatable to a native language.
Your "research" seems to be cherry-picked.
The difference in meanings and interpretations described in your quote was what my story was intended to illustrate. As I said, that was the whole point of the story.
Quoting Harry Hindu
Actually no, the standard 2-valued truth-functional implication truth-table was unanimously identical at every academic website that I checked.
But we've already been all over this several times.
Michael Ossipoff