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

• 1.2k
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.

Of course.

It is a universal quantifier: for all t.

Say it how you want. I said "at any particular time".

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.

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.. • 1.2k 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 No contradiction. It's a universally-agreed part of the truth-table for 2-valued truth-functional implication. ↪Michael Ossipoff So, all you've done is create an impossible scenario where someone actually receives the diamond? 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.

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.”? 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 • 1.2k 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's implication didn't say anything about the diamond being given without the money being given. The sign would have been true when the customer walked in because the customer had not yet given the clerk the money. Of course. 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?

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. [in "(not A) or (B)"] The word, "or" seems to separate the two statements It more than seems to. - making them independent of each other, which means that the conclusion doesn't necessarily follow the premise. Incorrect. "(not A) OR (B)" and A-> B are equivalent. They mean that B necessarily follows from A. 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). 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:

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 replied:

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?

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. Forget about the "truth" table. Just read the words. They contradict each other, which means that the first statement is never true - ever. 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 • 1.2k It has nothing to do with offering employment. 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? • 1.2k 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 • 1.2k The sign said "...the clerk will give the diamond to you, and at that time it will become yours" AT THAT TIME. Why is this codicil present? It's a rental! • 1.2k 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 • 861 t=the sign is true. t⟷((p→q)∧(¬p→(q∨¬q))) (¬p→(q∨¬q))¬→t • 1.2k 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. I'm talking about the logical implications of 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. So 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. • 1.2k No contradiction. It's a universally-agreed part of the truth-table for 2-valued truth-functional implication. 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. • 7.1k So you're admitting that there is more than one logical way to interpret the sign as the customer did. Of course, which would make the sign (and the sales-clerk) misleading, not false (or lying). There is also a classical logical rule that two statements that contradict each other are false. I don't understand the relevance of this. • 1.2k I'm talking about the implications of the truth table and how those p's and q's get translated into English words. An objection would have to be a lot more specific than that. Language is logical and they both need to be consistent with each other. 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 • 1.2k 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) 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. but it is not true for values of t more than or equal to t1. 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. .

Hence the statement is not true at time t1 because...

At t1, the payment hasn't been made, and so the implication-proposition's premise is false, and so the implication-premise is true.

it is universally quantified and it is not true for all values of t.

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

but it is not true for values of t more than or equal to t1

Yes it is. It's true until after you've made the payment, some unknown time after 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.

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
• 1.4k
Here is the sign:
"“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.
• 1.2k
under the rules of FOPL...

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
• 1.2k

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
• 1.4k
$\forall t2\ \forall t1\Bigg(\bigg(Pays(C,5000,t1) \wedge (t2 \ge t1)\bigg)\to GetsDiamond(C,t1+1)\Bigg)$
where $C$ is the customer.

Substituting 1002 (10:02am) for both $t2$, the 'any particular time', and $t1$, the time the money was paid, gives

$Pays(c,5000,1002) \wedge (1002 \ge 1002)\to GetsDiamond(C,1003)$

We observe that the money is paid at 10:02.
So both antecedents are true, so the consequent must be true, ie:

$GetsDiamond(C,1003)$

But observation shows this is false. So the original statement must be false.

Does anybody have a different formalisation to suggest?
• 1.2k

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
• 1.4k
and then writing a long, elaborate argument in those term
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?
• 1.2k
It can take a long time to wade through a crackpot argument, to explain each of its errors.

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
• 1.2k

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 • 1.4k 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.
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 $t2$ strictly exceeds $t1$. The money was paid at 10:01:30am.

Do you disagree with it? If so, with which bit?

$\forall t2\ \forall t1\Bigg(\bigg(Pays(C,5000,t1) \wedge (t2 \gt t1)\bigg)\to OwnsDiamond(C,t1+1)\Bigg)$
where $C$ is the customer.

Substituting 1002 (10:02am) for $t2$, the 'any particular time', and 1001.5 for $t1$, the time the money was paid, gives

$Pays(c,5000,1001.5) \wedge (1002 \gt 1001.5)\to OwnsDiamond(C,1002.5)$

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:

$OwnsDiamond(C,1002.5)$

But observation shows this is false. The customer does not own the diamond at 10:02:30. So the original statement must be false.
• 1.2k

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 • 1.2k Instead of T2>T1, I'd say: T1+1 > T2 > T1 Michael Ossipoff • 1.2k 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: OwnsDiamond(C,1002.5) But observation shows this is false. The customer does not own the diamond at 10:02:30. So the original statement must be false. 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 • 1.4k I think the following incorporates your amendment: $\forall t2\ \forall t1\Bigg(\bigg(Pays(C,5000,t1) \wedge (t1+1\gt t2 \gt t1)\bigg)\to OwnsDiamond(C,t1+1)\Bigg)$ where $C$ is the customer. Substituting 1002 (10:02am) for $t2$, the 'any particular time', and 1001.5 for $t1$, the time the money was paid, gives $Pays(c,5000,1001.5) \wedge (1002.5 \gt 1002 \gt 1001.5)\to OwnsDiamond(C,1002.5)$ 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: $OwnsDiamond(C,1002.5)$ 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. • 1.2k Oops! I mis-worded the sign. And my inequality doesn't help. 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:

---------------------------------------

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.

**************************************************************************************************************

------------------

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
• 1.2k
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:

OwnsDiamond(C,1002.5)

But observation shows this is false. The customer does not own the diamond at 10:02:30. So the original statement must be false.

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.

----------------------------------

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
• 1.2k
Of course, which would make the sign (and the sales-clerk) misleading, not false (or lying).

I don't understand the relevance of this.
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.
• 1.2k
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.
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.
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