• Michael
    5.6k


    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
    5.6k
    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
    991
    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
    5.6k
    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
    843
    “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
    966
    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
    966
    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
    966
    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
    966
    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
    966
    Make "if you have not given $5,000 to the sales-clerk" = pHarry 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
  • Michael Ossipoff
    966
    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.andrewk

    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..
  • Michael Ossipoff
    966
    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
    Harry Hindu

    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
  • Michael Ossipoff
    966
    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?
    Harry Hindu

    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
  • charleton
    843
    It has nothing to do with offering employment.charleton

    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?
  • Michael Ossipoff
    966
    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?
    charleton

    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
  • charleton
    843
    The sign said "...the clerk will give the diamond to you, and at that time it will become yours"Michael Ossipoff

    AT THAT TIME. Why is this codicil present?
    It's a rental!
  • Michael Ossipoff
    966
    AT THAT TIME. Why is this codicil present?
    It's a rental!
    charleton

    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
  • BlueBanana
    558
    t=the sign is true.

    t⟷((p→q)∧(¬p→(q∨¬q)))

    (¬p→(q∨¬q))¬→t
  • Harry Hindu
    991
    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.
    Michael
    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.Michael
    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.
  • Harry Hindu
    991
    No contradiction. It's a universally-agreed part of the truth-table for 2-valued truth-functional implication.Michael Ossipoff
    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.
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