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  1. #1
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    True/False Logic Test

    Which of the following statements are false, and which are true?

    1) Statements 1 and 2 have different answers
    2) Both statement 1 and statement 10 are true
    3) Statements 3 and 10 have the same answer
    4) At least one of the first three statements are true
    5) Statements 3 and 4 have different answers
    6) Statement 8 is false
    7) Of the first six statements, exactly one is true
    8) Of the first seven statements, exactly two are true
    9) If statement 9 is true, than statement 6 is true
    10) Statement 10 is true

    Good luck! Remember to use spoiler tags.
    Last edited by Castle; September 24th, 2008 at 05:12 PM. Reason: Edited to ensure uniqueness of solution; see post #56
    Freedom is you choosing for yourself. Law is the government choosing for you. The two are opposites.

    Pray - To ask that the laws of the universe be annulled in behalf of a single petitioner confessedly unworthy - Ambrose Bierce
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  2. #2
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    Re: True/False Logic Test

    1. F
    2. T
    3. T
    4. T
    5. F
    6. T
    7. F
    8. F
    9. T
    10. T

  3. #3
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    Re: True/False Logic Test

    Quote Originally Posted by cds69 View Post
    1. F
    2. T
    3. T
    4. T
    5. F
    6. T
    7. F
    8. F
    9. T
    10. T
    Incorrect (and your incorrectness has nothing to do with the error fix I just had to make to the test, so I feel less guilty about changing it).
    Freedom is you choosing for yourself. Law is the government choosing for you. The two are opposites.

    Pray - To ask that the laws of the universe be annulled in behalf of a single petitioner confessedly unworthy - Ambrose Bierce
    Faith - Belief without evidence in what is told by one who speaks without knowledge about things without parallel - Ambrose Bierce

  4. #4
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    Re: True/False Logic Test


    We are only talking decimal numbers, I suppose?



    _________________________________ Post Merged _________________________________



    only number 10 is true
    Last edited by AliceLiddell; September 17th, 2008 at 10:08 PM. Reason: Automerged Doublepost

    Contrariwise, if it was so, it might be; and if it were so, it would be; but as it isn't, it ain't. That's logic.

  5. #5
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    Re: True/False Logic Test

    Here's my answer:

    1. F
    2. F
    3. F
    4. F
    5. F
    6. F
    7. F
    8. F
    9. F
    10. T



    Explanation:

    1) Statements 1 and 2 have different answers
    2) Both statement 1 and statement 10 are true
    3) Statements 3 and 10 have the same answer
    4) At least one of the first three statements are true
    5) Statements 3 and 4 have different answers
    6) Statement 8 is true
    7) Of the first five statements, exactly one is true
    8) Of the first seven statements, exactly two are true
    9) If statement 9 is true, than statement 6 is true
    10) Statement 10 is true

    A. 1 and 2 cannot both be true (they would have the same answer, making 1 false), nor can 1 be false and 2 be true (1's falseness would contradict 2's assertion that 1 is true). It is possible that 1 is true and 2 is false, but only if 10 is also false (it would then be false that statements 1 AND 10 are true, even though statement 1 is true). Or they could both be false, and statement 10 is false as well. Or they could both be false, and statement 10 is true as well.

    B. If 3 is false, then 10 is true (if 10 was also false, then 3 and 10 would have the same answer, making 3 true). If 3 is true, then 10 is true. So 10 must be true, but 3 could be either false or true.

    C. 4 could either be false or true; if 3 is true, then 4 is true. If 3 is false, then 4 is false.

    D. 5 must be false; 3 and 4 must have the same answer (see C).

    E. 6, at this point, is either true or false depending on 8.

    F. 7 is false; 3 and 4 have the same answer, which means that either there are zero true statements among 1-5 (if 3 and 4 are false), or there are 2 true statements among 1-5 (if 3 and 4 are true).

    G. 8 is false; if 3 and 4 are true, that makes 8 true, which then makes 6 true, which then makes 8 false (since there are now 3 true statements among 1-7).

    H. 9 is false, since 8 cannot be true (see above), which means that 6 cannot be true.

    I. 10 is true (see B).
    If I am capable of grasping God objectively, I do not believe, but precisely because I cannot do this I must believe. - Soren Kierkegaard
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  6. #6
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    Re: True/False Logic Test

    Well, done Clive.

    Contrariwise, if it was so, it might be; and if it were so, it would be; but as it isn't, it ain't. That's logic.

  7. #7
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    Re: True/False Logic Test

    Holy cow, I didn't even get what I was doing until I read Clive's answer. Wow. I need to do these more often.
    "And that, my lord, is how we know the Earth to be banana-shaped." ~ Monty Python


  8. #8
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    Re: True/False Logic Test

    Quote Originally Posted by Castle View Post
    1) Statements 1 and 2 have different answers
    2) Both statement 1 and statement 10 are true
    3) Statements 3 and 10 have the same answer
    4) At least one of the first three statements are true
    5) Statements 3 and 4 have different answers
    6) Statement 8 is true
    7) Of the first five statements, exactly one is true
    8) Of the first seven statements, exactly two are true
    9) If statement 9 is true, than statement 6 is true
    10) Statement 10 is true

    Good luck! Remember to use spoiler tags.
    True: 1,2,4,5,10
    False:3,6,7,8,9

    1 and 2 have different answers, so 1 is true. Since 1 is true, 10 is true, so 2 is true. 3 and 10 have different statements, so 3 is false. At least one of the first three statements was true (1-2 are true) so 4 is true. 3-4 have different answers, so 5 is true. Statement 8 is false since of the first seven statements four of them are true, so both 8 and 6 are false. There's more than one true statement in the first five, so 7 is false. Since 6 is false, 9 is false. 10 is true.



    _________________________________ Post Merged _________________________________


    Well paint my ass red and call me wilson. I was WAY off.
    Last edited by Mr. Hyde; September 18th, 2008 at 07:25 AM. Reason: Automerged Doublepost
    But in your hearts revere Christ as Lord. Always be prepared to give an answer to everyone who asks you to give the reason for the hope that you have. But do this with gentleness and respect, keeping a clear conscience, so that those who speak maliciously against your good behavior in Christ may be ashamed of their slander.
    1 Peter 3:15-16

  9. #9
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    Re: True/False Logic Test

    Yeah, Hyde. Your very first sentence was wrong according to the answers you provided.

    Contrariwise, if it was so, it might be; and if it were so, it would be; but as it isn't, it ain't. That's logic.

  10. #10
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    Re: True/False Logic Test

    Quote Originally Posted by Galendir
    Thus, S9 is equivalent to "If statement 6 is false, then statement 9 is false."
    Wrong. It could be false that "if statement 6 is false, then statement 9 is false." Just as it could be false that "If statement 9 is true, then statement 6 is true." Statement 9 asserts an "if, then" relationship between 9 and 6; if it is false, such a relationship does not exist. 9 could be false, and 6 true (and thus it would be false that "if 6 is true, then 9 is true"). They could also both be true. They could also both be false; p->q does not imply ~p->~q, the classical example being "I am out in the rain, therefore I am wet; I am not out in the rain, so I am not wet" (even though you could be in a pool, or the shower).


    _________________________________ Post Merged _________________________________


    What the hell happened to Galendir's post?
    Last edited by CliveStaples; September 18th, 2008 at 08:31 AM. Reason: Automerged Doublepost
    If I am capable of grasping God objectively, I do not believe, but precisely because I cannot do this I must believe. - Soren Kierkegaard
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  11. #11
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    Re: True/False Logic Test

    Wrong, you lose, Clive.

    p-> q does imply ~q ->~p. I think your trouble is in minding your ps and qs.

    If it's true that If I am out in the rain then I am wet, then it's also true that if I am not wet then I am not in the rain.


    _________________________________ Post Merged _________________________________


    Quote Originally Posted by CliveStaples View Post
    p->q does not imply ~p->~q, the classical example being "I am out in the rain, therefore I am wet; I am not out in the rain, so I am not wet" (even though you could be in a pool, or the shower).
    This part is correct, but a false analogy to Galendir's statement.
    Last edited by AliceLiddell; September 18th, 2008 at 09:38 AM. Reason: Automerged Doublepost

    Contrariwise, if it was so, it might be; and if it were so, it would be; but as it isn't, it ain't. That's logic.

  12. #12
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    Re: True/False Logic Test

    Quote Originally Posted by AliceLiddell
    p-> q does imply ~q ->~p. I think your trouble is in minding your ps and qs.

    If it's true that If I am out in the rain then I am wet, then it's also true that if I am not wet then I am not in the rain.
    That's "~q, therefore ~p". Which "p, therefor q" does imply. I'm talking about
    "~p, therefor ~q."

    There's a difference between "I am not wet, therefor I am not in the rain" and "I am not in the rain, therefor I am not wet".

    Another example:

    I. p, therefor q: "If I get exactly 98% on my test, I will get an A."
    II. ~q, therefor ~p: "If I don't get an A, then I didn't get exactly 98% on my test."
    III. ~p, therefor ~q: "If I don't get exactly 98% on my test, I will not get an A."

    Obvious, III is not true. You could get 100% on your test--which is not "exactly 98%"--and still get an A.

    "~p, therefor ~q" is only true for biconditional statements ("if and only if")--that is, if p is both a necessary and sufficient condition of q.


    _________________________________ Post Merged _________________________________


    Quote Originally Posted by AliceLiddell
    This part is correct, but a false analogy to Galendir's statement.
    I was challenging Galendir's assertion, but I didn't want to attach spoiler tags to the quote, so I just put up the initial part of his reasoning. He claimed that 6 had to be true.
    Last edited by CliveStaples; September 18th, 2008 at 10:53 AM. Reason: Automerged Doublepost
    If I am capable of grasping God objectively, I do not believe, but precisely because I cannot do this I must believe. - Soren Kierkegaard
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  13. #13
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    Re: True/False Logic Test

    Quote Originally Posted by CliveStaples View Post
    What the hell happened to Galendir's post?
    He deleted it.
    "And that, my lord, is how we know the Earth to be banana-shaped." ~ Monty Python


  14. #14
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    Re: True/False Logic Test

    Quote Originally Posted by CliveStaples View Post
    That's "~q, therefore ~p". Which "p, therefor q" does imply. I'm talking about
    "~p, therefor ~q."
    I understand that's what you are talking about but Galendir's statement as you quoted it was "~q, therefore ~p", so he was right and you were arguing a strawman.

    There's a difference between "I am not wet, therefor I am not in the rain" and "I am not in the rain, therefor I am not wet".

    Another example:

    I. p, therefor q: "If I get exactly 98% on my test, I will get an A."
    II. ~q, therefor ~p: "If I don't get an A, then I didn't get exactly 98% on my test."
    III. ~p, therefor ~q: "If I don't get exactly 98% on my test, I will not get an A."

    Obvious, III is not true. You could get 100% on your test--which is not "exactly 98%"--and still get an A.
    When you read my post did you not think I understood this, or are you just trying to emphasize your own understanding?

    "~p, therefor ~q" is only true for biconditional statements ("if and only if")--that is, if p is both a necessary and sufficient condition of q.
    Agreed, but again, strawman.

    I was challenging Galendir's assertion, but I didn't want to attach spoiler tags to the quote, so I just put up the initial part of his reasoning. He claimed that 6 had to be true.
    I will take your word for it since we now have no record. I prefer when people just admit they made a mistake in their posts and ask other to disregard rather than deleting the entire post. Deleting a mistake you've made without acknowledging is bad form, IMHO.

    Contrariwise, if it was so, it might be; and if it were so, it would be; but as it isn't, it ain't. That's logic.

  15. #15
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    Re: True/False Logic Test

    Quote Originally Posted by AliceLiddell
    When you read my post did you not think I understood this, or are you just trying to emphasize your own understanding?
    My argument was that p -> q does not imply ~p -> ~q, an implication that Galendir based his assertion "statement 6 must be true" on.

    Your reply:
    Wrong, you lose, Clive.

    p-> q does imply ~q ->~p. I think your trouble is in minding your ps and qs.

    If it's true that If I am out in the rain then I am wet, then it's also true that if I am not wet then I am not in the rain.
    I assumed you had read Galendir's post (and that if you hadn't, you would be able to; I hadn't anticipated the deletion of his post); I quoted the beginning of Galendir's argument and disputed his ultimate conclusion.

    I wasn't sure whether you misunderstood my objection, so I thought I should clarify my position. That's all.
    If I am capable of grasping God objectively, I do not believe, but precisely because I cannot do this I must believe. - Soren Kierkegaard
    **** you, I won't do what you tell me

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  16. #16
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    Re: True/False Logic Test

    Quote Originally Posted by CliveStaples View Post
    Wrong. It could be false that "if statement 6 is false, then statement 9 is false." Just as it could be false that "If statement 9 is true, then statement 6 is true." Statement 9 asserts an "if, then" relationship between 9 and 6; if it is false, such a relationship does not exist. 9 could be false, and 6 true (and thus it would be false that "if 6 is true, then 9 is true"). They could also both be true. They could also both be false; p->q does not imply ~p->~q, the classical example being "I am out in the rain, therefore I am wet; I am not out in the rain, so I am not wet" (even though you could be in a pool, or the shower).
    _________________________________ Post Merged _________________________________

    What the hell happened to Galendir's post?
    I posted prematurely and deleted my own post.

    First, I didn't claim that a conditional statement is equivalent to its inverse (p->q = ~p->~q); I claimed that it is (logically) equivalent to its contrapositive (or contraposition) (p->q = ~q->~p). Which it is. Always.

    My first solution was the same as yours, but statement 9 proved problematic. The problem is that in classical sentential logic, in order to preserve truth-functionality, if the antecedent of a conditional statement is false then the conditional statement itself is true (by material implication), therefore S9 is true. The self-referential nature of S9 makes it tautological thus entailing its own truth. This is clearly problematic for a purely bivalent logic as any conditional sentence referencing itself as an antecedent would imply the truth of any and all consequents when interpreted according to such a logic. Self-referencing statements should thus not be so interpreted, but that is what this puzzles requires. (Bivalent meaning that every statement is strictly either true or false). So, classically, there is no way to resolve the set of statements without deriving a contradiction.

    More directly, it can be seen from the contrapositive of S9 that S6 can't be false without rendering S9 self-contradictory. As far as I can tell, even if employing a non-bivalent logic (allowing some statements to be neither true nor false, or in some cases merely undetermined) the contrapositive formulation of a conditional statement is always its logical equivalent. That is, it preserves truth-values for all cases.


    _________________________________ Post Merged _________________________________


    Quote Originally Posted by CliveStaples View Post
    My argument was that p -> q does not imply ~p -> ~q, an implication that Galendir based his assertion "statement 6 must be true" on.
    No, that is incorrect. The only statement of mine you quoted was:
    Quote Originally Posted by Galendir
    Thus, S9 is equivalent to "If statement 6 is false, then statement 9 is false."
    My reasoning to S6 being necessarily true follows directly form this statement which is the contrapositive (or contraposition if you prefer) of S9, not its inverse as you assert.
    Last edited by Galendir; September 18th, 2008 at 01:11 PM. Reason: Automerged Doublepost

  17. #17
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    Re: True/False Logic Test

    Quote Originally Posted by Galendir View Post
    I posted prematurely and deleted my own post.

    First, I didn't claim that a conditional statement is equivalent to its inverse (p->q = ~p->~q); I claimed that it is (logically) equivalent to its contrapositive (or contraposition) (p->q = ~q->~p). Which it is. Always.

    My first solution was the same as yours, but statement 9 proved problematic. The problem is that in classical sentential logic, in order to preserve truth-functionality, if the antecedent of a conditional statement is false then the conditional statement itself is true (by material implication), therefore S9 is true. The self-referential nature of S9 makes it tautological thus entailing its own truth. This is clearly problematic for a purely bivalent logic as any conditional sentence referencing itself as an antecedent would imply the truth of any and all consequents when interpreted according to such a logic. Self-referencing statements should thus not be so interpreted, but that is what this puzzles requires. (Bivalent meaning that every statement is strictly either true or false). So, classically, there is no way to resolve the set of statements without deriving a contradiction.

    More directly, it can be seen from the contrapositive of S9 that S6 can't be false without rendering S9 self-contradictory. As far as I can tell, even if employing a non-bivalent logic (allowing some statements to be neither true nor false, or in some cases merely undetermined) the contrapositive formulation of a conditional statement is always its logical equivalent. That is, it preserves truth-values for all cases.
    There's only a possible contradiction if 9 is actually true. If 9 is true, then 6 must be true, since if "if 9 is true, 6 is true" is true, then 6 must be true.

    But if 9 is false--that is, if 9 does not imply 6 is true (not "9 implies 6 is false")--then 6 can still be either true or false.

    The only contradiction would be if 9 is true and 6 is false. Other than that--if 9 is true and 6 is true, or if 9 is false and 6 is true, or if 9 is true and 6 is false--there is no contradiction.


    Quote Originally Posted by Galendir
    No, that is incorrect. The only statement of mine you quoted was:My reasoning to S6 being necessarily true follows directly form this statement which is the contrapositive (or contraposition if you prefer) of S9, not its inverse as you assert.
    First, I was quoting an innocuous and non-disclosing excerpt rather than your entire argument, mostly because I didn't want to bother with spoiler tags.

    Second, 9 states "If statement 9 is true, statement 6 is true." Which means, if true, that "if statement 6 is false, statement 9 is false." If false, it means that there does not exist a logical implication between the truth of statements 6 and 9.

    Statement 9 does not say "Statements 6 and 9 have the same answer." Rather, it asserts a logical implication between statements 6 and 9. The negation of which does not affirm the inverse; it might turn out that both are false, but not because "if statement 6 is false, statement 9 is false."
    If I am capable of grasping God objectively, I do not believe, but precisely because I cannot do this I must believe. - Soren Kierkegaard
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  18. #18
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    Re: True/False Logic Test

    Gah! Why can't I ever make up one of these things correctly?!?

    Galendir is correct that statement 9 is self-affirmingly true, and therefore 6 is true as well. Which means that there is no way to solve the puzzle. I apologize. 6 should read "statement 8 is false"; I've edited the OP to reflect this change.
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  19. #19
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    Re: True/False Logic Test

    Quote Originally Posted by CliveStaples View Post
    Second, 9 states "If statement 9 is true, statement 6 is true." Which means, if true, that "if statement 6 is false, statement 9 is false." If false, it means that there does not exist a logical implication between the truth of statements 6 and 9.
    Not exactly. Galendir is right on this one although it is unsatisfactory.

    s9: 9->6
    if we say that s9 is false, then we are asserting:
    ~(9->6)
    which is equivalent to
    ~(~9 or 6)
    which is equivalent to
    9 and ~6.
    which means that
    9 must be true.
    which violates our assumption that s9 is false.


    _________________________________ Post Merged _________________________________


    Doh! Of course, Castle beat me to it. Hopefully my proof helps.


    _________________________________ Post Merged _________________________________


    So do we start all over again, Castle?
    Last edited by AliceLiddell; September 18th, 2008 at 04:38 PM. Reason: Automerged Doublepost

    Contrariwise, if it was so, it might be; and if it were so, it would be; but as it isn't, it ain't. That's logic.

  20. #20
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    Re: True/False Logic Test

    Quote Originally Posted by Castle
    Gah! Why can't I ever make up one of these things correctly?!?

    Galendir is correct that statement 9 is self-affirmingly true, and therefore 6 is true as well. Which means that there is no way to solve the puzzle. I apologize. 6 should read "statement 8 is false"; I've edited the OP to reflect this change.
    I disagree. 9 states "If 9 is true, 6 is true."

    Thus:

    1.
    2.
    3.
    4.
    5.
    6. T
    7.
    8.
    9. T
    10.

    The only contradiction is:

    1.
    2.
    3.
    4.
    5.
    6. F
    7.
    8.
    9. T
    10.

    Since 9's truth implies 6's truth, 9 cannot be true if 6 is false.

    This does not entail contradiction:

    1.
    2.
    3.
    4.
    5.
    6. T
    7.
    8.
    9. F
    10.

    If 9 is false, then 6 is either true or false, without logical implication from 9.

    Nor does this entail a contradiction:

    1.
    2.
    3.
    4.
    5.
    6. F
    7.
    8.
    9. F
    10.

    Even though the contrapositive of 9 is "If 6 is false, 9 is false", denying the statement does not mean that 6 and 9 cannot be simultaneously false; all that it means is that the falsity of 9 does not depend on the falsity of 6.


    _________________________________ Post Merged _________________________________


    Quote Originally Posted by AliceLiddell
    s9: 9->6
    if we say that s9 is false, then we are asserting:
    ~(9->6)
    which is equivalent to
    ~(~9 or 6)
    which is equivalent to
    9 and ~6.
    which means that
    9 must be true.
    which violates our assumption that s9 is false.
    ~(p->q)

    does not assert that p is false, nor that q is false, nor that p is true, nor that q is true. It asserts that p does not imply q. Not that p->~q. Not that ~p->q. Not that ~p->~q.
    Last edited by CliveStaples; September 18th, 2008 at 04:56 PM. Reason: Automerged Doublepost
    If I am capable of grasping God objectively, I do not believe, but precisely because I cannot do this I must believe. - Soren Kierkegaard
    **** you, I won't do what you tell me

    HOLY CRAP MY BLOG IS AWESOME

 

 
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