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You over exert yourself working on logic puzzles and die of exhaustion. You are awoken by the omniscient deities True and False. True only speaks true statements and False only speaks false statements. They are willing to take you to logic paradise, but first you must say a true statement that True cannot say, a false statement that False cannot say, and a statement that has a conclusive truth value (either true or false) but that neither can say. All three statements must be different (you cannot give the same statement twice or a linguistically different statement which gives the same proposition) and you cannot use conjunctions to combine one of the previously given statements, i.e. no giving statement A and also “statement A and statement B”. The statements must give the same proposition no matter who speaks it, e.g. no using statements which rely on pronouns so that the proposition given by the statement changes depending on who speaks it. Which three statements can you give?

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  • $\begingroup$ Does "The statements must give the same proposition no matter who speaks it" mean that the truth value of the statement has to be the same if we ask either, or just that the interpretation of the statement must remain unchanged by who is asked? $\endgroup$ – Jonathan Allan Jul 5 '16 at 17:27
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    $\begingroup$ It means that that the interpretation of the statement must remain unchanged regardless of who speaks it. For example, a man named John could say "My name is John," which is a true statement. But True of course cannot say "My name is John." The point of the clause is to eliminate these types of statements, as True will interpret the pronoun as the proper noun that is intended by the pronoun, meaning when when John says "My name is John," True will understand it as the proposition "John's name is John," which he can say. $\endgroup$ – rjones Jul 5 '16 at 17:38
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    $\begingroup$ Enjoy the way this plays with the distinction between what cannot be said because of its truth value and what cannot be said because of implied contradiction $\endgroup$ – humn Jul 5 '16 at 23:06
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This answer has been expanded to consider 7 possible combinations of truth values and statement-making abilities, thanks to PellMel's reminder to account for indeterminate truth values, called “moot” here, from the outset.
  Bear in mind the difference between what a deity may (is allowed to) say— based on the truth of a statement — and what that deity can say— based on what the statement claims and implies.

    Statement S           True deity                   False deity
    truth value      may say S    can say S       may say S     can say S
    -----------      ---------    ---------       ---------     ---------
1.     true             may          can           may not       cannot
2.     true             may          moot          may not       cannot
3.     true             may         cannot         may not       cannot

4.     moot           may not       cannot         may not       cannot

5.     false          may not       cannot           may          can
6.     false          may not       cannot           may          moot
7.     false          may not       cannot           may         cannot 


True statement (A) that True cannot say:

A. True cannot make this statement.

The only consistent combination of values is that A is true and that True may say it, but cannot.

                 True deity vs A
      Assume       (suppose)         Then A's
       A is        may   can         claim is           Contradiction
      ------     ---------------    ----------     -----------------------
1.     true        may   can           false        A's claim is not true
2.     true        may   moot          moot         A's claim is not true
3.     TRUE  -->   may  CANNOT   -->   true   -->   NONE
4.     moot        not  cannot         true         A's claim is not moot
5-7.   false       not  cannot         true         A's claim is not false


False statement (B) that False cannot say:

B. False can make this statement.

The only consistent combination of values is that B is false and that False may say it, but cannot.

                 False deity vs B
      Assume       (suppose)          Then B's
       B is        may   can          claim is          Contradiction
      ------     ----------------    ----------    ------------------------
1-3.   true        not  cannot         false        B's claim is not true
4.     moot        not  cannot         false        B's claim is not moot
5.     false       may   can           true         B's claim is not false
6.     false       may   moot          moot         B's claim is not false
7.     FALSE -->   may  CANNOT   -->   false  -->   NONE                  


True statement (C), other than A, that neither deity can say:

These are effectively equivalent:
C'.   False has at least as much ability to make this statement as has True (in colloquial terms).
C''.   If True can make this statement then so can False.
C'''. True cannot make this statement or False can (or both, T cannot while F can).

Discussion plagiarized from PellMel's comment:   Suppose that True can say C. For this to be possible, C must be true, and thus at least one of the alternatives of C''' must be true. The first alternative is that True cannot say C, which contradicts the supposition. The other alternative is that False can say C, but C is true (thus definitely not false), hence False is not even allowed to say it. Thus we reject the proposition that True can say C, but this satisfies the first alternative of C''', so C is conclusively true all the same. Since C is true, False may not say it either.
(Click within any hidden area to reveal it permanently.)

   C  =  True cannot say C  or  False can say C        (C''', the easiest version to test)


           True deity vs C     False deity vs C
Assume       (suppose)           (suppose)         Then C's
 C is        may   can           may   can         claim is          Contradiction
------     ---------------     ----------------    --------    ---------------------------
 true        may   can           not  cannot        false       None of C's claim is true
 true        may   moot          not  cannot        moot        None of C's claim is true
 TRUE  -->   may  CANNOT   -->   not  CANNOT  -->   true   -->  NONE

 moot        not  cannot         not  cannot        true        None of C's claim is moot

 false       not  cannot         may   can          true        None of C's claim is false
 false       not  cannot         may   moot         true        None of C's claim is false
 false       not  cannot         may  cannot        true        1st part of C is not false


[Deleted: Side note that incompletely tabulated PellMel's solution]

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  • $\begingroup$ Your statement A and B and the 2nd version of C are what I had in mind as answers. $\endgroup$ – rjones Jul 6 '16 at 1:20
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    $\begingroup$ Glad to hear, @rjones. Funny how the three C versions serve differently. "As much ability" is (almost too) colloquial, if/then is the cleanest to state but would require explanation in a truth table, and or is the simplest for a truth table but begs clarification (inclusive/exclusive) when first stated. $\endgroup$ – humn Jul 6 '16 at 1:34
  • $\begingroup$ In considering statement C, it is essential to take into account the facts that (1) there are true statements that True cannot say and false statements that False cannot say, and (2) statements exist that are neither true nor false. I think C does satisfactorily answer the puzzle, but I don't think the argument given is conclusive or that the table presents all the possibilities. $\endgroup$ – PellMel Jul 7 '16 at 18:51
  • $\begingroup$ I would argue it this way: suppose True can make statement C. In that case, it follows that C is true. The first alternative of C is that True cannot make statement C, which contradicts the premise. The alternative is that False can make statement C, but C is true (thus definitely not false) hence False cannot make it. Thus we reject the proposition that True can make statement C, but that satisfies C's first alternative, therefore C is conclusively true all the same. Since C is true, False cannot say it either. $\endgroup$ – PellMel Jul 7 '16 at 19:00
  • $\begingroup$ Your comment is now the discussion for C, @PellMel, thanks again, and every statement survived a kind of all-out consideration of indeterminate truth values, also credited to you. (There is no single definitive way to consider three-value logic, wouldn't you know, so I picked one without really stating the rules.) Feel free to edit. $\endgroup$ – humn Jul 8 '16 at 4:10
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"True" cannot say:

True cannot say this statement.

"False" cannot say:

False can say this statement.

Neither may say:

True cannot say that False can say this statement.

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  • $\begingroup$ Note: pretty certain this answer is borderline on a couple of the rules. The basics are there though. $\endgroup$ – zzzzBov Jul 5 '16 at 17:18
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    $\begingroup$ Very good. The first two answers you’ve given are correct. Your "neither" answer follows the rules, but I believe that one of the deities can say it: (1) True cannot say statement (2). (2) False can say statement (1). If False can say statement (1), then it is not a statement that neither can say. If False cannot say statement (1), then this means (2) is false and therefore True cannot say statement (2), which means that True can say statement (1). $\endgroup$ – rjones Jul 5 '16 at 17:49
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    $\begingroup$ @rjones, True may not state the 3rd statement because it would invalidate the "True cannot say" clause. Because of this, the statement remains true, which means that False may not state the 3rd statement either. $\endgroup$ – zzzzBov Jul 5 '16 at 19:25
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    $\begingroup$ If we name your answer statement (1), and statement (2) is “False can say statement (1),” then (1) is equivalent to “True cannot say statement (2).” Either False can say (1) or he cannot. If he can say it, then it is not a statement neither True nor False can say. If he cannot say (1), then (2) is false and therefore True cannot say (2), which means that True can say statement (1). Statement (1) is not saying that True cannot say statement (1), but that he cannot say that False can say(1). If False cannot say (1), then True will have no problem saying he cannot say False can say it. $\endgroup$ – rjones Jul 5 '16 at 19:54
  • $\begingroup$ I don't think this works, if True can only say something if it is true, then he simply cannot say "I cannot say this statement". If uttering the phrase makes it false, then it's not true. Timing doesn't seem like it should be relevant if True's requirement is to say only things that are true. There's no requirement that True has to say anything, just that what he says is true. It seems a lot like "At the end of this sentence, I will have told you something false." $\endgroup$ – John Jul 5 '16 at 22:21
3
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How about

Say to True

As far as I am aware, this isn't a repetition of a statement.
(true when you say it*, false if True wants to say it)

Say to False

As far as I am aware, this is a repetition of a statement.
(false when you say it*, true if False wants to say it)

Say to both

I always lie

*

If you are aware of the statement you are to say to True/False having been said before, you would need to clarify it in such a way as to make it true/false.
For example, "As far as I am aware, this isn't/is a repetition of a slightly protracted statement".

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    $\begingroup$ Couldn't False say your number 3, since it is not true? $\endgroup$ – Daphne B Jul 5 '16 at 18:48
  • $\begingroup$ @DaphneB maybe a rewording is necessary, the "each of" is meant to make it verify the two facts independently, rather than the conjunction of the two. $\endgroup$ – Jonathan Allan Jul 5 '16 at 18:55
  • $\begingroup$ @DaphneB - thinking it through my updated question is not only clearer but also much simpler. $\endgroup$ – Jonathan Allan Jul 5 '16 at 19:30
  • $\begingroup$ Vote of approval, especially for how the first two work, but these might turn out to contradict the pronoun stipulation $\endgroup$ – humn Jul 5 '16 at 19:32
  • $\begingroup$ @humn which is why I asked about it. I am using proper nouns that do not change the interpretation of the statement (I think!!) $\endgroup$ – Jonathan Allan Jul 5 '16 at 19:34
3
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True cannot make this true statement:

True cannot utter this statement. (If he could make the statement then it would be false, and True cannot make false statements.)

False cannot make this false statement:

False can utter this statement. (Similar to the previous, if False could make the statement then it would be true, therefore he could not make it.)

Neither True nor False can make this false statement:

False can say that True can utter this statement. (True could utter the statement only if it were true, but in that case it would claim that False can utter a true statement, which is false. The statement is therefore conclusively false, and True cannot make it. But False cannot make it either because if he could then it would be true, and (i) False cannot make true statements, but also (ii) we have already proven the statement false.)

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    $\begingroup$ I do not believe your "neither" statement meets the criteria of having a conclusive truth value. If we assume it is true, then True can utter it and False cannot say that True can utter it, which means the statement is false. If we assume that it is false, then True cannot utter it and False can say so, which means that the statement is true. Therefore, it can neither be true nor false. It is a paradoxical statement. $\endgroup$ – rjones Jul 5 '16 at 22:54
  • $\begingroup$ Vote of approval for the Klein-bottle-like logic in any case $\endgroup$ – humn Jul 5 '16 at 22:58
  • $\begingroup$ @rjones, your logic is flawed. You reason that if the "neither" statement is false then it is one that False can make, but there are false statements that False cannot make. False cannot make the "neither" statement in particular because if he could then that statement would be true, and he cannot make true statements. Since he cannot make the statement, and the statement asserts that he can do, the statement is conclusively false. $\endgroup$ – PellMel Jul 6 '16 at 17:22
  • $\begingroup$ A tabular layout shows that the third statement (call it $\small\sf P$) does work in that neither deity can say it. But...$~~ \small\sf P$ is true after all, not false, and boils down to the following: $~ \small\sf P = true = \textsf{False-can-say-Q} ~~~$ & $~~~ \small\sf Q = false = \textsf{True-can-utter-P} $ . True _may_ say $\small\sf P$ because it is true but cannot say it because that would produce a contradiction through $\small\sf Q$. False cannot say $\small\sf P$ because it is true. (continued next comment) $\endgroup$ – humn Jul 7 '16 at 9:35
  • $\begingroup$ (continued) For $\small\sf P$ to be false, however, True would not be allowed to say it, so $\small\sf Q$ would be false as well, This would allow False to say $\small\sf Q$, so $\small\sf P$ would be true after all—a contradiction. I added the table for this to my post and will gladly delete it if you'd like to use it in some form. $\endgroup$ – humn Jul 7 '16 at 9:35
2
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Love logic puzzles (this one seems fun +1) but not great at them. Still think this might work though:

True

"True can't say this" -true until spoken making it unspeakable

False

"False can say this" -false until spoken making it unspeakable

Both

"Either True of False could say this, but not both." -false because false couldn't initially say it and if true were to say it, false could too making it false which stops true from saying it.

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  • $\begingroup$ I think we've made the same error - saying "I ..." is interpreted as "Brent Hackers ..." I now believe (although I could still be wrong) - but yours may be salvageable. $\endgroup$ – Jonathan Allan Jul 6 '16 at 0:36
  • $\begingroup$ @jonathanallan updated changing out the pronouns. I think some of the other answers are better anyway though. $\endgroup$ – Brent Hackers Jul 6 '16 at 4:34

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