True and... True?!?!

Logic Level 3

On an island, everyone is either a knight or a knave. Knights only tell the truth (only make true statements), and knaves always lie (only make false statements).

There exists a legal statement such that both it and its negation can always be said by a knight.

Is this true or false ?

Details of assumption :

By a legal statement is meant a sentence which does not lead to inconsistency . For example: ‘ $1+1=2$ ‘ and ‘This statement is more than $5$ characters long‘ and ‘I am a knight‘ are legal statements (which, in this case, can also always be said by a knight), while ‘This sentence is false‘ and ‘This sentence is true‘ are not legal statements.
‘ always ‘ can intuitively be interpreted as ‘without preparation, in any situation‘. For example: ‘My name is Steve‘ and ‘Max is happy‘ can not always (but possibly, sometimes) be said by a knight (whenever his name is Steve; whenever Max is happy). We say, however, that a knight can ‘always‘ say statements such as ‘ $1+1=2$ ‘.

Followup question

True False

5 solutions

Steve McMath
Sep 20, 2019

"This sentence has five words.". Negation is "This sentence does not have five words." which is also true. Thus either could be said by a knight at any time.

The negation of “This sentence has five words“ is ”The sentence <<This sentence has five words>> does not have five words“, which is false.

Simon Kaib - 1 year, 8 months ago

Also, this is assuming the knight can speak English. I'm just guessing this, but I would think there exists at least one language that those (or other similar) statements can be uttered with the same number of words (or characters, or whatever grapheme is used in that language). This is also assuming they speak a language that can be written (I know, it sounds bizarre, but ones that cannot be written do exist).

Michael Kelsey - 1 year, 8 months ago

How about "This sentence is a positive statement"?

Its negation is "This sentence is not a positive statement"

In both cases they'd be true and the knight can sleep easy knowing he hasn't betrayed his kin with his words.

Ofek Zelig - 1 year, 6 months ago

The negation is actually "The sentence 'This sentence is a positive statement‘ is not a positive statement". Which is false. The Negation of a statement P is true if and only if P is false. Your proposed statements are true at the same time, however. To convince yourself of this, you might want to take a look at the link I linked to 'negation' in the problem.

Simon Kaib - 1 year, 6 months ago

What about this "Either 1+1=2 OR 1+1=3"? one of the parts of the statement should always be true.

Marcus S. - 1 year, 5 months ago

The negation of that statement is not "Either $1+1 \neq 2$ OR $1+1\neq 3$ .". Note that the negation of a statement (by definition of negation ) has to be false if and only if the (original) statement is true. The negation proposed before does not satisfy that as it is true while the original sentence is true as well. Since the original sentence is always true, its negation is always false and is thus not a solution to this puzzle.

Note: the negation is (given you mean an exclusive or ) "((1+1 \neq 2) AND $1+1\neq 3$ ) OR ( $1+1 = 2$ AND $1+1 = 3$ )". The brackets indicate how the sentence is supposed to be read. You can check that this sentence is indeed false.

Simon Kaib - 1 year, 5 months ago

Elijah Frank
Dec 2, 2020

This is a dilemma of true and false that can be Either or Both, if I say “The sentence is true that This is false”. “The sentence, “The sentence is true that This is false” does not true” all of This have a contradiction of nonesense (False+False=True, True+True=True) (True=False not always but sometemies) (all of this nothing is a contradiction of everything is nothing)

Kyrie Lity
Nov 7, 2019

"This Sentence is positive" may be an example. Positive is a State in English Grammar where there is no negation, that is, Not having words like "NOT" or "CAN'T". That statement is true as it doesn't have any negation. Its negation is "This Sentence is not positive". This also is true due to the presence of not, making it negative. Also, if we take the world in account, then statements like "A Joker may tell the truth" or "A Coin flip may end in head" may turn into a required statement. This answer is not entirely correct because it may twist methods of negation.

This is a nice try, it doesn‘t work thought. The negation of "This sentence is positive" is not "This sentence is not positive". It is "The sentence <<This sentence is positive>> is not positive", which is false. (Click the link of negation in my problem).Your second example using vagueness is not a legal statement.

Simon Kaib - 1 year, 7 months ago

Ayush Singh
Oct 22, 2019

How about "this statement can be said by one of the knight or knave"

Note that the negation of that statement is "The statement <<This statement can be said by one of the knight or knave.>> can not be said by a knight and ...". If the statement can be said by a knight then its negation can clearly not be a said by a knight as it would be false. Have I missed something?

Simon Kaib - 1 year, 7 months ago

Danny Radden
Sep 21, 2019

Will you be posting the answer to this question?

The statement “I am saying this sentence.” works. Take a look at the solution I posted on the follow up question for further elaboration. There are similarities.

Simon Kaib - 1 year, 8 months ago

True and... True?!?!

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