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Are these propositions?
1.This sentence is true
2.This sentence is false

Aren’t these liar paradox?
in Mathematical Logic
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edited by
Both of these statements are Self Referential Statements.

This sentence is false :

It is a negative self referential sentence. A negative self referential sentence is always not a proposition since we cannot assign any truth value to it. These sentences keeps on oscillating between truth and false values.

This sentence is true :

It is a positive self referential sentence hence we can associate both the truth values with it but for a sentence to be a proposition it should have a single truth value associated to it which could be either true or false but not both at the same time.
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Suppose, There is a statement :-

S : This statement 'S' is false

Now, There are $2$ possible cases :-

Case $1)$ :- 'S' is true

If statement 'S' is true, It means statement "This statement 'S' is false " is True which means Statement 'S' is False  which is contradicting our assumption that 'S' is true. So, It is not a possible case which means statement 'S' can't be true.

Case $2)$ :- 'S' is false

If statement 'S' is false, It means statement "This statement 'S' is false " is False which means Statement 'S' is True which is again contradicting our assumption that 'S' is false. So, It is also not a possible case which means statement 'S' can't be false.

Now, Proposition is a declarative statement which is either true or false but not both. Here, statement "This statement 'S' is false"  is not getting any truth value either true or false. So, It is not a proposition. It is paradox. It is an example of Liar Paradox

Now,   Suppose, There is a statement :-

S : This statement 'S' is true

Again, there are $2$ possible cases :-

Case $1)$ :- 'S' is true

If statement 'S' is true, It means statement "This statement 'S' is true" is True which means Statement 'S' is True which is not contradicting our assumption that 'S' is true. So, It is a possible case.

Case $2)$ :- 'S' is false

If statement 'S' is false, It means statement "This statement 'S' is true" is False which means Statement 'S' is False which is again not contradicting our assumption that 'S' is false. So, It is also a possible case.

In both cases, statement  "This statement 'S' is true"  is getting both truth values i.e. true and false. So, according to the definition of proposition,  "This statement 'S' is true" is not a proposition and it is also not a paradox because we are not getting contradiction in both cases here.

4 Comments

@srestha I have no problem with the second sentence. 

 

@Satbir 

$x+2 = 5$ is not a proposition, because it contains a reference to an external variable $x$ which is not defined and hence it's not a proposition. 

@ankitgupta.1729

is the sentence "toxic is a boy"  a proposition? If it is, then what is the truth value of it? I don't think you can tell it's truth value, cause you don't know who toxic is, but you can tell that it's either true or false right? 

similarly, "this sentence is true" does not lead to contradicting results. Also, it's self referential, and does not contain any external variables to depend on as in the example $x+2=5$. 

so, just like If I assume "toxic is a boy" to be true / false, I get two distinct results that "toxic is a boy" / "toxic is not a boy" in order. 

Can I not assume "this statement is true" to be true / false, to get distinct results that "this statement is true." / "This statement is false"?

Correct me if I'm wrong. 

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@toxicdesire

x+2 =5 is not a proposition because it can be true for some case and false for some other case but not true and false at same time.

This example i gave in context of your comment

Am I doing something wrong here? I think "toxic is a boy" is a proposition as far as I know.  I think (but not sure) that "this statement is true" is a proposition, cause it has truth value of either true or false, but not both at the same time. 

Both the highlighted lines are same and are supporting opposite statements.

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@toxicdesire

yeah, it is self-referential but to become a proposition, It should have fixed truth value like other statements which are proposition.

2nd one is not the proposition. It is anti-paradox means opposite of paradox. Truth value can be true or false. It depends on us. If we get contradiction in either 1 case then it becomes the proposition because then it will either be true or false.

 https://math.stackexchange.com/questions/2671517/is-this-sentence-is-true-true-or-false-or-both-is-it-a-proposition   

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