Proposition Logic Question

Question

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

Aren’t these liar paradox?

Comments

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.

Answer 1

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.

Comments

This sentence is true is not a proposition right because it takes both T and F values and works fine for them.

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yes, I have written the same.

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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.

why?? not getting this.

According to ur ans  

 This statement 'S' is false- should be paradox

 This statement 'S' is True-Proposition

is it not??

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why?? not getting this.

"This statement is true" is a propositional statement if either it is true or false but should not be both.

Suppose, we don't know the truth value of statement  "This statement is true"

Now, Assume this statement is true means  "This statement is true" is true.It means statement is actually true. So, It is not contradicting what we assumed. So, It is possible that statement "This statement is true" has truth value True.

Now, Assume this statement is false means  "This statement is true" is false. It means statement is actually false. So, It is not contradicting what we assumed. So, It is possible that statement "This statement is true" has truth value False.

Since, "This statement is true" has both truth values true and false , So, It is not a proposition.

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@ankitgupta.1729 I think I'm on the same boat as @srestha here. 

"toxic is a boy" is a propositional statement if either it is true or false but should not be both.

Suppose, we don't know the truth value of statement  "toxic is a boy"

Now, Assume this statement is true means  "toxic is a boy" is true. It means toxic is actually a boy. So, It is not contradicting what we assumed. So, It is possible that statement "toxic is a boy"  has truth value True.

Now, Assume this statement is false means  "toxic is a boy" is false. It means toxic is not a boy. So, It is not contradicting what we assumed. So, It is possible that statement "toxic is a boy" has truth value False.

Since, "toxic is a boy" has both truth values true and false, So, It is not a proposition.

 

- 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. 

 

Correct me if I'm wrong. 

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 @toxicdesire 

coz it has a truth value of true or false but not both at same time.

x+2 =3 is not a proposition and it can also be true for some cases and false for some cases but not both at same time. 

proposition should be either true or false for all the cases.

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@toxicdesire

yes, proposition has truth value of either true or false, but not both at the same time. 

So, can you tell me the truth value of statement "This statement is true" ? It should be either true or false like when we say statement "New Delhi is the capital of India" has truth value "true". It should not be like sometimes true and sometimes false.

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@toxicdesire

I think 2 nd statement is a liar paradox but not 1st one

"This sentence is true"- this is not contradicting True or False value everytime

So, not liar paradox.

https://en.wikipedia.org/wiki/Liar_paradox

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@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