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.