I want to prove two self-referential sentences $S, S_1$ are proposition or not. My approaches are given below.
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 a paradox.
Now, Suppose, There is a statement :-
$S_1$ : This statement '$S_1$' is true
Again, there are 2 possible cases :-
Case 1):- '$S_1$' is true
If statement '$S_1$' is true, It means statement "This statement '$S_1$' is true" is True which means Statement '$S_1$' is True which is not contradicting our assumption that '$S_1$' is true. So, It is a possible case.
Case 2):- '$S_1$' is false
If statement '$S_1$' is false, It means statement "This statement '$S_1$' is true" is False which means Statement '$S_1$' is False which is again not contradicting our assumption that '$S_1$' is false. So, It is also a possible case.
In both cases, statement "This statement '$S_1$' is true" is getting both truth values i.e. true and false. So, according to the definition of proposition, "This statement '$S_1$' is true" is not a proposition and it is also not a paradox because we are not getting contradiction in both cases here.
I have tried for two cases of both self-referential sentences $S$ and $S_1$ to prove both sentences are proposition or not.
I don't know my approach is right or not. I haven't found any valid stuff in internet.
Best Answer
Like in most cases you're stating both your sentence $S$ (liar) and your sentence $S_1$ with quotation mark to mention itself via the same English language without object/meta language differentiation, since most natural languages are closed the liar case is like Russell's paradox its truth value is overdetermined thus cannot be determined logically, therefore it's paradoxical and cannot be a proposition. As noted in the wiki reference, there're many different solutions out of this paradox.
Your sentence $S_1$ apparently lacks grounding evidence to judge its truth condition (it may be either true or false as you rightly analyzed) thus its truth value cannot be determined empirically but without any logical paradox, therefore it's a proposition though its truth value is not known yet. This is not uncommon like $P=NP$ which is a well formed proposition attracting many people to try to determine its truth value though currently no one knows for sure yet. So in this natural language common case, your referenced website saying your second sentence is a true proposition is not totally right, but we can say for sure that it's a proposition unparadoxically.