In an earlier post I went over the necessity of three logical states, where you have 1 for true, -1 for false, but negatably true, and 0 for statements not negatably true.
To understand that system, consider the following.
One plus one equals two.
That sentence is true, but consider.
One plus one equals three.
That sentence is false, but negatably true, where by negation, I mean to negate one side of the equals:
One plus one does not equal three.
One plus one is not three.
To show the last logical state is a little harder, as I need a nonsense statement like:
The sjdlfy is green.
Here I've created the word "sjdlfy" by randomly typing, so it's not meaningful to give it a color!
Under the rules of 3 Logic its logical value can be rigorously determined to be 0 by trying to negate it, which gives:
The sjdlfy is not green.
As that is still nonsense, it has a logical value of 0.
I'll talk again about the Liar Paradox, as it's a well-known example, where 3 Logic does not give a contradiction, but the resolution can be confusing I think, as consider the sentence:
This sentence is false.
It has a logical value of -1, as it is negatable:
This sentence is not false.
Notice then the logical result that a sentence cannot truthfully declare itself to be false, even if some other sentence can declare it to be false, as true does not equal false, T does not equal F.
So while I can say, that sentence is false, a sentence cannot truthfully claim that it is false, as it could only do so, if truth and falsity were equals, but they are not.
You may say to yourself, "The sentence claims it is false, and I can say, yes, that sentence is false, so then the sentence is true!"
But at the start you note that the sentence is false!
The contradiction is in your own behavior. If the sentence is false, then THAT sentence is false, so it's not true, no matter what. In claiming it is false, it is attempting to equate truth with falsity, which is just false.
One way to look at it is as if a clever con-man were trying to convince you to part with your life's wages. This con-man needs to convince you that true is false, so that you will give up your money, so he just tells you that, true is false, right?
Nope. That won't work. He tries something more clever, where you assume truth, like with sentences, as given a sentence there is an assumption of truth.
So in claiming itself to be false, a sentence is relying on your own inner assumptions about sentences, which you can consider.
Now consider this example:
A sentence normally declares a truth as the purpose of sentences is to communicate information, but THIS sentence is going to tell you that it is false, so it is telling you that it is not communicating truth to you, but is in fact, not true, as I go ahead and just now say, this sentence is false.
Now negate it:
A sentence normally declares a truth as the purpose of sentences is to communicate information, and THIS sentence is not going to tell you that it is false, so it is telling you that it IS communicating truth to you, which is in fact, true, as I go ahead and just now say, this sentence is true.
And I did have to negate in several places as it is a compound sentence, but I think you should have some sense of what is happening with the original statement:
This sentence is false.
It's like a con-man saying: "Trust me, truth is falsehood, give me your money."
But, truth is not falsehood. Invest wisely.