For a quick description of the Liar, read this.
Some philosophers, including C.S. Peirce, have argued that the Liar is demonstrably false and not true. The argument is based on the premise that all statements implicitly assert their own truth.
At first glance, this seems plausible. If I tell someone, “I did your laundry,” it carries with it an implied “it is true that I did your laundry.” This would seem to hold for all assertions. So, Peirce argues, the Liar is really saying: “It is true that this sentence is false,” which essentially comes down to saying “this sentence is true and false.” This is no longer a paradox, but a plain contradiction, and so false. It is like saying, “I’m a cat owner that doesn’t own a cat.” That’s not a mystery, just a lie. What makes the Liar a paradox is that what it says is, on the surface of it, coherent. If it just asserts a flat-out contradiction, then it poses no problem.
That a proposition automatically asserts its own truth is an interesting notion, and it is not easy to say whether it is accurate or not. Peirce later in life argued that it was incorrect. Luckily, it is not necessary to determine whether it is accurate or not because, even if it is, it does not resolve the Liar.
It is not true that, if a proposition automatically asserts its own truth, then the Liar really says: “It is true that this sentence is false.” In that sentence, “this sentence” refers to that whole sentence. However, in the original Liar, “this sentence” refers only to “This sentence is false.” “This sentence” has changed what it refers to from one sentence to the other. Therefore, the purported truth-asserting version asserts the truth of a different statement than the Liar. This mistake happens because of how slippery “this sentence” can be if you’re not careful with it.
Consider, for example, the obviously true sentence “This sentence has five words.” If this sentence were really saying, “It is true that this sentence has five words,” it would then be false. But that’s a mistake. If the sentence asserts its own truth, then it says, “It is true that the sentence ‘This sentence has five words’ has five words.” This is still true.
Similarly, the accurate truth-asserting version of the Liar is: “It is true that the sentence ‘This sentence is false’ is false.” This is a sentence that says that some other sentence is true. That other sentence says that it itself is false. Neither sentence expresses a flat-out contradiction. Both still yield contradictions when we try to determine their truth-values. This is because, as we would expect, they really say the exact same thing. Automatic truth-assertion doesn’t add anything to the Liar; it certainly doesn’t solve it.