I don’t remember the first time I heard about the Liar’s Paradox, but it was definitely in college, because it came up in my computer theory classes in discussions on formal grammars. As such, I’ve been thinking about it on and off, for about 20 years. Wikipedia says that the first correct articulation of the Liar’s Paradox is attributed to a Greek philosopher named Eubulides, who stated it as follows: “A man says that he is lying. Is what he says true or false?” For purposes of this note, I’m going to distill that to the modern formulation of, “This statement is false.”

As an initial matter, we must accept that not all statements are capable of meaningful truth values. For example, “Shoe”. This is just a word, that does not carry any intrinsic truth value, nor is there any meaningful mechanical process that I can apply to the statement to produce a truth value. Contrast this with, “A AND B”, where we know that A = “true” and B = “false”, given the typical boolean “AND” operator. There is in this case a mechanical process that can be applied to the statement, producing the output “false”. Now, all of that said, there is nothing preventing us from concocting a lookup table where, e.g., the statement “Shoe” is assigned the value “true”.

Now consider the distilled Liar’s Paradox again: “This statement is false”. There is no general, mechanical process that will evaluate such a statement. However, it is plainly capable of producing a truth value, since it simply asserts one for itself, much like a lookup table. Typically, this is introduced as producing a paradox, because if we assume the truth value is false, then the truth value of false is consistent with the truth value asserted in the statement. Generally speaking, when assertion and observation are consistent, we say the assertion is true, and this is an instance of that. As such, the statement is true, despite the fact that the statement itself asserts that it is false. Hence, the famous paradox.

Now instead approach the problem from the perspective of solving for the truth value, rather than asserting the truth value. This would look something like, “This statement is A”, where . Now we can consider the two possible values of . If , then the statement asserts a truth value that is consistent with the assumed truth value, and there’s nothing wrong with that. If instead , then we have a contradiction, as noted above. Typically, when engaging in mathematics, contradictions are used to rule out possibilities. Applying that principle in this case yields the result that , which resolves the paradox.

In summary, not all statements are capable of mechanical evaluation, and only a subset of those mechanically evaluable statements resolve to true or false. This however does not prevent us from simply assigning a truth value to a statement, whether by a lookup table, or within the statement itself. However, if we do so, we can nonetheless apply basic principles of logic and mathematics, and if we adhere to them, we can exclude certain purported truth values that are the result of mere assertion. In this case, such a process implies that a statement that asserts its own truth value, is always true.