derosa8David Konietzko: Thanks very much for your brief reply above. I agree that when someone says, “This sentence is false” it is in no way clear what the content of the sentence is, so there is nothing of which anything is affirmed. However, just because it is UNCLEAR that the sentence has an intelligible content, does not necessitate that it HAS NO INTELLIGIBLE content. So, it seems you are begging the question when you say “this sentence doesn’t even have an intelligible content.” Do you agree? I know this may seem like word games, but I am sincerely interested in your response to that.
Dr. Casey: Thanks for a very well thought out and thorough response. I actually find your solution very satisfying (the one where you say the sentence implies contradiction either way and is therefore false). That has the ring of Russell’s paradox in Set Theory, and it certainly seems to be a viable argument.
The thought of ALL THOSE DIFFERENT LOGICS is quite disturbing. I thought since logic is the science of necessary inference (proper thinking) that we could rely on it to yield certain truths. But with so many logics to choose from, where does that leave us???
My attempt at Synthesis in Solving the Liar’s Paradox:
(1) Let L = “This sentence is false”
(2) It is not clear if L has intelligible content.
(3) If L has no intelligible content, then it is not a proposition since it contains nothing that is affirmed or denied of something else.
(4) If L has intelligible content, then employ Dr. Casey’s solution (i.e. If the sentence is false, then it is true [contradiction], and If it is true, then it is false [contradiction]) to show that the statement is meaningless. I.e. just stringing words together does not necessarily make a proposition, and in this case no proposition has been made. Just like in Set theory, merely defining a set with a property does not prove the set’s existence.
(5) Since L is not a proposition, it does not fall within the realm of the law of excluded middle.