In the fifth century BCE, a pagan mystic named Epimenides famously assessed that “All Cretans are liars.” This statement later became the basis of a so-called logical paradox known as “the liar paradox.” The supposed paradox is as follows:
“If Epimenides, himself a Cretan, states that ‘all Cretans are liars,’ then how can his statement be true, since he himself must be a liar as well? If we say it’s true, then it’s false that all Cretans are liars, since Epimenides here has told the truth, and if we say it’s false, then the statement itself must be true.”
Setting aside for now the obvious problems with viewing this “paradox” as anything other than a contextual quirk of language, it took on various forms throughout history and was discussed by various thinkers.
In a modern philosphical context, the “liar paradox” is commonly illustrated with a statement similar to:
“This sentence is false.”
For various reasons, this and similar statements have managed to bewilder various philosophers and have received an inordinate amount of attention from logicians as well. As a result, entire edifices of supposedly “logical” structures have been constructed in attempt to solve the apparent conundrum.
Before presenting the solution to the above “paradox,” it is essential to review the basic principles of logic – often called the “canons of logic” – and how they are substantiated as such. The basic principles of logic are three:
1. Identity, i.e. A is A
2. Non-contradiction, i.e. A is not ~A (not-A)
3. Excluded Middle, i.e. Either A or ~A
These principles led Ayn Rand to describe logic as “the art of non-contradictory identification.”
The Law of Identity (#1), viz. that every thing “is what it is,” is a fundamental principle of metaphysics, and it is based in objective reality. A rock is a rock and is not an egg. That is an objective fact of reality. No matter how much one may wish, hope, or will that a rock be an egg, it will never be possible to crack open a geode and make an omelette. This means that the validity of logic depends on its correspondence with reality. It also means that “A” (and any other variable or symbol of logical analysis) is void and meaningless without being properly representative of reality. Conversely, one cannot extrapolate facts of reality from an unrelated juxtaposition of logical symbols or variables, even if they happen to be arranged in a technically or syntactically correct fashion.
“This sentence is false” is not paradoxical precisely because it neither corresponds with nor references anything in reality. If we were, for example, to state the converse, i.e. “This sentence is true,” then we would immediately be faced with questions such as “How is it true?” and “What makes it true?” and “What is true about it?” There are no answers to these questions because simply stating something about a sentence which has no referent (i.e. no correspondence to reality) is meaningless. The same is true of “This sentence is false.” The fact that we jump to a seemingly paradoxical conclusion is simply an error of logic based on our being led astray by the word “false” contained in the phrase itself.
The only way in which “This sentence is false” could be meaningful is if it were actually referring to a sentence other than itself such as, “Dogs are a type of cat. This sentence (i.e. the preceding assertion) is false.” “Dogs are a type of cat” conveys meaning on “this sentence is false” because it references a sentence that is invalid due to its discontinuity with the facts of reality.
Those who conclude in favor of a real paradox make the mistake of constricting meaning to a simple accordance with grammatical rules. Sentences which are both grammatically correct and meaningless are abundant. For example, “These submarines are hominids” is completely meaningless yet grammatically correct.
I have only scratched the surface here, but I think the general schema of the issues as presented here is clear. Perhaps I will expound on this further in another post, but suffice it to say, “This sentence is false” presents no paradox.