One of the most famous paradoxes in philosophy that is still widely discussed to this day comes from the ancient Greek philosopher Eubulides of Miletus, from the fourth century B.C. Eubulides of Miletus states the following:

“A man says that he is lying. Is what he says true or false?”

No matter how one answers this question, problems arise because the result is always a contradiction.

If we say the man is telling the truth, that would mean that he is lying, which would then mean that the statement is false.

If we say the man’s statement is false, that would mean that he is not lying, and therefore what he says is true.
However, it is not possible to have a statement be both true and false.

EXPLAINING THE LIAR PARADOX

The problem of the liar paradox goes beyond the simple scenario of the lying man that Eubulides portrayed. The liar paradox has very real implications.

Over the years, there have been several philosophers that have theorized about the meaning of the liar paradox. The liar paradox shows that contradictions can arise from common beliefs regarding truth and falsity, and that the notion of truth is a vague one. Furthermore, the liar paradox shows the weakness of language.

While the liar paradox is grammatically sound and adheres to the rules of semantics, the sentences produced from the liar paradox have no truth value. Some have even used the liar paradox to prove that the world is incomplete, and therefore there is no such thing as an omniscient being.

To understand the liar paradox, one must first understand the various forms it can take.

The Simple-Falsity Liar

The most basic form of the liar paradox is the simple-falsity liar. This is stated as such:

FLiar: “This sentence is false.”

If FLiar is true, then that means “This sentence is false” is true, so therefore FLiar has to be false. FLiar is both true and false, creating a contradiction and a paradox.

If FLiar is false, then that means “This sentence is false” is false, and so FLiar has to be true. FLiar is both false and true, creating a contradiction and a paradox.

The Simple-Untruth Liar

The simple-untruth liar does not work from falsehood, and instead constructs a paradox based on the predicate “not true.” The simple- untruth liar appears as:

ULiar: “ULiar is not true.”

Like the simple-falsity liar, if ULiar is not true, then it is true; and if it is true, then it is not true. Even if ULiar is neither true nor false, that means it is not true, and since that is precisely what ULiar states, ULiar is true. Thus, another contradiction appears.

LIAR CYCLES

Up until now, we’ve only seen examples of liar paradoxes that are self-referential. However, even removing the self-referential nature of the paradoxes still creates contradictions. The liar cycles is stated as:

• “The next sentence is true.”
• “The previous sentence is not true.”

If the first sentence is true, then the second sentence is true, which would make the first sentence not true, thus creating a contradiction. If the first sentence is not true, then the second sentence is not true, which would make the first sentence true, thus creating a contradiction.

POSSIBLE RESOLUTIONS TO THE LIAR PARADOX

The liar paradox has been a source of philosophical debate. Over time, philosophers have created several well-known solutions that allow one to “get out of” the liar paradox.

Arthur Prior’s Solution

Philosopher Arthur Prior claimed the liar paradox was not a paradox at all. To Prior, every statement has its own implied assertion of truth. Therefore, a sentence like “This sentence is false” is actually the same as saying, “This sentence is true, and this sentence is false.” This creates a simple contradiction, and because you cannot have something be true and false, it has to be false.

Alfred Tarski’s Solution

According to philosopher Alfred Tarski, the liar paradox can only arise in a language that is “semantically closed.” This refers to any language where there is the ability to have one sentence assert the truth or falsity of itself or another sentence. In order to avoid such contradictions, Tarski believed there should be levels of languages, and that truth or falsity could only be asserted by language that is at a higher level than that sentence. By creating a hierarchy, Tarski was able to avoid self-referential contradictions. Any language that is higher up in the hierarchy may refer to language that is lower; however, not vice versa.

Saul Kripke’s Solution

According to Saul Kripke, a sentence is only paradoxical depending on contingent facts. Kripke claimed that when the truth value of a sentence is tied to a fact about the world that can be evaluated, this sentence is “grounded.” If the truth value cannot be linked to an evaluable fact about the world, it is “ungrounded,” and all ungrounded statements have no truth value. Liar statements and statements similar to liar statements are ungrounded and, therefore, contain no truth value.

Jon Barwise’s and John Etchemendy’s Solution

To Barwise and Etchemendy, the liar paradox is ambiguous. Barwise and Etchemendy make a distinction between “negation” and “denial.” If the liar states, “This sentence is not true,” then the liar is negating himself. If the liar states, “It is not the case that this sentence is true,” then the liar is denying himself. According to Barwise and Etchemendy, the liar that negates himself can be false without contradiction, and the liar that denies himself can be true without any contradiction.

Graham Priest’s Solution

Philosopher Graham Priest is a proponent of dialetheism, the notion that there are true contradictions. A true contradiction is one that is simultaneously true and false. In believing this to be the case, dialetheism must reject the well-known and accepted principle of explosion, which states all propositions can be deduced from contradictions, unless it also accepts trivialism, the notion that every proposition is true. However, because trivialism is instinctively false, the principle of explosion is almost always rejected by those who subscribe to dialetheism.