The Liar Paradox

1) What is the Paradox?

The classic liar paradox, also known as the liar’s paradox or antinomy of the liar, is the claim made by a liar that they are lying, such as “I am lying.” If the person who is lying is, in fact, stating the truth, then the person who is lying has just lied. The contradiction is enhanced in “this sentence is a lie” so that it may be subjected to more in-depth logical investigation.

Despite abstraction being formed directly from the liar making the assertion, it is nevertheless commonly referred to as the “liar paradox.” A contradiction results from attempting to give this statement—the strengthened liar—a traditional binary truth value. The sentence “this sentence is false” states that it is untrue, thus if it were true, it would also be true that “this sentence is false,” and vice versa.

2) About Epimenides:

Greek philosopher-poet Epimenides of Cnossos was a semi-mythical seer and poet from Knossos or Phaistos who lived in the seventh or sixth century BC. It’s unclear when Epimenides started to be linked to the liar paradox variant known as the Epimenides paradox. When he said, “Cretans, always liars,” Epimenides himself does not appear to have had any irony or paradox in mind. One of themselves, even a prophet of their own, warned that the Cretians are always liars, wicked beasts, and slow eaters in the epistle to Titus. Numerous variations of the liar dilemma were examined in the Middle Ages under the umbrella term of insolubilia, but these were not connected to Epimenides.

3) Other Thinkers on The Liar Paradox:

Alfred Tarski:

According to Alfred Tarski, the paradox only occurs in “semantically closed” languages, which are those in which one statement can predicate the truth (or untruth) of another sentence in the same language (or even of itself). When considering truth values, it is required to visualise tiers of languages, each of which can only predicate truth (or falsehood) of languages at a lower level in order to avoid self-contradiction. Therefore, it is semantically higher when one statement relates to the truth value of another.

The referenced sentence is seen as a component of the “object language,” whereas the referring phrase is regarded as a component of the “meta-language” with regard to the object language. Sentences from “languages” higher on the semantic hierarchy may refer to sentences from “languages” lower on the semantic hierarchy, but not vice versa. This stops a system from turning inward on itself.

This system is not complete, though. A statement like “For every statement in level of the hierarchy, there is a statement at level +1 which says that the first statement is false” would be nice to be able to say. This is a true, significant claim about the hierarchy that Tarski outlines, but as it relates to claims at every level of the hierarchy, it must be above all of the levels, and as a result, it is not possible within the hierarchy (although bounded versions of the sentence are possible). It is acknowledged as a general issue in hierarchical languages, and Saul Kripke is credited with detecting this incompleteness in Tarski’s hierarchy in his widely renowned paper “Outline of a theory of truth.”

Arthur Prior:

According to Arthur Prior, the liar paradox is not paradoxical. He contends that every statement contains an implicit claim to the validity of that statement (which he credits to Charles Sanders Peirce and John Buridan). Because the phrase “it is true that…” is always included, for instance, the statement “It is true that two plus two equals four” contains no more information than the sentence “It is true that two plus two equals four.” Additionally, “it is true that” is comparable to “this entire statement is true and…” in the self-referential spirit of the Liar Paradox.

Since both “This statement is false” and “This statement is true and this statement is false” are true, they are equivalent. The latter is untrue since it is a straightforward contradiction of the type “A and not A.” Because the assertion that this two-conjunct liar is false does not lead to acontradiction, there is no paradox.

Saul Kripke:

According to Saul Kripke, contingent facts may determine whether a phrase is paradoxical or not. If Jones only says these three things about Smith: “Smith is a huge spender,” “Smith is soft on crime,” and “Everything Smith says about me is true,” then Smith will only say, “A majority of what Jones says about me is false.” Both Smith’s comment about Jones and Jones’ last comment about Smith are paradoxical if Smith actually is a large spender but is not soft on crime.

Kripke suggests the following approach as a resolution. A proposition is “grounded” if its truth value is eventually connected to some measurable reality about the outside world. Otherwise, that claim is “ungrounded.” Unfounded assertions lack a truth value. Liar remarks and words that resemble liars are unsupported and have no bearing on truth.

Jon Barwise and John Etchemendy:

The liar phrase, which Jon Barwise and John Etchemendy claim is synonymous with the Strengthened Liar, is said to be ambiguous. This conclusion is based on their distinction between a “denial” and a “negation.” The liar is denying oneself if they mean, “It is not the case that this statement is true.” It negates itself if it means, “This statement is not true.” They continue by arguing that the “denial liar” can be truthful without contradiction whereas the “negation liar” can be untrue without contradiction based on situation semantics. Their 1987 work heavily relies on shaky set theory.

4) Its Importance:

The Liar Paradox has occasionally been used as evidence to demonstrate a significant philosophical point. For instance, Grim (1991) stated that it proves there cannot be an omniscient being because it ultimately reveals the world to be “incomplete” in some sense. The Liar Paradox, according to McGee (1991) and others, exposes the concept of truth as being nebulous. While Eklund (2002) contends that The Liar Paradox reveals significant information on the nature of semantic competence and the languages we use, Glanzberg (2001) contends that it reveals significant information about the nature of context dependence in language. According to Gupta and Belnap (1993), it displays significant characteristics of the generic notion of definition. Other lessons have also been drawn, as well as variations on those teachings.

What the Liar Paradox demonstrates to us regarding the fundamental rules controlling truth and logic is of greater immediate significance. In a pessimistic spirit, Tarski himself (1935, 1944) appears to have believed that the Liar Paradox demonstrates that the conventional notion of truth is incoherent and needs to be replaced with a more credible one from the standpoint of science. The notion that the fundamental rules controlling truth are more nuanced than the T-schema represents is more prevalent and may perhaps be the overarching theme in the solutions to the Liar Paradox.

The Liar Paradox has also served as the central argument against classical logic since some crucial aspects of it allow for capture and release to produce absurdity. The justifications for paracomplete and paraconsistent logics stand out among them. Ripley contends that classical logic can be preserved while eschewing the aforementioned aspects