In "this sentence is a lie", the paradox is strengthened in order to make it amenable to more rigorous logical analysis.
Trying to assign to this statement, the strengthened liar, a classical binary truth value leads to a contradiction.
The paradox's name translates as pseudómenos lógos (ψευδόμενος λόγος) in Ancient Greek.
One version of the liar paradox is attributed to the Greek philosopher Eubulides of Miletus, who lived in the 4th century BC.
"[2] The paradox was once discussed by Jerome of Stridon in a sermon: "I said in my alarm, Every man is a liar!"
[3]The Indian grammarian-philosopher Bhartrhari (late fifth century AD) was well aware of a liar paradox which he formulated as "everything I am saying is false" (sarvam mithyā bravīmi).
Naṣīr al-Dīn al-Ṭūsī could have been the first logician to identify the liar paradox as self-referential.
[6] The problem of the liar paradox is that it seems to show that common beliefs about truth and falsity actually lead to a contradiction.
Sentences can be constructed that cannot consistently be assigned a truth value even though they are completely in accord with grammar and semantic rules.
[7] This response to the paradox is, in effect, the rejection of the claim that every statement has to be either true or false, also known as the principle of bivalence, a concept related to the law of the excluded middle.
But, a simpler version is possible, by assuming that the single word 'true' bears a truth value.
The analogue to the paradox is to assume that the single word 'false' likewise bears a truth value, namely that it is false.
[11][12] Saul Kripke is credited with identifying this incompleteness in Tarski's hierarchy in his highly cited paper "Outline of a theory of truth,"[12] and it is recognized as a general problem in hierarchical languages.
His claim (which he attributes to Charles Sanders Peirce and John Buridan) is that every statement includes an implicit assertion of its own truth.
[15] Saul Kripke argued that whether a sentence is paradoxical or not can depend upon contingent facts.
[16] Graham Priest and other logicians, including J. C. Beall and Bradley Armour-Garb, have proposed that the liar sentence should be considered to be both true and false, a point of view known as dialetheism.
Chief among these is that since dialetheism recognizes the liar paradox, an intrinsic contradiction, as being true, it must discard the long-recognized principle of explosion, which asserts that any proposition can be deduced from a contradiction, unless the dialetheist is willing to accept trivialism – the view that all propositions are true.
Andrew Irvine has argued in favour of a non-cognitivist solution to the paradox, suggesting that some apparently well-formed sentences will turn out to be neither true nor false and that "formal criteria alone will inevitably prove insufficient" for resolving the paradox.
[7] The Indian grammarian-philosopher Bhartrhari (late fifth century AD) dealt with paradoxes such as the liar in a section of one of the chapters of his magnum opus the Vākyapadīya.
[citation needed] Bhartrhari's solution fits into his general approach to language, thought and reality, which has been characterized by some as "relativistic", "non-committal" or "perspectivistic".
But we can keep apart the warring sides of the contradiction by the simple expedient of temporal contextualisation: what is 'true' with respect to one point in time need not be so in another ...
The overall force of the 'Austinian' argument is not merely that 'things change', but that rationality is essentially temporal in that we need time in order to reconcile and manage what would otherwise be mutually destructive states.
"[18] According to Robert's suggestion, it is the factor "time" which allows us to reconcile the separated "parts of the world" that play a crucial role in the solution of Barwise and Etchemendy.
[16]: 188 The capacity of time to prevent a direct confrontation of the two "parts of the world" is here external to the "liar".
In these terms, the Gödel sentence states that no natural number exists with a certain, strange property.
The liar paradox is occasionally used in fiction to shut down artificial intelligences, who are presented as being unable to process the sentence.
BOSS tries to figure it out but cannot and eventually decides the question is irrelevant and summons security.
However, lacking the intelligence to realize the statement is a paradox, he simply responds, "Um, true.
Humorously, all other AIs present barring GLaDOS, all of which are significantly less sentient and lucid than both her and Wheatley, are still killed from hearing the paradox.
In the seventh episode of Minecraft: Story Mode, titled "Access Denied", the main character Jesse and their friends are captured by a supercomputer named PAMA.