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Many early Islamic philosophers and logicians discussed the liar paradox. Their work on the subject began in the 10th century and continued to Athīr al-Dīn al-Abharī and Nasir al-Din al-Tusi of the middle 13th century and beyond. Although the Liar paradox has been well known in Greek and Latin traditions, the works of Arabic scholars have only recently been translated into English.〔 Each group of early Islamic philosophers discussed different problems presented by the paradox. They pioneered unique solutions that were not influenced by Western ideas. == Athīr and the Liar paradox == Athīr al-Dīn Mufaḍḍal (b. ʿUmar Abharī, d. 663/1264) was a Persian philosopher, astronomer and mathematician from the city of Abhar in Persia. There is some speculation that his works on the Liar paradox could have been known to Western logicians, and in particular to Thomas Bradwardine. He analyzed the Liar sentence as follows: In other words, Athīr says that if the Liar sentence is false, which means that the Liar falsely declares that all he says at the moment is false, then the Liar sentence is true; and, if the Liar sentence is true, which means that the Liar truthfully declares that all he says at the moment is false, then the Liar sentence is false. In any case, the Liar sentence is both true and false at the same time, which is a paradox. Athīr offers the following solution for the paradox: According to the traditional idealization that presumably was used by Athīr, the sentence as an universal proposition is false only, when "either it has a counter-instance or its subject term is empty". *Other examples of a counter-instance include: it is false to say that all birds could fly because there are some that could not, like for example penguins.〔 *Other examples of an empty subject term include: it is false to say that all flying carpets have four corners, and not only because some carpets are round or have three corners, but rather because there are no flying carpets at all.〔 The Liar sentence, however, has neither an empty subject nor counter-instance. This fact creates obstacles for Athīr's view, who must show what is unique about the Liar sentence, and how the Liar sentence still could be only true or false in view of the "true" and "false" conditions set up in the universal proposition's description. Athīr tries to solve the paradox by applying to it the laws of negation of a conjunction and negation of a disjunction.〔 Ahmed Alwishah, who has a Ph.D. in Islamic Philosophy and David Sanson, who has a Ph.D. in Philosophy explain that Athīr actually claims that: (1) "It is not the case that, if the Liar Sentence is not both true and false, then it is true." Alwishah and Sanson continue: "The general principle behind (1) is clear enough: the negation of a conjunction does not entail the negation of a conjunct; so from not both true and false you cannot infer not false and so true. Abharī appears to be saying that the Liar rests on an elementary scope fallacy! But, of course, Abharī is not entitled to (1). In some cases, the negation of a conjunction does entail the negation of a conjunct: 'not both P and P' for example, entails 'not P'. As a general rule, the negation of a conjunction entails the negation of each conjunct whenever the conjuncts are logically equivalent, i.e., whenever the one follows from the other and vice verse. So Abharī is entitled to (1) only if he is entitled to assume that ‘The Liar Sentence is true’ and ‘The Liar Sentence is false’ are not logically equivalent."〔 The Liar sentence is a universal proposition (The Liar says All I say ...), so "if it is (non–vacuously) false it must have a counter–instance".〔 But in this case scenario, when the only thing that the liar is saying is the single sentence declaring that what he is saying at the moment is false, the only available counter–instance is the Liar sentence itself. When staging the paradox Abharī said: "if it is not true, then it is necessary that one of his sentences at this moment is true, as long as he utters something. But, he says nothing at this moment other than this sentence. Thus, this sentence is necessarily true and false"〔 So the explanation provided by Abharī himself demonstrates that both "'The Liar Sentence is false' and 'The Liar Sentence is true' are logically equivalent. If they are logically equivalent, then, contrary to (1), the negation of the conjunction does entail the negation of each conjunct. Abharī’s 'solution; therefore fails." 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Liar paradox in early Islamic tradition」の詳細全文を読む スポンサード リンク
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