Liar paradox in early Islamic tradition

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 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:

One of the difficult fallacies is the conjunction of the two contradictories (Jamʿal-naqīḍyan) when someone says, “All that I say at this moment is false”. This sentence (qawl) is either true or false. If it is true, then it must be true and false. And 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.

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.

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