Nasir al-Din al-Tusi on the Liar paradox

Statement

Naṣīr al-Dīn al-Ṭūsī was a Persian polymath and prolific writer: an astronomer, biologist, chemist, mathematician, philosopher, physician, physicist, scientist, theologian and Marja Taqleed. He adhered to the Ismaili, and subsequently Twelver Shī‘ah Islamic belief systems. The Arab scholar Ibn Khaldun (1332–1406) considered Tusi to be the greatest of the later Persian scholars.

Ṭūsī's work on the paradox begins with a discussion of the paradox and the solution offered by Abharī, with which Ṭūsī disagrees. As Alwishah and Sanson (2009) point out, "Ṭūsī argues that whatever fancy thing (conjunction, conditional) Abharī wants to identify as the truth condition for the Liar Sentence, it will not matter, because pace Abharī, we can generate the paradox without inferring, from the negation of a complex truth condition, the negation of one of its parts. We can argue directly that its being false entails the negation of its being false, and so entails its being true."

Ṭūsī then prepares a stage for his own solution of the Liar paradox, writing that: If a declarative sentence, by its nature, can declare-something-about anything, then it is possible that it itself can declare-something-about another declarative sentence.

He does not see a reason that could prevent a declarative sentence to declare something about another declarative sentence.

With an example of two declarative sentences, (D1) "It is false" and (D2) "Zayd is sitting", Ṭūsī explains how one declarative sentence (D1) can declare another declarative sentence (D2) to be false: "It is false that Zayd is sitting". There is no paradox in the above two declarative sentences because they have different subjects. To generate a paradox a declarative sentence must declare something about itself. If (D1) falsely declares itself to be not (D1) then this false declaration referencing to itself as being "false" creates a paradox.

Ṭūsī writes: Moreover, if the first declarative sentence declares itself to be false, then [both] its being true, insofar as it is a declarative sentence, and its being false, insofar as it is that-about-which-something-is-declared, are concomitant. Thus, the following paradox can be generated: The first declarative sentence, which is a declaration (khabar) about itself, namely that it is false, is either false or true. If it is true, then it must be false, because it declares itself to be false. If it is false, then it must be true, because if it is said falsely, then it will become true, which is absurd.

The above conclusions are very important to the history of Liar Paradox. Alwishah and Sanson point out: "It is hard to overemphasize how remarkable this passage is. The contemporary reader will be familiar with the idea that the Liar Paradox is a paradox of selfreference. But Ṭūsī is, as far as we know, the first person to express this idea. This passage has no precedent in any tradition. Ṭūsī has performed three remarkable feats in short order. First, his Liar Sentence is singular: its subject is itself, and it declares itself to be false. Gone, then, is the choice between universal or particular Liar Sentence, and the associated problem of adding further assumptions to generate a genuine paradox. Second, he has characterized the paradox as one of self-reference. Third, he has identified a key assumption that might be responsible for generating the entire problem: the assumption that a declarative sentence, by its nature, can declare-something-about anything."

Recognizing that, if a declarative sentence that declares itself being false, is false, this does not necessitate it being true. Ṭūsī says that it would be absurd to say that this declarative sentence is true only because it is not false. Ṭūsī writes:. . . its being false, insofar as it is a declarative sentence, does not necessitate its being true. Instead, its being false necessitates the denial of its being false, insofar as it is that-about-which-something-is-declared, and [necessitates] its being false, insofar as it is a declarative sentence. Hence, we should not concede that, in this way, the denial of its being false necessitates its being true.

Ṭūsī then interprets the definitions of "true" and "false", in an attempt to prove that those definitions should not be taken into consideration when dealing with a declarative sentence that declares itself, as its own subject, to be false.

Al-Baghdādī's definition of "truth" and "falsity" says that: "truth is an agreement with the subject, and falsity is the opposite of that". Ṭūsī argues that this definition cannot be applied to a declarative sentence that declares its own subject to be false because then there are at least two opposite parts that are in disagreement with each other. The same subject cannot be in disagreement with itself. Therefore, a self–referenced declarative sentence that declares itself to be false is neither false nor true, and truth/falsity definitions are not applicable to those sentences.

the result of a judgment that applies truth and falsity to something to which they in no way apply, and to apply them in any way is the misuse of a predicate.

Ṭūsī stopped short from offering a solution for the Liar sentences discussed by Āmidī "All that I say at this moment is false". This sentence presents a different case scenario because it can be interpreted as declaring something about itself, and something about another sentence. The solution for this paradox is absent from Ṭūsī's papers.