Liar paradox in early Islamic tradition
Encyclopedia
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.
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".
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."
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 Islam
ic 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 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:
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:
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:
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.
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.
Logicians
The Logicians or School of Names was a Chinese philosophical school that grew out of Mohism in the Warring States Period 479–221 BCE....
discussed the liar paradox
Liar paradox
In philosophy and logic, the liar paradox or liar's paradox , is the statement "this sentence is false"...
. Their work on the subject began in the 10th century and continued to Athīr al-Dīn al-Abharī and Nasir al-Din al-Tusi
Nasir al-Din al-Tusi
Khawaja Muḥammad ibn Muḥammad ibn Ḥasan Ṭūsī , better known as 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...
of the middle 13th century and beyond. Although the Liar paradox has been well known in Greek
Greeks
The Greeks, also known as the Hellenes , are a nation and ethnic group native to Greece, Cyprus and neighboring regions. They also form a significant diaspora, with Greek communities established around the world....
and Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
traditions, the works of Arabic scholars have only recently been translated into English
English language
English is a West Germanic language that arose in the Anglo-Saxon kingdoms of England and spread into what was to become south-east Scotland under the influence of the Anglian medieval kingdom of Northumbria...
.
Each group of early Islamic philosophers discussed different problems presented by the paradox. They pioneered unique solutions that were not influenced by Western
Western philosophy
Western philosophy is the philosophical thought and work of the Western or Occidental world, as distinct from Eastern or Oriental philosophies and the varieties of indigenous philosophies....
ideas.
Athīr and the Liar paradox
Athīr al-Dīn Mufaḍḍal (b. ʿUmar Abharī, d. 663/1264) was a PersianPersian people
The Persian people are part of the Iranian peoples who speak the modern Persian language and closely akin Iranian dialects and languages. The origin of the ethnic Iranian/Persian peoples are traced to the Ancient Iranian peoples, who were part of the ancient Indo-Iranians and themselves part of...
philosopher, astronomer
Astronomer
An astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...
and mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
from the city of Abhar
Abhar
Abhar is a city in and capital of Abhar County, Zanjan Province, Iran. At the 2006 census, its population was 70,836, in 19,136 families.Abhar is located between Qazvin and Zanjan, near the Abhar River. It is the second-largest city in the province after Zanjan...
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
Thomas Bradwardine
Thomas Bradwardine was an English scholar, scientist, courtier and, very briefly, Archbishop of Canterbury. As a celebrated scholastic philosopher and doctor of theology, he is often called Doctor Profundus, .-Life:He was born either at Hartfield in Sussex or at Chichester, where his family were...
.
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
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
.
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 penguinPenguinPenguins are a group of aquatic, flightless birds living almost exclusively in the southern hemisphere, especially in Antarctica. Highly adapted for life in the water, penguins have countershaded dark and white plumage, and their wings have become flippers...
s.
- Other examples of an empty subject term include: it is false to say that all flying carpetsMagic carpetA magic carpet, also called a flying carpet, is a legendary carpet that can be used to transport persons who are on it instantaneously or quickly to their destination.-In literature:...
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
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
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."
Nasir al-Din al-Tusi on the Liar paradox
Naṣīr al-Dīn al-Ṭūsī was a Persian
Persian people
The Persian people are part of the Iranian peoples who speak the modern Persian language and closely akin Iranian dialects and languages. The origin of the ethnic Iranian/Persian peoples are traced to the Ancient Iranian peoples, who were part of the ancient Indo-Iranians and themselves part of...
polymath
Polymath
A polymath is a person whose expertise spans a significant number of different subject areas. In less formal terms, a polymath may simply be someone who is very knowledgeable...
and prolific writer: an astronomer
Islamic astronomy
Islamic astronomy or Arabic astronomy comprises the astronomical developments made in the Islamic world, particularly during the Islamic Golden Age , and mostly written in the Arabic language. These developments mostly took place in the Middle East, Central Asia, Al-Andalus, and North Africa, and...
, biologist
Biologist
A biologist is a scientist devoted to and producing results in biology through the study of life. Typically biologists study organisms and their relationship to their environment. Biologists involved in basic research attempt to discover underlying mechanisms that govern how organisms work...
, chemist, mathematician
Islamic mathematics
In the history of mathematics, mathematics in medieval Islam, often termed Islamic mathematics or Arabic mathematics, covers the body of mathematics preserved and developed under the Islamic civilization between circa 622 and 1600...
, philosopher
Early Islamic philosophy
Early Islamic philosophy or classical Islamic philosophy is a period of intense philosophical development beginning in the 2nd century AH of the Islamic calendar and lasting until the 6th century AH...
, physician
Islamic medicine
In the history of medicine, Islamic medicine, Arabic medicine or Arabian medicine refers to medicine developed in the Islamic Golden Age, and written in Arabic, the lingua franca of Islamic civilization....
, physicist
Islamic physics
Physics in medieval Islam is the development of physics in the medieval Islamic world in the history of physics. In the course of the expansion of the Islamic world, Muslim scholars encountered the science, mathematics, and medicine of antiquity through the works of Aristotle, Archimedes, Galen,...
, scientist
Islamic science
Science in the medieval Islamic world, also known as Islamic science or Arabic science, is the science developed and practised in the Islamic world during the Islamic Golden Age . During this time, Indian, Iranian and especially Greek knowledge was translated into Arabic...
, theologian
Kalam
ʿIlm al-Kalām is the Islamic philosophical discipline of seeking theological principles through dialectic. Kalām in Islamic practice relates to the discipline of seeking theological knowledge through debate and argument. A scholar of kalām is referred to as a mutakallim...
and Marja Taqleed. He adhered to the Ismaili
Ismaili
' is a branch of Shia Islam. It is the second largest branch of Shia Islam, after the Twelvers...
, and subsequently Twelver Shī‘ah Islam
Islam
Islam . The most common are and . : Arabic pronunciation varies regionally. The first vowel ranges from ~~. The second vowel ranges from ~~~...
ic belief systems. The Arab scholar Ibn Khaldun
Ibn Khaldun
Ibn Khaldūn or Ibn Khaldoun was an Arab Tunisian historiographer and historian who is often viewed as one of the forerunners of modern historiography, sociology and economics...
(1332–1406) considered Tusi to be the greatest of the later Persian
Persian people
The Persian people are part of the Iranian peoples who speak the modern Persian language and closely akin Iranian dialects and languages. The origin of the ethnic Iranian/Persian peoples are traced to the Ancient Iranian peoples, who were part of the ancient Indo-Iranians and themselves part of...
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 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:
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:
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
Self-reference
Self-reference occurs in natural or formal languages when a sentence or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding...
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:
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.
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.