Abstract

A paradox in its original Greek means a statement which goes beyond the accepted opinion; a logical paradox is a statement which involves its contradictory, i.e. if asserted as true, must necessarily be false and vice versa. A classical example of such paradoxes is the Eulibides Paradox, Known as the Liar's Paradox. Eulibides, a Cretan, says that "All Cretans are liars". Is his statement true? If true, it must necessarily be false and if false, it must necessarily be true.The Liar's Paradox was Known as the Riddle of the Incommensurable Root among Muslim logicians and over a hundred treatises were written to solve the logical contradiction involved in the riddle, of which several are extant.Abhari considers the fallacy in the riddle the result of the coincidence of two contradictories. One can assume a person who has said a false statement as saying that "every statement of mine is false". His statement can be either false or true. If true, then every single one of his statements including this very statement must necessarily be false. But if not, the contrary of his statement must be true.We can sum up Abhari's solution to the riddle as follows: one cannot assume that if a statement such as the above statement is false, it logically follows that some of the individuals of that statement are necessarily true. What is certain is the falsehood of this and his other statements. Falsehood can be realized in a statement only when the truth of the statement ensues its falsehood.Khwajeh Nasiruddin Tusi, another famous philosopher-logician of Islam, first quotes and then analyzes the paradox. His solution rests on the distinction between a statement and what is stated. Analyzing the logical structure of the Liar's Paradox, Tusi says it is in the very structure of a logical statement to be a logical statement, to state something about anything. A statement, moreover, can be a statement about another statement so the second statement can be considered in two different aspects. In one respect it is a statement and in another, it is the thing stated and one should not logically confuse these two logical aspects. If the first statement, P, states that a second statement, Q, is false then the truth of P and the falsehood of Q are concomitant.The riddle has drawn the attention of many other logicians including Katebi Qazwini, Allameh Helli and Samarghandi each of whom has looked at it from a special angle.