Logical Reasoning and the Torn Shirt

The story of Prophet Yūsuf in the Qur’ān includes an episode that presents an interesting exercise in logical reasoning. In the incident where the ‘Azīz of Egypt finds his wife and Prophet Yūsuf making conflicting claims, a witness speaks up and presents a series of statements that make it possible to reach a conclusion about what happened using a combination of observation (of Prophet Yusuf’s shirt) and logic.

Verses 25 to 28 of Surah #12 Yūsuf mention the following:

… She said, ‘What is to be the requital of him who has evil intentions for your wife except imprisonment or a painful punishment?’ He said, ‘It was she who solicited me.’ A witness of her own household testified [thus]: ‘If his shirt is torn from the front, she tells the truth and he lies. But if his shirt is torn from behind, then she lies and he tells the truth.’ So when he saw that his shirt was torn from behind…

Minor paraphrasing of these verses gives us the following argument that summarizes the information available to the ‘Azīz:

1. She told the truth if and only if he intended to violate her.
2. He told the truth if and only if she tried to seduce him.
3. If his shirt is torn from the front, then she told the truth and he lied.
4. If his shirt is torn from the back, then she lied and he told the truth.
5. The shirt was torn from the back (and not the front).

Let’s introduce the following notation so we can represent the argument symbolically.

• Z = She (the wife of the ‘Azīz) told the truth
• Y = He (Prophet Yūsuf) told the truth
• V = He intended to violate her
• S = She tried to seduce him
• F = The shirt was torn from the front
• B = The shirt was torn from the back

The argument then becomes

1. Z ↔ V.
2. Y ↔ S.
3. F → ( Z ∧ ~Y ).
4. B → ( ~Z ∧ Y ).
5. B ∧ ~F.

It doesn’t require too much effort (either by symbolic manipulation and deductions, or via a large truth table) to deduce the only logically consistent truth assignment for the six variables:

• B, Y, and S are all True.
• F, Z, and V are all False.

Therefore, the ‘Azīz can conclude that Prophet Yūsuf told the truth and that his wife had tried to seduce him, and that she lied and Prophet Yūsuf never intended to violate her. Indeed, he reaches that very conclusion, as the verse continues:

… He said, ‘This is [a case] of you women’s guile! Your guile is indeed great! Yūsuf, let this matter alone, and you, woman, plead for forgiveness for your sin, for you have indeed been erring.’