All the King's philosophers: part II

(See part I.)

Al-Khwārizmī spent the night at the royal palace. Very early the next morning he looked for a cool and quiet place on a veranda by the palace garden, asked for a few sheets of the famous Samarkand paper and some ink and put himself to work on the quest for the Philosopher Rule. The following is a mildly adapted version of the annotations made by al-Khwārizmī during that day.

"To simplify things, let us assume that questions posed to the Council are of the form that can be asked simply by saying yes or no, as in 'Must we go to war against the Bactrians?' or 'Should taxes be raised?'. To every such question, the Council will be divided into those philosophers answering affirmatively and those whose reply is negative. The Philosopher Rule must then decide whether the final answer is yes or no based on the distribution of individual replies across the Council. The Rule will be complete if it can take care of every possible answer distribution."

"Now, as philosophers pride themselves with having an answer to everything, we know that each of the Council members will either reply yes or no to every conceivable query they might be confronted with, that is, they never remain silent at a question. So, to a given question the group of negative philosophers is exactly the complementary of the group of affirmative philosophers, and we can in a sense concentrate only on the affirmative party. The Philosopher Rule can be then regarded as playing the following role: for each group of affirmative philosophers the Rule must decree whether the group is authoritative, that is, whether the final judgment on a question must be taken to be affirmative if it is this precise group of philosophers that reply in the positive."

"We cannot choose at random which groups are authoritative, because such disposition would likely yield inconsistent results when the Council was consulted over time. Clearly, some constraints must apply to the way in which we choose the list of authoritative groups. I think the art of Logic will help us here, fortunately I brought with me some books on the subject."

That was how far the mathematician arrived the first day. The second day, al-Khwārizmī ordered a fresh load of paper and sat on the veranda from dusk till dawn furiously scribbling symbols and consulting a heap of books by the school of the Stoics. When the last light of day vanished beyond the palace walls and the first crickets began chirping in the garden, the mathematician arose from his seat with a triumphal smile, as he thought he had the foundations of the problem basically laid down, and only some routinely calculations were necessary to reach a solution. This is a summary of his conclusions that day:

"We explore the constraints imposed on eligible authoritative groups by studying their connection with the basic laws of Logic by which every wise man must abide. For it is our goal that the decisions taken by the Rule be free of contradiction much as those of any philosopher are bound to be."

• "If the entire Council supports a decision (though according to the King this event has not ever happened), it is only sensible for the Rule to approve the decision as well. So, the group formed by all philosophers in the Council is an authoritative group."
• "If a group A is deemed authoritative, then the complementary group, i.e. that comprising the philosophers outside A, cannot be authoritative. Otherwise, if we asked a question which is supported by A and then submitted exactly the opposite question, which would be supported by the complementary of A, the Rule would end up holding both a position and its negation."
• If a certain group A is considered authoritative, so will be the case with any other group including A: having more philosophers supporting the decision can only make us more confident on the result.
  
• "Let us suppose we pose a question, like 'Should we raise taxes?' to which an authoritative group A responds affirmatively, and then some other question, like 'Should we offer a sacrifice to the gods?', which is supported by another authoritative group B. If we had submitted the combined question 'Should we raise taxes and offer a sacrifice?' the laws of Logic teach us that the philosophers answering positively would be exactly those belonging to both A and B. Hence the Rule must have the intersection of authoritative groups as authoritative."
  

"The only work left to do is finding a list of authoritative groups that satisfy the restrictions. This I will do tomorrow through some fairly easy if tedious calculations."

Al-Khwārizmī had a light dinner and went to sleep with the peace of mind enjoyed by those who feel success within reach of their hand.