Sunday, February 7, 2010

logic for waiters

You are in a restaurant with your parents, and you have ordered three dishes: Fish, Meat, and Vegetarian. Now a new waiter comes back from the kitchen with three dishes. What will happen?

. . . Indeed, two questions plus one inference are all that is needed
J. F. A. K. van Benthem, "Logic and Reasoning: Do the Facts Matter," 2008.

Of course, the answer is indeed "two questions plus one inference," but the example is misleading. Stating the problem with only three diners underspecifies the type of reasoning employed. The waiter could use a simple elimination of alternatives, for example. But two questions plus one inference is also the solution to the n-person case.

The reason, of course, is that waiters can rely on regularities in the structure of the situation. In particular, the vast majority of eating arrangements are topologically equivalent to a circle. If the waiter who takes the orders writes them in sequence, the waiter delivering the food can arrange the plates in order on his tray. Then, one question to orient himself on the circle, and a second to establish the direction in which the order was taken, is enough for the waiter to complete distribution of dishes. I have seen this practice employed frequently for tables of four to five people, but in principle it could be applied to any size table (so long as the waiter can fit the dishes (or drinks!) in sequence on his tray).

Strictly speaking, a further convention could establish an even more efficient distribution system. If the two waiters agree previously on a directionality (say, clockwise), a single question will be sufficient to establish the order. Most likely this further convention is not employed due to the restriction it places on the order-taking waiter. He must be free to submit to customers' whim, and removing a degree of freedom from his order-taking practice would greatly reduce that ability.

In fact, this method for delivering dishes is noticed more often in the breach, when the delivering waiter attempts to deliver a dish without asking and discovers it is wrong. Frequently, the cause is a violation of convention in the order on the ticket caused initially by the unwillingness of the customers themselves to follow the order of order-taking preferred by the initial waiter.

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