Incompatible Food Triad

The Incompatible Food Triad is a puzzle that allegedly originated with the philosopher Wilfrid Sellars, and has been spread by some of his former colleagues and students, including Nuel Belnap and George Hart, who keeps a page on it at: http://www.georgehart.com/triad.html. To date no solution has been forthcoming.

The problem is this: Find three foods, such that any two of them go together, but all three do not.

We understand "go together" in any reasonable sense of the expression, as it is ordinarily applied to foods.

One way of seeing the problem is this: given three foods that don't go together, it's usually because two of them don't go together. For example, Richard Feynman's famous example of accidentally requesting milk and lemon in his tea is not a solution: (1) Milk, tea, and lemon do not go together. (2)(a) Tea and lemon do go together, (b) Tea and Milk do go together, but (c) Milk and lemon do not go together. For the solution to work milk and lemon would have to go together as well. Most attempted solutions (so far, according to Hart) tend to overlook one of the three pairs.

The problem can also be formulated thus: Find a counter example to either of the following alleged theorems (where R(x,y,...) means "x, y, ... all go together") :

(1) Given any three foods A, B, and C, if [R(A,B), R(A,C) and R(B,C)] then R(A,B,C)

(2) Given any three foods A, B, and C, if ~R(A,B,C) then [~R(A,B) or ~R(A,C) or ~R(B,C)].

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