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 W. Hart.

The puzzle

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.

Possible solutions

Potential solutions on Hart's webpage include:
#Salted cucumbers, sugar, yogurt.
#Orange juice, gin, tonic.
#Lemon, cocoa, curry.

False solutions

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. While tea and lemon "do" go together, and tea and milk "do" go together, milk and lemon do "not" go together. For this solution to work, milk and lemon would have to go together as well.

According to Hart, most attempted solutions tend to overlook one of the three pairs. Issues of personal taste and preparation complicate the issue, as combinations some consider acceptable sound unpalatable to others, and problems such as milk curdling with the addition of lemon juice can potentially be overcome if a cheesemaking process is employed.

Formula

Find a counter example to either of the following (equivalent) alleged theorems (where R(x,y,...) means "x, y, ... all go together") :
#Given any three foods A, B, and C, if [R(A,B), R(A,C) and R(B,C)] then R(A,B,C)
#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)] .

External links

* [http://www.georgehart.com/triad.html The Incompatible Food Triad by George W. Hart]