Multiple analogical proportions

Abstract

Analogical proportions are statements of the form “a is to b as c is to d”, denoted a : b : : c : d, that may apply to any type of items a, b, c, d. Analogical proportions, as a building block for analogical reasoning, is then a tool of interest in artificial intelligence. Viewed as a relation between pairs ( a , b ) and ( c , d ), these proportions are supposed to obey three postulates: reflexivity, symmetry, and central permutation (i.e., b and c can be exchanged). The logical modeling of analogical proportions expresses that a and b differ in the same way as c and d, when the four items are represented by vectors encoding Boolean properties. When items are real numbers, numerical proportions – arithmetic and geometric proportions – can be considered as prototypical examples of analogical proportions. Taking inspiration of an old practice where numerical proportions were handled in a vectorial way and where sequences of numerical proportions of the form x 1 : x 2 : ⋯ : x n : : y 1 : y 2 : ⋯ : y n were in use, we emphasize a vectorial treatment of Boolean analogical proportions and we propose a Boolean logic counterpart to such sequences. This provides a linear algebra calculus of analogical inference and acknowledges the fact that analogical proportions should not be considered in isolation. Moreover, this also leads us to reconsider the postulates underlying analogical proportions (since central permutation makes no sense when n ⩾ 3) and then to formalize a weak form of analogical proportion which no longer obeys the central permutation postulate inherited from numerical proportions. But these weak proportions may still be combined in multiple weak analogical proportions.