A Discussion of Analogical-Proportion Based Inference

The Boolean expression of an analogical proportion, i.e., a statement of the form “a is to b as c is to d”, expresses that “a differs from b as c differs from d, and vice-versa. This is the basis of an analogical inference principle, which is shown to be a particular instance of the analogical “jump”: from P (s), P (t), and Q(s), deduceQ(t). Roughly speaking, an analogical proportion sounds like a sort of qualitative derivative. A counterpart of a first order Taylor-like formula indeed exists for affine Boolean functions. Affine functions can be predicted without error by means of analogical proportions. These affine functions are essentially the constants, the projections, the xor-based functions, and their complements. We discuss how one might take advantage of this state of fact for refining the scope of application of the analogical-proportion based inference to subparts of a Boolean function that may be assumed to be “locally” linear.