From Default to Analogical and Paralogical Reasoning. Logics of Pairs and Their Multiple-Valued Extensions

The paper pursues the study of consequence relations associated with logical proportions. The latter are special expressions that combine conjunctively two equivalences between comparison indicators pertaining to two pairs (a, b) and (c, d) of Boolean variables. Analogical proportions, namely statements of the form a is to b as c is to d, are a special case of logical proportions in the Boolean case. There are three kinds of consequence relations in this setting. One is known for a long time and linked to default reasoning, another has been introduced last year and deals with the gain or lost of properties. The third one, associated with a logical proportion named paralogy (which states that what a and b have in common, c and d have it also and vice-versa) is studied in this paper together with its interplay with analogical proportions. Moreover, the paper presents multiple-valued extensions of the consequence relations.