Logical definition of analogical proportion and its fuzzy extensions

An analogical proportion is a statement of the form "A is to B as C is to D". In a logical setting, items A,B,C and D are Boolean vectors. This notion is at the core of analogical reasoning. This paper proposes a sound definition of analogical proportion, based on a logical expression that holds true for each vector component if and only if the analogical proportion holds true. The analogical equation, where D is unknown, is also discussed. The logical expression of the analogical proportion has several equivalent forms, which may lead to distinct extensions when the vector components takes its values in the unit interval, depending on the choice of the multiple-valued connectives. Applications to case-based and approximate reasoning, and to learning are outlined.