A relationship between applications of a concept in a judgment such that the applications are in one respect like, and in another respect unlike, each other; thus, it is a relationship that is a kind of mean between univocity (total likeness) and equivocity (total unlikeness). More succinctly, it could be called a relationship of imperfect or inexact likeness. Analogy is not only a conceptual or logical matter but has a basis in reality. 

Analogy of Attribution (pros hen analogy).— The analogy between different applications of the notion of an attribute that belongs intrinsically to one thing and extrinsically to others. The notion is only applied to the others because of the relationship they bear to the attribute as it belongs to the thing that possesses it intrinsically and, therefore, the likeness of application is inexact. The application of the notion to the latter is called the “primary analogate” and its applications to the former are called the “secondary analogates.” One might regard, e.g., a book as containing knowledge either because it causes knowledge in those who read it, or because it expresses the knowledge of the author. But knowledge is in persons intrinsically because they really do possess knowledge and in books extrinsically as signs or causes of knowledge.  According to Aristotle, such analogy is, in fact, a form of equivocation. 

Analogy of Proper Proportionality.— The analogy between different applications of a notion that belongs intrinsically to all the things covered by the analogy. The attribute is inexactly alike across the different cases but, again, is intrinsic to each thing. Thus, the notion applies to each thing considered in itself and not only because of the relationship that thing has toward the given attribute in something else which is distinct from that thing. In analogy of proper proportionality, therefore, the notions of primary and secondary analogates have a different meaning than in the analogy of attribution.  Consider that, e.g., goodness belongs intrinsically to a horse that has some perfection as a horse; a human act that has some perfection as a morally perfect act; a virtue as a quality of soul insofar as it is perfective as a quality; a piece of art has goodness insofar as it is fittingly made; and God is the Uncreated Good. The notion of “goodness” as applied to these various “analogates” has an inexact likeness, for although they all possess some perfection in themselves, the perfection of a horse is not the same as that of human moral action, virtuous qualities, artistic fabrications, or God.  Insofar as proper proportionality utilizes order to explicate the relations among particular analogates, the analogy of proper proportionality thereby uses, in a unique and new way, the analogical ordering used in analogy of attribution.  The shared notion in analogies of proper proportionality is said to “imperfectly abstract” from the inferior “analogates” to which it applies.  

Analogy of Improper Proportionality (also called Metaphor).— The analogy between different applications of the notion of an attribute that belongs to one of the things covered by the analogy intrinsically and in reality but does not belong to the other in reality. Despite this, there is, nevertheless, a reason for applying the attribute to the latter. E.g., one might say that a certain person is a bulldog but not because he really is a bulldog but because he has a tenacity that is akin to that of bulldogs. There is, thus, an inexact likeness in how the notion “bulldog” is applied to a human being and to an actual bulldog.