Analogy is traditionally regarded as one of the three main factors responsible for language change, along with sound change and borrowing. Whereas sound change is understood to be phonetically motivated and blind to structural patterns and semantic and functional relationships, analogy is licensed precisely by those patterns and relationships. In the Neogrammarian tradition, analogical change is regarded, at least largely, as a by-product of the normal operation (acquisition, representation, and use) of the mental grammar. Historical linguists commonly use proportional equations of the form A : B = C : X to represent analogical innovations, where A, B, and C are (sets of) word forms known to the innovator, who solves for X by discerning a formal relationship between A and B and then deductively arriving at a form that is related to C in the same way that B is related to A. Along with the core type of analogical change captured by proportional equations, most historical linguists include a number of other phenomena under the analogy umbrella. Some of these, such as paradigm leveling—the reduction or elimination of stem alternations in paradigms—are arguably largely proportional, but others such as contamination and folk etymology seem to have less to do with the normal operation of the mental grammar and instead involve some kind of interference among the mental representations of phonetically or semantically similar forms. The Neogrammarian approach to analogical change has been criticized and challenged on a variety of grounds, and a number of important scholars use the term “analogy” in a rather different sense, to refer to the role that phonological and/or semantic similarity play in the influence that forms exert on each other.