# Hypostatic abstraction

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Hypostatic abstraction in mathematical logic, also known as hypostasis or subjectal abstraction, is a formal operation that transforms an assertion to a relation; for example "Honey is sweet" is transformed into "Honey has sweetness". Hypostasis changes a a propositional formula of the form X is Y to another one of the form X has the property of being Y or X has Y-ness. This abstraction process takes an element of information about the properties of a subject, and conceives it to consist in the relation between the original subject and the property seen as a second subject.

The existence of the latter subject, here Y-ness, consists solely in the truth of those propositions that have the corresponding concrete term, here Y, as the predicate. The object of discussion or thought thus introduced may also be called a hypostatic object.

The above definition is adapted from the one given by Charles Sanders Peirce (CP 4.235, "The Simplest Mathematics" (1902), in Collected Papers, CP 4.227–323).

The way that Peirce describes it, the main thing about the formal operation of hypostatic abstraction, insofar as it can be observed to operate on formal linguistic expressions, is that it converts an adjective or some part of a predicate into an extra subject, upping the arity, also called the adicity, of the main predicate in the process.

For example, a typical case of hypostatic abstraction occurs in the transformation from "honey is sweet" to "honey possesses sweetness", which transformation can be viewed in the following variety of ways:

The grammatical trace of this hypostatic transformation tells of a process that abstracts the adjective "sweet" from the main predicate "is sweet", thus arriving at a new, increased-arity predicate "possesses", and as a by-product of the reaction, as it were, precipitating out the substantive "sweetness" as a new second subject of the new predicate, "possesses".