Hypostatic abstraction

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Hypostatic abstraction in mathematical logic, also known as hypostasis or subjectal abstraction, is a formal operation that transforms a predicate into a relation; for example "Honey is sweet" is transformed into "Honey has sweetness". The relation is created between the original subject and a new term that represents the property expressed by the original predicate.

Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.

In mathematical logic, a formal calculation is a calculation which is systematic, but without a rigorous justification. This means that we are manipulating the symbols in an expression using a generic substitution, without proving that the necessary conditions hold. Essentially, we are interested in the form of an expression, and not necessarily its underlying meaning. This reasoning can either serve as positive evidence that some statement is true, when it is difficult or unnecessary to provide a proof, or as an inspiration for the creation of new definitions.

In mathematical logic, a predicate is commonly understood to be a Boolean-valued function P: X→ {true, false}, called the predicate on X. However, predicates have many different uses and interpretations in mathematics and logic, and their precise definition, meaning and use will vary from theory to theory. So, for example, when a theory defines the concept of a relation, then a predicate is simply the characteristic function of a relation. However, not all theories have relations, or are founded on set theory, and so one must be careful with the proper definition and semantic interpretation of a predicate.

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Hypostasis changes 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. The logical functioning of the second object Y-ness consists solely in the truth-values of those propositions that have the corresponding abstract property Y as the predicate. The object of thought introduced in this way may be called a hypostatic object and in some senses an abstract object and a formal object.

In propositional logic, a propositional formula is a type of syntactic formula which is well formed and has a truth value. If the values of all variables in a propositional formula are given, it determines a unique truth value. A propositional formula may also be called a propositional expression, a sentence, or a sentential formula.

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). As Peirce describes it, the main point about the formal operation of hypostatic abstraction, insofar as it operates on formal linguistic expressions, is that it converts a predicative adjective or predicate into an extra subject, thus increasing by one the number of "subject" slots—called the arity or adicity—of the main predicate.

Charles Sanders Peirce American philosopher, logician, mathematician, and scientist who founded pragmatism

Charles Sanders Peirce was an American philosopher, logician, mathematician, and scientist who is sometimes known as "the father of pragmatism". He was educated as a chemist and employed as a scientist for thirty years. Today he is appreciated largely for his contributions to logic, mathematics, philosophy, scientific methodology, semiotics, and for his founding of pragmatism.

In logic, mathematics, and computer science, the arity of a function or operation is the number of arguments or operands that the function takes. The arity of a relation is the dimension of the domain in the corresponding Cartesian product. The term springs from words like unary, binary, ternary, etc. Unary functions or predicates may be also called "monadic"; similarly, binary functions may be called "dyadic".

The transformation of "honey is sweet" into "honey possesses sweetness" can be viewed in several ways:

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

The abstraction of hypostasis takes the concrete physical sense of "taste" found in "honey is sweet" and gives it formal metaphysical characteristics in "honey has sweetness".

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