Hypostatic abstraction
Encyclopedia
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".
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, also known as hypostasis or subjectal abstraction, is a formal operation
Formal calculation
In mathematical logic, a formal calculation is sometimes defined as 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...
that transforms an assertion
Logical assertion
A logical assertion is a statement that asserts that a certain premise is true, and is useful for statements in proof. It is equivalent to a sequent with an empty antecedent....
to a relation; for example "Honey is sweet" is transformed into "Honey has sweetness". Hypostasis changes a a propositional formula
Propositional formula
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...
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
Arity
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...
, 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".
See also
- AbstractionAbstractionAbstraction is a process by which higher concepts are derived from the usage and classification of literal concepts, first principles, or other methods....
- Abstraction in computingAbstraction (computer science)In computer science, abstraction is the process by which data and programs are defined with a representation similar to its pictorial meaning as rooted in the more complex realm of human life and language with their higher need of summarization and categorization , while hiding away the...
- Abstraction in mathematicsAbstraction (mathematics)Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalising it so that it has wider applications or matching among other abstract...
- AnalogyAnalogyAnalogy is a cognitive process of transferring information or meaning from a particular subject to another particular subject , and a linguistic expression corresponding to such a process...
- Category theoryCategory theoryCategory theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
- Continuous predicateContinuous predicateContinuous predicate is a term coined by Charles Sanders Peirce to describe a special type of relational predicate that results as the limit of a recursive process of hypostatic abstraction....
- ReificationReificationReification generally refers to bringing into being or turning concrete.Specifically, reification may refer to:*Reification , making a data model for a previously abstract concept...
- E-primeE-PrimeE-Prime is a version of the English language that excludes all forms of the verb to be. E-Prime does not allow conjugations of to be , archaic forms E-Prime (short for English-Prime, sometimes denoted E′) is a version of the English language that excludes all forms of the verb to be. E-Prime does...