Functor (logic)
In logic , a functor is usually understood to be an operator that - depending on arity - is applied to one or more singular terms ( variables , nominators or functorial terms) and in turn generates a singular term. For example, "the father of ..." is usually interpreted as a one-digit functor, the arithmetic symbol "+" as a two-digit functor. "The water level from ... on ... in ..." would be an example of a three-digit functor (substitutions for the "digits" marked by an ellipsis could be: "Rhein", "08/20/07", "Bingen").
That an interpretation as a functor (or, alternatively, as a label ) is obvious here is evident from the fact that a structure like "Hans's father" relates to exactly one object and, like a nominator, can occupy a position of a predicator ; for example "Hans's father is identical to Franz", "2 + 2 = 4". Often functors are symbolized by lower case letters in order to be able to distinguish them more easily from predicators: "the-father-of (x)" or "v (x)" in contrast to "is-father-of (x, y)" or " V (x, y) "or" xVy ". Due to the recursive structure of standard languages, "the-father-of (the-father-of (a))" is also a valid expression.
More precisely: Applying an n-place functor φ (...) to n terms θ 1 to θ n results in a functorial term. If θ 1 to θ n are all closed (variable-free), then φ (θ 1 , ..., θ n ) closed. For every term θ i in φ (θ 1 ,…, θ n ) the following applies: If θ i is open in a variable ξ, then φ (θ 1 ,…, θ n ) in ξ is also open.
The above father example makes it clear that n-place functors can be defined by using n + 1-place predicators, according to the scheme:
Also marking Terme suitable for definitional introduction of functors:
When introducing a functor, it must be guaranteed that the resulting functional term refers to exactly one object.
The word "functor" was coined by the German philosopher Rudolf Carnap (1891-1970) in his book Logical Syntax of Language (1934) and was sometimes used in a wider sense than the one described above, which also included predicators. In Introduction to Symbolic Logic and its Applications (1958) he defined an n-digit functor as "any sign whose full expressions (involving n arguments) are not sentences". This means that predicators are not functors, which is what is common today.
Classifications
The functors are classified according to different points of view:
1st level functor - 2nd level functor
Level 1 functors are functors which, when supplemented, form a complete expression or sentence.
Example: {the capital of} is supplemented with "Germany" to express "the capital of Germany".
{is running} is supplemented with "Hans" to the sentence "Hans is running."
Second level functors are primarily the quantifiers that form complete expressions or sentences by inserting first level functors.
Classification according to the syntactic category of the arguments
According to the syntactic category of their arguments, functors are divided into:
(1) "Name-determining functors" (e.g. "runs"; "is less than")
(2) "statement-determining functors" (e.g. "not", "or" ...)
(3) “functor-determining functors” (e.g. “very” in “the child is very beautiful” (the argument here is “beautiful”)).
Classification according to the syntactic category of the resulting expression
According to the syntactic category of the molecular expression, which consists of the functor and its arguments, functors are divided into:
(1) Name-generating functors (ex .: "a bad" in "a bad example");
(2) statement-generating functors (eg: "he runs or stands" is again a statement);
(3) Functors generating functors (e.g. “shrill” in “the bell rings shrill”, here “shrill” with its argument “rings” is another functor).
Classification according to the number of arguments
According to the number of their arguments, the functors can be divided into:
(1) single-digit (monadic) functors (“yawns”);
(2) two-digit (dyadic) functors (“steals”);
(3) three-digit (triadic) functors (".. gives .. the book ..");
(4) n-place functors.
Individual evidence
- ^ Rudolf Carnap: Logical Syntax of Language . Vienna 1934. 2nd edition Vienna / New York 1968.
- ^ Rudolf Carnap: Introduction to Symbolic Logic with Applications . Dover 1958.
- ↑ Hügli / Lübcke, Philosophielexikon (1991) / functional expression / functor
- ↑ a b Bochenski, The contemporary thinking methods, 10th edition (1993), p. 53
- ↑ Bochenski, The contemporary thinking methods, 10th edition (1993), p. 53 f.