# Predicate (logic)

In modern predicate logic, a predicate (from the Latin praedicare ' to speak' ) is called that part of an atomic statement that is truth-functional . A predicate has one or more argument positions, a complete statement is created by inserting individual constants in the argument position (s) or by inserting variables and their binding through a preceding quantification . In common philosophical interpretation, a single-digit predicate is an expression for a property . In an atomic statement, the term corresponding to the property is assigned to the object represented by the individual symbol , or predicted by it . Multi-digit predicates are also referred to as relations, single-digit predicates as terms. The simplest formal-logical system that operates with (certain) predicates is first-order predicate logic.

The concept of predicate differs from the understanding of modern logic in traditional logic . The traditional concept of a predicate was founded by Aristotle and prevailed until the 19th century. According to this, a logical predicate is generally what is said about a subject. In modern logic, since Gottlob Frege , the logical predicate is what is said about one or more objects, or an expression that contains a space, "unsaturated expression", which is completed by other expressions to form an expression for a sentence . In expressions for predicates of the first level in the Frege sense, the space is filled with “proper names” or with bound variables.

## The predicate in traditional logic

In traditional logic (see also syllogistics ), when analyzing statements (traditionally called categorical judgments ), a distinction is made between what is said about something (the subject ) and what is said about it (the predicate ). The subject is the object about which something is said and predicate that which is ascribed to it in the statement, for example a property . The part of the statement that refers to the object is the subject term and the part of the statement that ascribes the predicate to the subject is the predicate term . In fact, however, “subject” is usually also used in the sense of “subject term” and “predicate” in the sense of “predicate term”. The speech act of ascription itself is predication .

Examples of simple statements are:

1. Socrates is a human.
2. My neighbor's dog is sleeping.
3. Socrates loves to discuss philosophy over long wine evenings.

In examples 1 and 3, the person Socrates is the subject, the expression "Socrates" (the first part of statements 1 and 3) the subject term. In Example 2, my neighbor's dog (the animal that barks at me every morning) is the subject and the phrase "my neighbor's dog" is the subject term.

The predicates in the example sentences are the characteristics of being human , of sleeping and of loving to discuss philosophy over long evenings with wine . The predicate terms are “human”, “sleeps” and “loves to discuss philosophy over long wine evenings”.

The first and last example show that the logical predicate (more precisely: the predicate term) does not have to agree with the grammatical predicate (“is” or “loves”): grammatically, “a person” is a nominative of equations, “for long wine evenings about philosophy to discuss “an accusative object.

As parts of a simple statement, the predicate term and the subject term are incomplete and are not statements themselves. They cannot be true or false for themselves.

In example 1, the predicate term is composed of two parts: the copula “is” and the predicate noun “der Mensch”. In syllogistics it has become common practice to also write the predicates in Examples 2 and 3 in this form, because they can only then be used directly in the context of formal, syllogistic inference. So roughly:

• My neighbor's dog is sleeping.
• Socrates is a lover of discussing philosophy over long evenings with wine.

Taking the traditional subject and predicate concepts as a basis, Immanuel Kant differentiates between analytical judgments , in which the predicate is already contained in the subject (e.g. the statement "All circles are round") and synthetic judgments, in which the predicate relates something to the subject add (for example, when saying “The dog is sleeping”). For this Kantian distinction see Synthetic Judgment a priori .

## The modern term of predicate

### crossing

Traditionally, the term “predicate” has been used both for an expression and for its content. It was only Gottlob Frege who consistently made the distinction between “term expression” and “concept”. This exacerbates the epistemological question of whether the “predicative structure of statements” is primarily a property of thinking or language. In logical practice, this distinction is occasionally ignored, since logic can also be operated as an uninterpreted formal calculus using expressions. The logical predicate is also referred to as a predicator or general term , in order to distinguish it terminologically from both the grammatical predicate and singular terms (object names). However, under the influence of modern logic, some newer grammatical predicate theories are based on the logical concept of predicate.

### Predicates as concepts and sentence functions

A predicate expression in the sense of modern logic is “an expression from which one can form a sentence by inserting individual names for individual variables”. In other words: a predicate is the expression that is left over when you delete the names that appear in a sentence.

Fundamental to the modern concept of a predicate is Frege's insight that the assessable content of a statement is a whole “that can be logically broken down in different ways, but always in such a way that relationships or properties are stated about an object”. In applying and expanding the function concept of analysis, the statement is no longer broken down into subject and predicate, but into function and argument. The subject-predicate scheme of colloquial language is replaced for logic by an argument-function scheme. The argument expression represents an object of which certain properties or relationships apply, which are expressed by a function expression.

The predicate in the sense of modern logic is thus a sentence function , which is also referred to as a proposition function, proposition form or (English) propositional function . Every propositional function is required to give a truth value for every argument (every singular expression) that is inserted into it.

A recent introduction says: “n-place predicates are actually n-place functions whose function values ​​have nothing to do with numbers. Rather, they give truth values. Single-digit predicates take individual objects as arguments and give truth values. Two-place predicates take ordered pairs of objects as arguments and give truth values ​​... in short: n-place predicates take n-tuples as arguments and give truth values. ”The extension of an n-place predicate is the set of n-tuples for which the predicate results in the truth value "true".

### Predicates, relations and statements about existence

The modern term of predicate clears the way to also logically adequately grasp relationships as well as statements about existence.

#### Relations

The modern concept of predicate enables multiple digits of the predicate and thus a logical treatment of relations.

• Example: "Socrates is a disciple of Plato"

Note: In truth, Plato was a student of Socrates. The example is only intended to show that false statements, just like true statements, contain logical elements.

traditional analysis : "Socrates" (subject) "is" (copula) "a student of Plato" (predicate)

modern analysis : the relationship of “being a student of” is analyzed as the predicate term “_ 1 is a student of _ 2 ”; the expressions "_ 1 " and "_ 2 " mark the places where the individuals are named, about which this relationship is to be stated - in the example these are the individuals (arguments, objects) Socrates and Plato.

In the case of "_ 1 is a student of _ 2 ", there is a relationship between two objects, which is why the predicate (or the predicate term) has two digits. Depending on the number of objects between which a relationship is stated, one also speaks of three-, four-, etc. predicates, or more generally of n- or indefinitely multiple-digit predicates.

#### Existence statements

The modern term of predicate also enables statements about existence to be grasped more adequately.

• Example: (1) “There are purple ants”; (2) "Some ants are purple"; (3) "Violet ants exist"

traditional logic : In (1), “it” is grammatically a pseudo-subject, which is a problem for traditional logic. If one rephrases (1) and (3) as “violet ants are existing”, this sentence can be analyzed as “violet ants” (subject) + “are” (copula) and “existing” (predicate). This sentence differs from (2): “some ants” (subject) + “are” (copula) + “violet” (predicate).

Modern logic : For modern logic the sentences (1) - (3) are synonymous and “exist” is a predicate only in a grammatical, but not in a logical sense. The details are controversial. According to Frege , existence is the property of a concept to have a non-empty scope. Russell's essay On Denoting (1905) is regarded as the locus classicus for the modern conception of existence .

### Terms as meanings of predicates

If one sees propositional functions in predicates with Frege , one uses the expression "concept" with him in a purely logical sense, and one sees the meaning of predicates in concepts, one arrives at his classic conceptual definition: "A concept is a function, its value is always a truth value ".

This is considered to be "the first stable concept of the term in the history of European philosophy".

### Predicates as names for properties and relations

A strict distinction must be made between predicates as linguistic expressions and their meanings. So z. B. the predicate “is white” the quality of being white, and the predicate “being friend” the relationship of friendship. The meaning of n-place predicates is also known as n-place terms .

Predicates are “names for properties and relations that are to be predicated of the individuals”. Single-digit predicates are "a character for a single-digit attribute (i.e. a property)". Depending on the terminology used in relation to logic, n-place predicates can also be referred to as single-place relational expressions.

### Predicate concept and ontology

Usually predicates are identified with properties of objects. However, this equation is to be restricted, since it only applies to a limited extent to atomic predicates of the first level.

For the Aristotelian concept of predicate, it says in summary: “The relation of subject and predicate in the sentence reflects the basic relationship of reality: the substance (subject) with its properties (predicates). Every true judgment reflects a relationship of being. "

It is not necessary here to go into the extent to which classical ontology with its substance and accident thinking necessitates the classical concept of predicate.

### Classifications of the predicates

Arity

Depending on the number of usable individual names (arguments), a distinction can be made between one-digit and multi-digit predicates. A predicate with n spaces is called an n-place predicate.

Instead of one-, two- or three-digit predicates, monadic, dyadic, triadic predicates are also used. Multi-digit predicates (predicators) are sometimes also called relators . A word can be the expression of predicates with different numbers of digits.

• Example ( lying )
• (1) single-digit ( f (a) ): "Anton is" (= "... is", (Anton));
• (2) two-digit ( f (a, b) ): Anton lies under an oak (= "... lies under ...", (Anton, Eiche)).
• (3) three-digit ( f (a, b, c) ): Anton lies between an oak and a birch (= "... lies between ... and ..." (Anton, Eiche, Birke))

"Incidentally, in every multi-digit predicate there is also one with fewer spaces and always a single digit." That means, Anton lies between an oak and a birch can also be analyzed as (= "... lies between an oak and a birch", " Anton "). In other terminology, the blanks of the predicate correspond to its syntactic valence .

Atomic and molecular predicates

An atomic predicate (semantic building block; semantic primitive; English: semantic primitive) is a predicate that does not contain any joiners . A molecular predicate is a "predicate that has arisen through the connection of several atomic predicates using junctions".

In the tradition of Gottlob Frege , a distinction is made between first-degree and second-degree predicates . First-level predicates are predicates whose scope encompasses objects that are identified with individual constants. For second-order predicates, only first-order predicates are possible arguments.

Empty / non-empty predicate

“A predicate is called empty if it does not apply to any individual.” (Example: __ is a unicorn ). The opposite is a non-empty predicate.

## Formalization of the predicate in mathematical logic

In contrast to traditional syllogistics, modern mathematical logic does not investigate logical reasoning with the help of normal-language sentences, but rather the reasoning in precisely described formal languages ​​or systems. For predicate calculi , the described expressions of the language include one-digit and multi-digit predicate symbols , also called predicate constants , predicate letters or predicators , often written as uppercase letters, followed by the arguments of the predicate or by spaces as placeholders for such arguments. The arguments are often put in brackets and separated from one another by commas. For example, a one-digit predicate with the predicate symbol “P” would be written as “P_” or as “P (_)”, a two-digit predicate with the predicate symbol “S” would be written as “S_ 1 _ 2 ” or as “S (_ 1 , _ 2 ) “is written. The single-digit predicate symbols correspond to the predicate terms of syllogistic logic.

In the interpretation of a formal language of a predicate calculus, each single-digit predicate symbol is assigned the set of individuals (objects, entities in the broadest sense) to which the predicate applies; for each two-digit predicate symbol the set of ordered pairs of individuals to which the predicate applies; and in general for each n -place predicate symbol the set of all n - tuples (in mathematics also referred to as relation ) of individuals to which the respective predicate applies. The entirety of all objects that are mentioned in the interpretation under consideration is called the universe of discourse or domain .

The term predicate is formally defined as a function in the set of truth values : An n -place predicate is an n -place function from the n -fold Cartesian product of the discourse universe D - that is, from the set of all n -tuples of individuals - in the set of truth values. Thus, every n -place predicate symbol P (_ 1 , _ 2 , ... _ n ) can be assigned such a function - a predicate - P (x 1 , x 2 , ... x n ) , so that P (x 1 , x 2 , ... x n )  = true if and only if the n tuple (x 1 , x 2 , ... x n ) is an element of the set of n tuples assigned to the predicate symbol , in other words: ${\ displaystyle \ prod _ {i = 1} ^ {n} D = \ underbrace {D \ times ... \ times D} _ {n \ times}}$

For all x 1 , x 2 ,… x nD :
(x 1 , x 2 ,… x n ) ∈ P ⇔ P (x 1 , x 2 ,… x n ) = true

For this reason, the statements (x 1 , x 2 ,… x n ) ∈ P and P (x 1 , x 2 ,… x n ) are also used synonymously.

A love triangle as a simple example. The universe of discourse U consists of Ulrich, Heiner and Anna:

U = {Ulrich, Heiner, Anna}

We have two predicate symbols F () (single digit) and L (,) (double digit). We assign the predicate symbol F () to the one-digit relation (i.e. a subset of U ) {Anna}. The predicate symbol L (,) of the two-digit relation {(Anna, Heiner), (Heiner, Anna), (Ulrich, Anna)}. Our predicates are F ( x ) and L ( x 1 , x 2 ). F ( x ) is true if and only if x = Anna. In our interpretation the following applies: F (Anna).

## Individual evidence

1. Cf. Paul Ruppen: Entry into formal logic. 1996, p. 157: "A predicate is an expression that we express about one or more objects."
2. Verena E. Mayer: The value of thoughts. 1989, p. 40 f. Footnote 25.
3. ^ Paul Hoyningen-Huene : Logic. 1998, p. 171.
4. z. B. in Albert Menne: Logic. 6th edition. 2001, p. 58; Helmut Seiffert: Introduction to logic. 1973, p. 23.
5. ^ So Willard Van Orman Quine after Ernst Tugendhat , Ursula Wolf: Logisch-semantische Propädeutik. 1983, p. 94.
6. ^ Eike von Savigny: Basic course in logical reasoning. 2nd Edition. 1984, p. 85.
7. ^ Cf. (for the simple sentence) Franz von Kutschera , Albert Breitkopf: Introduction to modern logic. 8th edition. 2007, ISBN 978-3-495-48271-1 , p. 84.
8. a b Verena E. Mayer: The value of thoughts. 1989, p. 70.
9. Verena E. Mayer: The value of thoughts. 1989, p. 68.
10. Gottlob Frege: “Foreword” to the term writing, in: Uwe Meixner, (Ed.): Philosophy of logic. 2003, pp. 27, 31.
11. ^ Niko Strobach: Introduction to Logic. 2005, p. 83.
12. Cf. Niko Strobach: Introduction to Logic. 2005, p. 83.
13. Example based on Ernst Tugendhat, Ursula Wolf: Logisch-semantische Propädeutik 1983, p. 94.
14. Ernst Tugendhat, Ursula Wolf: Logisch-semantische Propädeutik 198, p. 185.
15. Elena Tatievskaya: Introduction to Propositional Logic 2003, p. 48.
16. Ernst Tugendhat, Ursula Wolf: Logical-semantic propaedeutics. 1983, p. 191 ff.
17. See Rudolf Haller, "Term", in: HWPH Vol. 1 1971, Col. 780 (785)
18. ^ A b c Franz von Kutschera, Albert Breitkopf: Introduction to modern logic. 8th edition. 2007, ISBN 978-3-495-48271-1 , p. 85.
19. Gottlob Frege: Function and Concept. [1891].
20. ^ So Günther Patzig : Language and Logic , 2nd ed. 1981, p. 97
21. ^ Rudolf Carnap : Introduction to Symbolic Logic. 3. Edition. 1968, pp. 4-5.
22. So z. B. Robert Kirchner, Wilhelm Karl Essler, Rosa F. Martinez Cruzado: Basics of logic. Volume I, 4th edition. (1991), p. 174.
23. ^ Patrick Brandt, Rolf-Albert Dietrich, Georg Schön: Linguistics. 2nd Edition. 2006, p. 49.
24. Helmut Seiffert: Logic. 1973, p. 28.
25. ^ Paul Hoyningen-Huene: Logic. 1998, p. 173, there is also the following example.
26. ^ David Hilbert , Wilhelm Ackermann: Fundamentals of theoretical logic. 6th edition. Berlin u. a. 1972, ISBN 3-540-05843-5 , p. 69.
27. So Hadumod Bußmann (Ed.): Lexicon of Linguistics. 3rd, updated and expanded edition. Kröner, Stuttgart 2002, ISBN 3-520-45203-0 , (argument).
28. Hadumod Bußmann (Ed.): Lexicon of Linguistics. 3rd, updated and expanded edition. Kröner, Stuttgart 2002, ISBN 3-520-45203-0 , (atomic predicate).
29. Maximilian Herberger, Dieter Simon: Theory of Science for Jurists. P. 94.
30. ^ Rudolf Carnap: Introduction to Symbolic Logic. 3. Edition. 1968, pp. 65-68; see. also Albert Menne: logic. 6th edition. 2001, p. 68: "Predicators 1st level" = "Predicators that individuals have as arguments". Of Carnap individuals characters as characters zero level referred.
31. Albert Menne: Logic. 6th edition. 2001, p. 61.