Predicate (mathematical logic)

Predicate (from the Latin praedicare, award ') is called in modern logic the part of an atomic statement by which a property is predicated of an object. The corresponding property of the term is attributed to the object, or predicated of him. The simplest formal logical system, which operates with ( some ) predicates, the predicate logic of first order. There are two ways to win from predicates declarative sentences: on the one hand by the fact that applying the predicate " to some individual items or to specific n- tuple of objects, or the fact that it replaces matching vacancies of the predicate by individuals name or by quantifying binds.

Here, the term " predicate " is often used for both an expression as well as for its content. Only Gottlob Frege introduced the distinction between " term term" and " word" consistently. This exacerbated the epistemological question of whether the " predicative structure of statements " primary is a property of thought or language. In logical practice, the distinction is sometimes omitted because logic can also be operated as uninterpreted formal calculus on expressions. The logical predicate is called in role as a predictor or general term to it terminologically delineate both the grammatical predicate as well as singular terms (object name). However, under the influence of modern logic, some newer grammatical predicate theories based on logical predicate term.

From the understanding of modern logic, the predicate term is different in traditional logic. The traditional predicate term was founded by Aristotle and ruled until the 19th century. After logical predicate is generally that which is predicated of a subject. In modern logic, the logical predicate is since Frege that which is predicated of one or more objects, or an expression that contains a space, " unsaturated term", which is completed by other terms, an expression for a set. In expressions for predicates of the first stage in the sense of Frege the vacancy will be filled with proper names or with bound variables.

The predicate in the traditional logic

In traditional logic ( syllogistic see also ) is ( traditionally called categorical judgments ) in the analysis of statements drawn between what something is said ( the subject ), and what is said about it ( the predicate ). The subject is the object, is predicated on the something and predicate what is attributed to him in the statement, for example, a property. The part of the statement that refers to the object that is the subject term and the part of the statement which ascribes to the subject, the predicate, the predicate term. In fact, however, usually "subject" in the sense of "subject term" and " predicate " is used in the sense of " predicate term ." The speech act of attribution itself is the predication.

Examples of simple statements:

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

The predicates in the example sentences are the qualities to be a man, to sleep and love to discuss with long wine evenings on philosophy. The predicate terms are " human ", " sleeping " and " likes to discuss during long evenings of wine on philosophy ."

Show the first and last example that the logical predicate (more precisely, the predicate term) not with the grammatical predicate ( "is" or " love " ) must match: Grammatically, is " a man," a Gleichsetzungsnominativ, " with long wine evenings on philosophy to discuss " a direct object.

As parts of a simple statement predicate term and subject term are incomplete and not themselves statements. You can not be true or false in itself.

In Example 1, the predicate term of two parts is composed: the copula "is" and the predicate noun "man". In the syllogistic, it has become common to write the predicates in examples 2 and 3 in this form, because they in the formal, syllogistic closing are to be directly usable. So for example:

• The dog of my neighbors is a sleeper.
• Socrates is a lover of disputing about philosophy with long wine evenings.

On the basis of the traditional subject and predicate term Immanuel Kant distinguishes between analytic judgments in which the predicate is already contained in the subject ( eg, the statement " All circles are round" ) and synthetic judgments in which the predicate to the subject something adding (for example, the statement " the dog is sleeping "). At this Kantian distinction, see synthetic judgment a priori.

The modern predicate term

Predicates as concepts and record functions

A predicate expression in the sense of modern logic is " an expression from which one can form by inserting individual names for individual variables a sentence. ". Put another way: A predicate is an expression that is left by parting occurring in his name in a sentence.

Fundamental to the modern predicate term is the insight of Frege that the assessable content of a statement is a whole, " which can be logically broken down in various ways, but always so that from an object relations or properties are predicated. ". In application and extension of the concept of function of analysis the statement is broken down not into subject and predicate, but in function and argument. The subject-predicate scheme of colloquial language is replaced by an argument for the logic function schema. Here, the argument expression represents an object, apply by which certain properties or relations, which are expressed by a function expression.

The predicate in the sense of modern logic is thus a set function, which is also referred to as a statement function statement or form (English ) propositional function. From each set function is called that, a truth value is obtained for each argument ( each singular term ), which is used in them.

So says a recent introduction: " n-ary predicates are actually n- ary functions, the function values ​​have nothing to do with numbers. Rather, they give truth values ​​. Predicates take arguments as individual items and give truth values ​​. Two-digit predicates take as arguments ordered pairs of objects and give truth values ​​... in short. N-ary predicates take n- tuple as arguments and return truth values ​​" The extension of a n-ary predicate is the set of n- tuples for which the predicate the truth value " true " results.

Predicates, relations and existence statements

The modern predicate concept paves the way to capture and relations (relations) as well as existential statements logically adequate.

Relations

The modern concept of predicate allows Mehrstelligkeit of the predicate and thus a logical treatment of relations.

• For example, " Socrates is a student of Plato "

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

Traditional analysis, " Socrates " (subject) "is" ( copula ) " a student of Plato " ( predicate )

Modern analysis: The relation of the "Student One of ' is a predicate term " _1 _2 student is of "analyzes; the terms " _1" and " _2 " thereby mark the points at which the individuals are designated through this relationship should be stated - in the example, these are the individuals ( arguments, objects) Socrates and Plato.

In the case of "_1 _2 student is " there is a relationship between two objects, which is why the predicate (or predicate term) is called the double digits. Depending on the number of items between which a relationship is said, one also speaks of three -, four -, etc. -place predicates, or more generally multi-digit by n-or indeterminate.

Existence results

The modern concept of predicate also allows existential statements to capture adequate.

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

Traditional logic: If ( 1) is "it" grammatically a bogus subject, which is a problem for the traditional logic. Formulated man (1 ) and ( 3) in " Violette ants are existent " to, you can set this in " Violette ants " (subject) "are" analyze ( copula ) and " existent " ( predicate ). This set differs from (2): " Some ants " (subject) "are" ( copula ) " violet " ( predicate ).

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

Concepts as meanings of predicates

Apart Frege in predicates set of functions that you use the term " concept " with him in a only logical sense, and can be seen in terms the meaning of predicates, one comes to his classical concept definition: " a term is a function whose value always a truth value is ".

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

Predicates as names for properties and relations

Between the predicates as linguistic expressions and their meanings is to be distinguished strictly. So, for example, means the predicate " is white ," the property of being white, and " his friend " the predicate the relationship of friendship. The importance of n-ary predicates are also called n-ary terms.

Predicates are " names for properties and relations that are to be predicated of individuals." Predicates are " a sign of a digit attribute (ie a property ) ." Depending on the relations of logical terminology can be n-ary predicates as unary relationship terms refer.

Predicate term and ontology

Usually predicates with properties of objects are identified. However, this equation is less reliable because it is only partly true of atomic predicates of the first stage.

For the Aristotelian predicate term it means summing up: " The relation of subject and predicate in a sentence reflects the fundamental relationship of reality: the substance (subject) and their properties ( predicates ) Every true judgment reflects an ontological relationship. . "

It is not deepen here to what extent the classical ontology with their substance and commercial thinking the classical predicate term is necessary.

Classifications of predicates

According to the available number of individuals name (arguments) can be made between single-and multi -digit predicates. A predicate with n vacancies are called n- ary predicate.

Instead of one-, two - or three -place predicates ( predictors ) is also spoken by monadic, dyadic, triadic predicates. Multi-digit predicates ( predictors ) are sometimes referred to as relators. A word can be an expression of predicates of different number of digits.

• Example ( lie ) ( 1) digit ( f (a) ): " Anton is located " (= " .. lies ," ("Alpha "));
• ( 2) two letters ( f (a, b)): Anton is located under an oak tree ( = " .. below .. " ("Alpha ", " oak " ) ).
• ( 3) digits long ( f (a, b, c) ): Anton is located between an oak and a birch ( = " .. between .. and .. " ("Alpha ", " oak ", " birch " ) )

" For the rest, is in every multi-digit predicate also one with less voids and always a single digit. " That is Anton is located between an oak and a birch can also be analyzed as ( = " .. is between an oak and a birch " ("Alpha "). The vacancy of the predicate correspond in other terminology, its syntactic valence.

An atomic predicate ( semantic component; semantic primitive; engl. Semantic primitive ) is a predicate that contains no connectives. A molecular predicate is a " predicate that is created by the combination of several atomic predicates by logical connectives ".

In the tradition of Gottlob Frege distinguishes between predicates and predicates first stage second stage. Predicates are predicates first stage, the scope of items that are designated individuals fare includes antenna. For predicates second stage first stage only predicates are used as arguments in question.

" A predicate is called empty if it applies to no individual. " (Example: __ is a unicorn ). The opposite is a non-empty predicate.

Formalization of the predicate in mathematical logic

Unlike the traditional syllogistic modern mathematical logic does not examine logical reasoning using normalsprachiger rates, but closing in precisely described formal languages ​​or systems. For predicate calculi are among the terms described the language of single and multi- ary predicate symbols, and predicate constants, predicate letters or called predictors, often written as capital letters, followed by the arguments of the predicate or of vacancies as a placeholder for such arguments. Often the arguments are enclosed in parentheses and separated by commas. For example, a predicate with the predicate symbol "P " as " P_ " or " P ( _) " would be written, a two -place predicate with the predicate symbol "S" would be considered " S_1_2 " or " S ( _1, _2 ) " written. The digit predicate symbols correspond to the predicate terms of syllogistic logic.

In the interpretation of a formal language of predicate calculus each digit predicate symbol ( objects, entities in the broadest sense) assigned to the set of individuals which have satisfied the affected predicate; each two place predicate symbol, the set of ordered pairs of individuals which have satisfied the predicate; and generally any n-ary predicate symbol the set of all n- tuples ( in Mathematics also called relation) of individuals who have satisfied the relevant predicate. The totality of all objects of which in the considered interpretation is the speech is called the universe of discourse (german universe of discourse or domain).

The term predicate is formally defined as a function in the set of truth values: An n- ary predicate is an n-ary function from the n-fold Cartesian product of the universe of discourse D - that is, from the set of all n- tuples of individuals - in the set of truth values ​​. Thus, each n-ary predicate symbol P ( _1, _2, ... _n ) such a function - a title - P ( x1, x2, ... xn ) be assigned, so that P ( x1, x2, ... xn ) = true if and only if the n- tuple ( x1, x2, ... xn ) is an element of the predicate symbol associated set of n -tuples, in other words,

For this reason, the statements of (x1, x2, ... xn) ∈ P and P ( x1, x2, ... xn ) is also used interchangeably.

As a simple example of a love triangle. The universe of discourse U consisting of Ulrich, Heiner and Anna:

We have two predicate symbols "F ()" ( digit) and "L (,) " ( two digits). We will arrange the predicate symbol "F ()" the digit ratio (ie, a subset of U) { } to Anna. The predicate symbol "L (,) " the binary relation { (Anna, Heiner ), ( Heiner, Anna ) (Ulrich, Anna ) }. Our predicates F (x ) and L ( x1, x2). F ( x ) is true if and only if x = Anna. In our interpretation, ie: F (Anna ).