Form of statement

The expression form of expression is ambiguous. He describes:

• an expression which contains a variable and which is converted into a statement by binding the variable to a quantifier (statement form in terms of mathematics and predicate logic );
• an expression in which a propositional variable occurs (proposition form in the sense of propositional logic ).

Usually in the first case one speaks of a predicate and only in the second case of a statement form. (see predicate logic )

In the prevailing meaning, the expression "statement form" stands for the statement form in the sense of propositional logic. From a mathematical point of view, statement forms are functions on Boolean values ​​with Boolean values ​​as the result.

Examples

• From mathematics: the statement form or the predicate “A (x)” = “x + 5 = 10” is converted into a sentence by inserting certain values. For x = 5 the proposition is true, for the proposition is false. ${\displaystyle x\neq 5}$
• from the predicate logic:
  • the one-digit predicate “x laughs” = “L (x)” (true for x = laughing Peter and false for x = crying Jörg);
  • the two-digit predicate "x admires y" = "B (x, y)";
  • the two-digit predicate "x is an actor and y admires x" = "SCH (x) and B (x, y)".

properties

In the first-level predicate logic, the variable must be an object variable ( individual variable ). In a strict sense, one speaks of a statement form only if the expression in question contains at least one free object variable.

Due to the indeterminacy of the free variable, statement forms have no determinable truth value and are therefore not statements (in the technical sense).

The statement form can be transformed into a statement in two ways: (a) by inserting constants for the variables or (b) by binding the free variables with quantifiers .

Forms of statements with a free variable are often understood to express terms and properties (“x is a person”, “x is pink”), i. H. Are predicates.

Forms of statements with several free variables are often understood as relations, for example “x is greater than y”, “x and y have a common child z”, “x + 1 = y and y + 1 = z”.

The relation of the concept of the statement form to that of the logical formula depends on the definition of the logical formula because of its ambiguity.

In contrast to the (mathematical) formula , relations , logical junctions and quantification are allowed in the statement form .

In contrast to the type of a tuple in a logical structure , the statement form is a purely syntactic representation that can be defined independently of a model. Formally, a type is a form of statement.

In the first-level predicate logic, forms of statement can be defined inductively via their structure:

• If are terms and a one -place relational symbol , then applies ${\displaystyle t_{1},\ldots ,t_{k}}$ ${\displaystyle R}$ ${\displaystyle k}$
  • ${\displaystyle t_{1}=t_{2}}$ is an (atomic) statement form,
  • ${\displaystyle R(t_{1},\ldots ,t_{k})}$ is an (atomic) statement form

with all the variables of the terms as free variables in it,

• if there are statements forms, then applies ${\displaystyle \varphi ,\varphi _{1},\varphi _{2}}$
  • ${\displaystyle \neg \varphi }$ is a (compound) statement form,
  • ${\displaystyle \varphi _{1}\wedge \varphi _{2}}$ is a (compound) statement form,
  • ${\displaystyle \varphi _{1}\vee \varphi _{2}}$ is a (compound) statement form;
  • ${\displaystyle \varphi _{1}\rightarrow \varphi _{2}}$ is a (compound) statement form,
  • ${\displaystyle \varphi _{1}\leftrightarrow \varphi _{2}}$ is a (compound) statement form;

with all free variables of the as free variables, ${\displaystyle \varphi _{i}}$

• if a free variable is in a proposition form , then holds ${\displaystyle x}$ ${\displaystyle \varphi }$
  • ${\displaystyle \exists x\varphi }$ is a (compound) statement form,
  • ${\displaystyle \forall x\varphi }$ is a (compound) statement form

with all free variables from except as free variables. ${\displaystyle \varphi }$ ${\displaystyle x}$

See also

• Statement scheme
• Type (model theory)
• Signature (model theory)

literature

• Duden - basic school knowledge , mathematics high school diploma. 2003, p. 11
• Hilbert, Ackermann: Basic features of mathematical logic . 6th edition. 1972, p. 9
• Menne, Logic, 6th ed. (2001), p. 59
• Statement form and statement scheme . In: Regenbogen, Meyer: Dictionary of Philosophical Terms . 2005.

Web links

Wiktionary: statement form  - explanations of meanings, word origins, synonyms, translations

Wikibooks: Math for non-freaks: form of expression and substitution  - learning and teaching materials

• Eric Weisstein: Sentential Formula . In: MathWorld (English).