Universal quantification
This article or subsequent section is not sufficiently supported by evidence (e.g., anecdotal evidence). Information without sufficient evidence may be removed in the near future. Please help Wikipedia by researching the information and adding good supporting evidence.
An all-statement is a statement about all elements of a particular subject area, for example, the statement "All men are mortal." Synonymously, modern terms such as universal statement, universal proposition, all-statement, generalization, or generalization (as a result) are used. In traditional logic, all-statements are called universal, universal or general judgements - for this, see Categorical judgement.
The logical properties of all-statements are treated modernly in predicate logic and have traditionally been treated as universally affirmative judgments in syllogistics.
In the natural language of German, all-phrases are expressed mainly with words like "alle/s", "jede/r/s", "immer" and "überall" or with in definite constructions ("Ein voller Bauch studiert nicht gern", "Menschen sind sterblich"). In the formal language of predicate logic, all-statements are formed by quantifying over predicates or statement forms with the help of the all-quantifier
To falsify a general statement, it is sufficient to specify a single object from the subject area to which the statement does not apply. To verify an all-statement, on the other hand, one must generally examine every item in the subject domain. If the set of objects in the object domain is inaccessible or infinitely large, only a more or less good confirmation can generally be considered. An exception, however, are all-statements in the formal sciences such as logic and mathematics, for example those about infinite sets of numbers, which can be verified by methods such as complete induction.
In syllogistics, all-statements (universal judgements) were accompanied by an existential presupposition, i.e. a universal judgement was considered meaningful only if the terms occurring in it applied to at least one object. In predicate logic it is also possible to deal with empty terms or predicates, but there is a weakened existential presupposition in the sense that the range of objects must not be empty as a whole. In alternative logical systems such as free logic, this restriction is also removed.
See also
- Existence statement
