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For example, in Zermelo–Fraenkel set theory, variables range over all sets. In this case, guarded quantifiers can be used to mimic a smaller range of quantification. Thus in the example above, to express For every natural number n, n·2 = n + n. in Zermelo–Fraenkel set theory, one would write For every n, if n belongs to N, then n·2 = n + n,
In particular the logical depth of a graph is defined to be the minimum level of nesting of quantifiers (the quantifier rank) in a sentence defining the graph. [17] The sentence outlined above nests the quantifiers for all of its variables, so it has logical depth n + 1 {\displaystyle n+1} .
Bound variables within nested quantifiers are handled by increasing the subscript by one for each successive quantifier. This leads to rule 4, which must be applied after the other rules since rules 1 and 2 produce quantified variables.
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, when used together with a predicate variable, is called an existential quantifier (" ∃ x " or " ∃( x ...
In this logic, quantifiers may only be nested to finite depths, as in first-order logic, but formulas may have finite or countably infinite conjunctions and disjunctions within them. Thus, for example, it is possible to say that an object is a whole number using a formula of L ω 1 , ω {\displaystyle L_{\omega _{1},\omega }} such as
For example, the first-order formula "if x is a philosopher, then x is a scholar", is a conditional statement with "x is a philosopher" as its hypothesis, and "x is a scholar" as its conclusion, which again needs specification of x in order to have a definite truth value. Quantifiers can be applied to variables in a formula.
Set-builder notation can be used to describe a set that is defined by a predicate, that is, a logical formula that evaluates to true for an element of the set, and false otherwise. [2]
For example, the quantifier ∀ A, which can be viewed as set-theoretic inclusion, satisfies all of the above except [symmetry]. Clearly [symmetry] holds for ∃ A while e.g. [contraposition] fails. A semantic interpretation of conditional quantifiers involves a relation between sets of subsets of a given structure—i.e. a relation between ...