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Existential quantification is distinct from universal quantification ("for all"), which asserts that the property or relation holds for all members of the domain. [2] [3] Some sources use the term existentialization to refer to existential quantification. [4] Quantification in general is covered in the article on quantification (logic).
A succinct, equivalent formulation which avoids these problems uses universal quantification: For each natural number n, n · 2 = n + n. A similar analysis applies to the disjunction, 1 is equal to 5 + 5, or 2 is equal to 5 + 5, or 3 is equal to 5 + 5, ... , or 100 is equal to 5 + 5, or ..., etc. which can be rephrased using existential ...
In symbolic logic, the universal quantifier symbol (a turned "A" in a sans-serif font, Unicode U+2200) is used to indicate universal quantification. It was first used in this way by Gerhard Gentzen in 1935, by analogy with Giuseppe Peano's (turned E) notation for existential quantification and the later use of Peano's notation by Bertrand Russell.
This sort of quantification is known as uniqueness quantification or unique existential quantification, and is often denoted with the symbols "∃!" [ 2 ] or "∃ =1 ". For example, the formal statement
Quantifier symbols: ∀ for universal quantification, and ∃ for existential quantification; Logical connectives: ∧ for conjunction, ∨ for disjunction, → for implication, ↔ for biconditional, ¬ for negation. Some authors [11] use Cpq instead of → and Epq instead of ↔, especially in contexts where → is used for other purposes.
It is generally given as a quantification rule for the universal quantifier but it can also be encoded in an axiom schema. It is one of the basic principles used in quantification theory. Example: "All dogs are mammals. Fido is a dog. Therefore Fido is a mammal." Formally, the rule as an axiom schema is given as
In predicate logic, existential generalization [1] [2] (also known as existential introduction, ∃I) is a valid rule of inference that allows one to move from a specific statement, or one instance, to a quantified generalized statement, or existential proposition.
The universal (dually, existential) quantifier interprets the interior operator; All open (or closed) elements are also clopen. A more concise axiomatization of monadic Boolean algebra is (1) and (2) above, plus ∀(x∨∀y) = ∀x∨∀y (Halmos 1962: 21). This axiomatization obscures the connection to topology.