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This includes both quantification of the form "exactly k objects exist such that …" as well as "infinitely many objects exist such that …" and "only finitely many objects exist such that…". The first of these forms is expressible using ordinary quantifiers, but the latter two cannot be expressed in ordinary first-order logic. [4]
When the meaning depends on the syntax, a symbol may have different entries depending on the syntax. For summarizing the syntax in the entry name, the symbol is used for representing the neighboring parts of a formula that contains the symbol. See § Brackets for examples of use. Most symbols have two printed versions.
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)" or ...
In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics.
There exists an x such that ... For at least one x, .... Keywords for uniqueness quantification include: For exactly one natural number x, ... There is one and only one x such that .... Further, x may be replaced by a pronoun. For example, For every natural number, its product with 2 equals to its sum with itself. Some natural number is prime.
In the theory of partial orders with one relation symbol ≤, one could define s = t to be an abbreviation for s ≤ t t ≤ s. In set theory with one relation ∈, one may define s = t to be an abbreviation for ∀x (s ∈ x ↔ t ∈ x) ∀x (x ∈ s ↔ x ∈ t). This definition of equality then automatically satisfies the axioms for equality.
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However, Razborov and Rudich showed that if one-way functions exist, P and NP are indistinguishable to natural proof methods. Although the existence of one-way functions is unproven, most mathematicians believe that they do, and a proof of their existence would be a much stronger statement than P ≠ NP.