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There are several rules of inference which utilize the existential quantifier. Existential introduction (∃I) concludes that, if the propositional function is known to be true for a particular element of the domain of discourse, then it must be true that there exists an element for which the proposition function is true. Symbolically,
! says “there exists exactly one such that has property .” Only ∀ {\displaystyle \forall } and ∃ {\displaystyle \exists } are part of formal logic. ∃ ! x {\displaystyle \exists !x} P ( x ) {\displaystyle P(x)} is an abbreviation for
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]
There exists a natural number s such that for every natural number n, s = n 2. This is clearly false; it asserts that there is a single natural number s that is the square of every natural number. This is because the syntax directs that any variable cannot be a function of subsequently introduced variables.
For example, the sentence "kangaroos live in Australia" is true because there are kangaroos in Australia; the existence of these kangaroos is the truthmaker of the sentence. Truthmaker theory states there is a close relationship between truth and existence; there exists a truthmaker for every true representation. [129]
For example, consider the sentence "There exists x such that x is a philosopher." This sentence is seen as being true in an interpretation such that the domain of discourse consists of all human beings, and that the predicate "is a philosopher" is understood as "was the author of the Republic." It is true, as witnessed by Plato in that text.
The ∃ sign stands for "there exists", which is known as existential quantification. So for example, () is read as "there exists an x such that P(x) ". {() [=]} is a notational variant for the same set of even natural numbers.
From the other direction, there has been considerable clarification of what constructive mathematics is—without the emergence of a 'master theory'. For example, according to Errett Bishop's definitions, the continuity of a function such as sin(x) should be proved as a constructive bound on the modulus of continuity, meaning that the existential content of the assertion of continuity is a ...