- Source: Existential generalization
In predicate logic, existential generalization (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. In first-order logic, it is often used as a rule for the existential quantifier (
∃
{\displaystyle \exists }
) in formal proofs.
Example: "Rover loves to wag his tail. Therefore, something loves to wag its tail."
Example: "Alice made herself a cup of tea. Therefore, Alice made someone a cup of tea."
Example: "Alice made herself a cup of tea. Therefore, someone made someone a cup of tea."
In the Fitch-style calculus:
Q
(
a
)
→
∃
x
Q
(
x
)
,
{\displaystyle Q(a)\to \ \exists {x}\,Q(x),}
where
Q
(
a
)
{\displaystyle Q(a)}
is obtained from
Q
(
x
)
{\displaystyle Q(x)}
by replacing all its free occurrences of
x
{\displaystyle x}
(or some of them) by
a
{\displaystyle a}
.
Quine
According to Willard Van Orman Quine, universal instantiation and existential generalization are two aspects of a single principle, for instead of saying that
∀
x
x
=
x
{\displaystyle \forall x\,x=x}
implies
Socrates
=
Socrates
{\displaystyle {\text{Socrates}}={\text{Socrates}}}
, we could as well say that the denial
Socrates
≠
Socrates
{\displaystyle {\text{Socrates}}\neq {\text{Socrates}}}
implies
∃
x
x
≠
x
{\displaystyle \exists x\,x\neq x}
. The principle embodied in these two operations is the link between quantifications and the singular statements that are related to them as instances. Yet it is a principle only by courtesy. It holds only in the case where a term names and, furthermore, occurs referentially.
See also
Inference rules
References
Kata Kunci Pencarian:
- Existential generalization
- Universal generalization
- Existential quantification
- Universal instantiation
- List of rules of inference
- Faulty generalization
- Modus tollens
- Disjunctive syllogism
- Distributive property
- Rule of inference
