Existential generalization

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. In first-order logic, it is often used as a rule for the existential quantifier (${\displaystyle \mathrm{\exists }}$) 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:

${\displaystyle Q(a)\to \mathrm{\exists }xQ(x),}$

where ${\displaystyle Q(a)}$ is obtained from ${\displaystyle Q(x)}$ by replacing all its free occurrences of ${\displaystyle x}$ (or some of them) by ${\displaystyle a}$.^[3]

According to Willard Van Orman Quine, universal instantiation and existential generalization are two aspects of a single principle, for instead of saying that ${\displaystyle \mathrm{\forall }xx=x}$ implies ${\displaystyle \text{Socrates}=\text{Socrates}}$, we could as well say that the denial ${\displaystyle \text{Socrates}\ne \text{Socrates}}$ implies ${\displaystyle \mathrm{\exists }xx\ne x}$. 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.^[4]

• Existential instantiation
• List of rules of inference
• Universal generalization

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