Existential introduction

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Existential generalization
TypeRule of inference
FieldPredicate logic
StatementThere exists a member x {\displaystyle x} {\displaystyle x} in a universal set with a property of Q {\displaystyle Q} {\displaystyle Q}
Symbolic statement Q ( a ) →   ∃ x Q ( x ) , {\displaystyle Q(a)\to \ \exists {x}\,Q(x),} {\displaystyle Q(a)\to \ \exists {x}\,Q(x),}

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

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

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} {\displaystyle \forall x\,x=x} implies Socrates = Socrates {\displaystyle {\text{Socrates}}={\text{Socrates}}} {\displaystyle {\text{Socrates}}={\text{Socrates}}}, we could as well say that the denial Socrates ≠ Socrates {\displaystyle {\text{Socrates}}\neq {\text{Socrates}}} {\displaystyle {\text{Socrates}}\neq {\text{Socrates}}} implies ∃ x x ≠ x {\displaystyle \exists x\,x\neq 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.[4]

See also

References

  1. Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall.
  2. Hurley, Patrick (1991). A Concise Introduction to Logic 4th edition. Wadsworth Publishing. ISBN 9780534145156.
  3. pg. 347. Jon Barwise and John Etchemendy, Language proof and logic Second Ed., CSLI Publications, 2008.
  4. Willard Van Orman Quine; Roger F. Gibson (2008). "V.24. Reference and Modality". Quintessence. Cambridge, Massachusetts: Belknap Press of Harvard University Press. OCLC 728954096. Here: p.366.