# Universal instantiation

__: Rule of inference in predicate logic__

**Short description**Type | Rule of inference |
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Field | Predicate logic |

Transformation rules |
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Propositional calculus |

Rules of inference |

Rules of replacement |

Predicate logic |

In predicate logic, **universal instantiation**^{[1]}^{[2]}^{[3]} (**UI**; also called **universal specification** or **universal elimination**,^{[citation needed]} and sometimes confused with *dictum de omni*)^{[citation needed]} is a valid rule of inference from a truth about each member of a class of individuals to the truth about a particular individual of that class. It is generally given as a quantification rule for the universal quantifier but it can also be encoded in an axiom schema. It is one of the basic principles used in quantification theory.

Example: "All dogs are mammals. Fido is a dog. Therefore Fido is a mammal."

Formally, the rule as an axiom schema is given as

- [math]\displaystyle{ \forall x \, A \Rightarrow A\{x \mapsto t\}, }[/math]

for every formula *A* and every term *t*, where [math]\displaystyle{ A\{x \mapsto t\} }[/math] is the result of substituting *t* for each *free* occurrence of *x* in *A*. [math]\displaystyle{ \, A\{x \mapsto t\} }[/math] is an **instance** of [math]\displaystyle{ \forall x \, A. }[/math]

And as a rule of inference it is

- from [math]\displaystyle{ \vdash \forall x A }[/math] infer [math]\displaystyle{ \vdash A \{ x \mapsto t \} . }[/math]

Irving Copi noted that universal instantiation "...follows from variants of rules for 'natural deduction', which were devised independently by Gerhard Gentzen and Stanisław Jaśkowski in 1934."^{[4]}

## 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*" implies "Socrates = Socrates", we could as well say that the denial "Socrates ≠ Socrates" implies "∃*x* *x* ≠ *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.^{[5]}

## See also

## References

- ↑ Irving M. Copi; Carl Cohen; Kenneth McMahon (Nov 2010).
*Introduction to Logic*. Pearson Education. ISBN 978-0205820375.^{[page needed]} - ↑ Hurley, Patrick. A Concise Introduction to Logic. Wadsworth Pub Co, 2008.
- ↑ Moore and Parker
^{[full citation needed]} - ↑ Copi, Irving M. (1979).
*Symbolic Logic*, 5th edition, Prentice Hall, Upper Saddle River, NJ - ↑ Willard Van Orman Quine; Roger F. Gibson (2008). "V.24. Reference and Modality".
*Quintessence*. Cambridge, Mass: Belknap Press of Harvard University Press. OCLC 728954096. Here: p. 366.

Original source: https://en.wikipedia.org/wiki/Universal instantiation.
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