# Universal instantiation

In predicate logic **universal instantiation**[1][2][3] (**UI**; also called **universal specification** or **universal elimination**, and sometimes confused with *dictum de omni*) 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. It is one of the basic principles used in quantification theory.

Transformation rules |
---|

Propositional calculus |

Rules of inference |

Rules of replacement |

Predicate logic |

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

In symbols the rule as an axiom schema is

for some term *a* and where is the result of substituting *a* for all *free* occurrences of *x* in *A*. is an **instance** of

And as a rule of inference it is

- from ⊢ ∀
*x**A*infer ⊢*A*(*a*/*x*),

with *A*(*a*/*x*) the same as above.

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]

## References

- Irving M. Copi; Carl Cohen; Kenneth McMahon (Nov 2010).
*Introduction to Logic*. Pearson Education. ISBN 978-0205820375. - Hurley
- Moore and Parker
- 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. Here: p. 366.