Universal instantiation

{{Short description|Rule of inference in predicate logic}}

{{Infobox mathematical statement

| name = Universal instantiation

| type = Rule of inference

| field = Predicate logic

| statement =

| symbolic statement = $\backslash forall\; x\; \backslash ,\; A\; \backslash Rightarrow\; A\backslash \{x\; \backslash mapsto\; t\backslash \}$

}}

{{Transformation rules}}

In predicate logic, universal instantiation{{cite book|author1=Irving M. Copi |author2=Carl Cohen |author3=Kenneth McMahon |title=Introduction to Logic | date = Nov 2010 | isbn=978-0205820375 |publisher=Pearson Education}}{{page needed|date=November 2014}}Hurley, Patrick. A Concise Introduction to Logic. Wadsworth Pub Co, 2008.Moore and Parker{{full citation needed|date=November 2014}} (UI; also called universal specification or universal elimination,{{cn|reason=Give a reference for each synonym.|date=June 2022}} and sometimes confused with dictum de omni){{cn|date=June 2022}} 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

: $\backslash forall\; x\; \backslash ,\; A\; \backslash Rightarrow\; A\backslash \{x\; \backslash mapsto\; t\backslash \},$

for every formula A and every term t, where $A\backslash \{x\; \backslash mapsto\; t\backslash \}$ is the result of substituting t for each free occurrence of x in A. $\backslash ,\; A\backslash \{x\; \backslash mapsto\; t\backslash \}$ is an instance of $\backslash forall\; x\; \backslash ,\; A.$

And as a rule of inference it is

:from $\backslash vdash\; \backslash forall\; x\; A$ infer $\backslash vdash\; A\; \backslash \{\; x\; \backslash mapsto\; t\; \backslash \}\; .$

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."Copi, Irving M. (1979). Symbolic Logic, 5th edition, Prentice Hall, Upper Saddle River, NJ