Universal instantiation
| Type | Rule of inference |
|---|---|
| Field | Predicate logic |
| Transformation rules |
|---|
| 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.
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