Universal instantiation: Difference between revisions
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| statement = For all <math>A</math> implies there is a free |
| statement = For all <math>A</math> implies there is a free occurrence of a member <math>a</math> that exists in the set of A |
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| symbolic statement = <math>\forall x \, A \Rightarrow A\{x \mapsto a\}</math> |
| symbolic statement = <math>\forall x \, A \Rightarrow A\{x \mapsto a\}</math> |
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Revision as of 15:28, 16 June 2022
Type | Rule of inference |
---|---|
Field | |
Statement | For all implies there is a free occurrence of a member that exists in the set of A |
Symbolic statement |
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 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
for every formula A and every term a, where is the result of substituting a for each free occurrence of x in A. is an instance of
And as a rule of inference it is
- from infer
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.