Universal instantiation: Difference between revisions
GrahamHardy (talk | contribs) m - defsort |
|||
| Line 1: | Line 1: | ||
{{Transformation rules}} |
{{Transformation rules}} |
||
In [[predicate logic]] '''universal instantiation'''<ref>{{cite book| |
In [[predicate logic]] '''universal instantiation'''<ref>{{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}}</ref><ref>Hurley{{full citation needed|date=November 2014}}</ref><ref>Moore and Parker{{full citation needed|date=November 2014}}</ref> ('''UI''', also called '''universal specification''' or '''universal elimination''', and sometimes confused with [[Dictum de omni et nullo|Dictum de omni]]) is a [[validity|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]]. |
||
Example: "All dogs are mammals. Fido is a dog. Therefore Fido is a mammal." |
Example: "All dogs are mammals. Fido is a dog. Therefore Fido is a mammal." |
||
Revision as of 19:37, 6 June 2016
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.
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 occurrences of x in A.
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
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[full citation needed]
- ^ 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. Here: p.366.