Universal instantiation: Difference between revisions
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{{Short description|Rule of inference in predicate logic}} |
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{{Infobox mathematical statement |
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| name = Universal instantiation |
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| type = [[Rule of inference]] |
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| field = [[Predicate logic]] |
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| statement = |
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| symbolic statement = <math>\forall x \, A \Rightarrow A\{x \mapsto t\}</math> |
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}} |
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{{Transformation rules}} |
{{Transformation rules}} |
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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 |
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, Patrick. A Concise Introduction to Logic. Wadsworth Pub Co, 2008.</ref><ref>Moore and Parker{{full citation needed|date=November 2014}}</ref> ('''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 et nullo|dictum de omni]]''){{cn|date=June 2022}} is a [[Validity (logic)|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]]. |
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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." |
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Formally, the rule as an axiom schema is given as |
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: <math>\forall x \, A |
: <math>\forall x \, A \Rightarrow A\{x \mapsto t\},</math> |
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for |
for every formula ''A'' and every term ''t'', where <math>A\{x \mapsto t\}</math> is the result of [[substitution (logic)|substituting]] ''t'' for each ''free'' occurrence of ''x'' in ''A''. <math>\, A\{x \mapsto t\}</math> is an '''instance''' of <math>\forall x \, A.</math> |
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And as a rule of inference it is |
And as a rule of inference it is |
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:from <math>\vdash \forall x A</math> infer <math>\vdash A \{ x \mapsto t \} .</math> |
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:from ⊢ ∀''x'' ''A'' infer ⊢ ''A''(''a''/''x''), |
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with ''A''(''a''/''x'') the same as above. |
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[[Irving Copi]] noted that universal instantiation "...[[logical consequence|follows from]] variants of rules for '[[natural deduction]]', which were devised independently by [[Gerhard Gentzen]] and [[Stanisław Jaśkowski]] in 1934." |
[[Irving Copi]] noted that universal instantiation "...[[logical consequence|follows from]] variants of rules for '[[natural deduction]]', which were devised independently by [[Gerhard Gentzen]] and [[Stanisław Jaśkowski]] in 1934."<ref>Copi, Irving M. (1979). ''Symbolic Logic'', 5th edition, Prentice Hall, Upper Saddle River, NJ</ref> |
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== Quine == |
== Quine == |
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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 [[Quantification (logic)|quantification]]s 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 [[Reference#Semantics|referentially]].<ref>{{cite book |author1=Willard Van Orman Quine |author1-link=Willard Van Orman Quine|author2=Roger F. Gibson |title=Quintessence |contribution= V.24. Reference and Modality |location=Cambridge, Mass |publisher=Belknap Press of Harvard University Press |year=2008 | |
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 [[Quantification (logic)|quantification]]s 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 [[Reference#Semantics|referentially]].<ref>{{cite book |author1=Willard Van Orman Quine |author1-link=Willard Van Orman Quine|author2=Roger F. Gibson |title=Quintessence |contribution= V.24. Reference and Modality |location=Cambridge, Mass |publisher=Belknap Press of Harvard University Press |year=2008 |oclc=728954096}} Here: p. 366.</ref> |
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==See also== |
==See also== |
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*[[Existential instantiation]] |
*[[Existential instantiation]] |
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*[[Existential generalization]] |
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*[[Existential quantification]] |
*[[Existential quantification]] |
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*[[Inference rules]] |
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==References== |
==References== |
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Latest revision as of 10:12, 25 January 2024
| Type | Rule of inference |
|---|---|
| Field | Predicate logic |
| Symbolic statement |
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
for every formula A and every term t, where is the result of substituting t 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[edit]
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[edit]
References[edit]
- ^ 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.