Existential instantiation: Difference between revisions
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In [[predicate logic]], '''existential instantiation''' is a [[validity|valid]] [[rule of inference]] whose use in a [[formal proof|proof]] has the restriction that the [[existential name]] which is introduced by the rule, must be a new name that has not occurred earlier in the proof. Failure to observe this restriction results in the [[existential fallacy]], a [[Fallacy|logical fallacy]]. Existential instantiation is the [[inference]] that if there exists an object that is such that it has a certain property, that, for the purposes of our proof, we can name this object using an existential name. In the case of a formal language, a symbol may be used called an [[instantial letter]]. Whatever name we use, it cannot be a part of the conclusion of the proof. It is only a hypothetical name, and we cannot assume that such an object really has the name that we have assigned it in our proof. |
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In formal language: |
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:{{exists}}(''x'')<math>\mathcal{F}</math>''x'' :: <math>\mathcal{F}</math>''a'' |
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Where ''a'' is an arbitrary name that has not been a part of our proof thus far. What is not allowed is: |
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:{{exists}}(''x'')<math>\mathcal{F}</math>''x'' :: <math>\mathcal{F}</math>''y'' |
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<ref>Hurley, Patrick. A Concise Introduction to Logic. Wadsworth Pub Co, 2008.</ref> |
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An example of an argument whose proof requires existential instantiation: |
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:All doctors are college graduates. |
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:Some doctors are golfers. |
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:Therefore, some golfers are college graduates. |
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{{reflist}} |
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[[Category:Rules of inference]] |
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[[Category:Predicate logic]] |
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Revision as of 13:34, 18 February 2012
In predicate logic, existential instantiation is a valid rule of inference whose use in a proof has the restriction that the existential name which is introduced by the rule, must be a new name that has not occurred earlier in the proof. Failure to observe this restriction results in the existential fallacy, a logical fallacy. Existential instantiation is the inference that if there exists an object that is such that it has a certain property, that, for the purposes of our proof, we can name this object using an existential name. In the case of a formal language, a symbol may be used called an instantial letter. Whatever name we use, it cannot be a part of the conclusion of the proof. It is only a hypothetical name, and we cannot assume that such an object really has the name that we have assigned it in our proof.
In formal language:
- Template:Exists(x)x :: a
Where a is an arbitrary name that has not been a part of our proof thus far. What is not allowed is:
- Template:Exists(x)x :: y
An example of an argument whose proof requires existential instantiation:
- All doctors are college graduates.
- Some doctors are golfers.
- Therefore, some golfers are college graduates.
- ^ Hurley, Patrick. A Concise Introduction to Logic. Wadsworth Pub Co, 2008.