Predicate abstraction: Difference between revisions
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In [[logic]], '''predicate abstraction''' is the result of creating a [[predicate]] from |
In [[logic]], '''predicate abstraction''' is the result of creating a [[Predicate (logic)|predicate]] from a [[formula (logic)|formula]]. If Q is any formula then the predicate abstract formed from that sentence is (λx.Q), where λ is an [[abstraction operator]] and in which every occurrence of x that is free in Q is bound by λ in (λx.Q). The resultant predicate (λx.Q(x)) is a monadic predicate capable of taking a [[term (logic)|term]] t as argument as in (λx.Q(x))(t), which says that the object denoted by 't' has the property of being such that Q. |
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The ''law of abstraction'' states ( |
The '''{{vanchor|law of abstraction}}''' states ( λx.Q(x) )(t) ≡ Q(t/x) where Q(t/x) is the result of replacing all free occurrences of x in Q by t. This law is shown to fail in general in at least two cases: (i) when t is irreferential and (ii) when Q contains [[modal operator]]s. |
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In [[modal logic]] the "''de re'' / ''de dicto'' distinction" is stated as |
In [[modal logic]] the "''[[de re and de dicto|de re'' / ''de dicto]]'' distinction" is stated as |
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1. (DE DICTO): <math>\Box A(t)</math> |
1. (DE DICTO): <math>\Box A(t)</math> |
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2. (DE RE): <math>(\lambda x.\Box A(x))(t)</math>. |
2. (DE RE): <math>(\lambda x.\Box A(x))(t)</math>. |
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In (1) the modal operator applies to the formula A(t) and the term t is within the scope of the modal operator. In (2) t is ''not'' within the scope of the modal operator. |
In (1) the modal operator applies to the formula A(t) and the term t is within the scope of the modal operator. In (2) t is ''not'' within the scope of the modal operator. |
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==References== |
==References== |
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For the semantics and further philosophical developments of predicate abstraction see Fitting and Mendelsohn, ''First-order Modal Logic'', [[Springer Science+Business Media|Springer]], |
For the semantics and further philosophical developments of predicate abstraction see [[Melvin Fitting|Fitting]] and Mendelsohn, ''First-order Modal Logic'', [[Springer Science+Business Media|Springer]], 1999. |
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[[Category:Modal logic]] |
[[Category:Modal logic]] |
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[[Category:Philosophical logic]] |
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{{semantics-stub}} |
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Latest revision as of 14:18, 29 September 2023
In logic, predicate abstraction is the result of creating a predicate from a formula. If Q is any formula then the predicate abstract formed from that sentence is (λx.Q), where λ is an abstraction operator and in which every occurrence of x that is free in Q is bound by λ in (λx.Q). The resultant predicate (λx.Q(x)) is a monadic predicate capable of taking a term t as argument as in (λx.Q(x))(t), which says that the object denoted by 't' has the property of being such that Q.
The law of abstraction states ( λx.Q(x) )(t) ≡ Q(t/x) where Q(t/x) is the result of replacing all free occurrences of x in Q by t. This law is shown to fail in general in at least two cases: (i) when t is irreferential and (ii) when Q contains modal operators.
In modal logic the "de re / de dicto distinction" is stated as
1. (DE DICTO):
2. (DE RE): .
In (1) the modal operator applies to the formula A(t) and the term t is within the scope of the modal operator. In (2) t is not within the scope of the modal operator.
References
[edit]For the semantics and further philosophical developments of predicate abstraction see Fitting and Mendelsohn, First-order Modal Logic, Springer, 1999.
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