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In [[logic]], '''predicate abstraction''' is the result of creating a [[Predicate (logic)|predicate]] from a [[sentence (linguistics)|sentence]]. If Q is any formula then the predicate abstract formed from that sentence is (λy.Q), where λ is an [[abstraction operator]] and in which every occurrence of y occurs bound by λ in (λy.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 operator]]s. | |||
In [[modal logic]] the "''de re'' / ''de dicto'' distinction" is stated as | |||
1. (DE DICTO): <math>\Box A(t)</math> | |||
2. (DE RE): <math>(\lambda x.\Box A(x))(t)</math>. | |||
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== | |||
For the semantics and further philosophical developments of predicate abstraction see Fitting and Mendelsohn, ''First-order Modal Logic'', [[Springer Science+Business Media|Springer]], 1999. | |||
[[Category:Modal logic]] | |||
[[Category:Philosophical logic]] | |||
Revision as of 06:48, 4 February 2014
In logic, predicate abstraction is the result of creating a predicate from a sentence. If Q is any formula then the predicate abstract formed from that sentence is (λy.Q), where λ is an abstraction operator and in which every occurrence of y occurs bound by λ in (λy.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
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
For the semantics and further philosophical developments of predicate abstraction see Fitting and Mendelsohn, First-order Modal Logic, Springer, 1999.