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 operators.
In modal logic the de re / de dicto distinction is stated as
1. (DE DICTO): \Box A(t)
2. (DE RE): (\lambda x.\Box A(x))(t).
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 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 operators.
In modal logic the de re / de dicto distinction is stated as
1. (DE DICTO): \Box A(t)
2. (DE RE): (\lambda x.\Box A(x))(t).
In (1) the modal operator applies to the formula A(t) and the term t is within the scope...