Predicate abstraction
Encyclopedia
In logic
, predicate abstraction is the result of creating a predicate from an open sentence
. If Q(x) is any formula with x free then the predicate formed from that sentence is (λx.Q(x)), where λ is an abstraction operator. 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 t in Q by x. 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):
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.
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
, predicate abstraction is the result of creating a predicate from an open sentence
Open sentence
In mathematics, an open sentence is described as "open" in the sense that its truth value is meaningless until its variables are replaced with specific numbers, at which point the truth value can usually be determined...
. If Q(x) is any formula with x free then the predicate formed from that sentence is (λx.Q(x)), where λ is an abstraction operator. 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 t in Q by x. 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
Modal operator
In modal logic, a modal operator is an operator which forms propositions from propositions. In general, a modal operator has the "formal" property of being non-truth-functional, and is "intuitively" characterised by expressing a modal attitude about the proposition to which the operator is applied...
s.
In modal logic
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
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.