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Predicate abstraction

 

 
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Predicate abstraction
 
 
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In logic
Logic

Logic is the study of the principles of valid demonstration and inference. Logic is a branch of philosophy, a part of the classical Trivium . The word derives from Greek language ?????? , fem....
, predicate abstraction is the result of creating a predicate
Predicate (logic)

Sometimes it is inconvenient or impossible to describe a set by listing all of its elements. Another useful way to define a set is by specifying a property that the elements of the set have in common....
 from an open sentence
Open sentence

A 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 (?x.Q(x)), where ? is an abstraction operator. The resultant predicate (?x.Q(x)) is a monadic predicate capable of taking a term t as argument as in (?x.Q(x))(t), which says that the object denoted by 't' has the property of being such that Q.

The law of abstraction states ( ?y.Q(x) )(t) = Q(t/x) where Q(t/x) is the result of replacing all free occurrences of t in Q by x.






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In logic
Logic

Logic is the study of the principles of valid demonstration and inference. Logic is a branch of philosophy, a part of the classical Trivium . The word derives from Greek language ?????? , fem....
, predicate abstraction is the result of creating a predicate
Predicate (logic)

Sometimes it is inconvenient or impossible to describe a set by listing all of its elements. Another useful way to define a set is by specifying a property that the elements of the set have in common....
 from an open sentence
Open sentence

A 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 (?x.Q(x)), where ? is an abstraction operator. The resultant predicate (?x.Q(x)) is a monadic predicate capable of taking a term t as argument as in (?x.Q(x))(t), which says that the object denoted by 't' has the property of being such that Q.

The law of abstraction states ( ?y.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 function, and is "intuitively" characterised by expressing a modal attitude about the proposition to which the operator is applied....
s.

In modal logic
Modal logic

A modal logic is any system of mathematical logic#Formal logic that attempts to deal with notions of possibility and necessity. Traditionally, there are three "modes" or "moods" or "modalities" of the Copula to be, namely, Logical possibility, probability, and Necessary_and_sufficient_conditions#Necessary_conditions....
 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.