17 Slides--Quantification

17 Slides--Quantification - CS103A HO#17 Quantification...

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CS103A HO#17 Quantification 1/30/08 1 CS103A 1/30/08 Midterm Exam Tues., Feb. 12 7 - 9 pm Location TBA Quantification So far we have used -- propositions and connectives -- identity -- predicates Propositional calculus Quantification -- propositions and connectives -- identity -- predicates -- quantifiers FOL Predicate calculus Quantification We are after more expressive power. "There is a cube in the world." "There are no tetrahedrons in the world." "All cubes in the world are large." Quantification The first thing we need are variables. We want to say that x is a variable and write Cube(x) Quantification The first thing we need are variables. We want to say that x is a variable and write Cube(x) That is not a sentence in FOL. Since it does not specify what x refers to, we can’t say whether it is true or false. We say that x is a "free variable" in the formula.
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CS103A HO#17 Quantification 1/30/08 2 Quantification We can make sentences out of formulas with variables: "There is an object x such that Cube(x)." "For every object x, Cube(x)." "For every object x, ¬Cube(x)." "It is not the case that there is an object x such that Cube(x)." Quantification To write these sentences in FOL, we need some notation. x Cube(x) is the Existential Quantifier x Cube(x) is the Universal Quantifier Quantification To write these sentences in FOL, we need some notation. x Cube(x) is the Existential Quantifier For some object x, Cube(x) x Cube(x) is the Universal Quantifier For every object x, Cube(x) We say that the quantifiers are "binding operators". Once we apply them, we have a sentence that is true or false. Quantification
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17 Slides--Quantification - CS103A HO#17 Quantification...

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