18 Intro to Quantification

18 Intro to Quantification - Handout #18 January 30, 2008...

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Handout #18 CS103A January 30, 2008 Robert Plummer Introduction to Quantification Predicates Revisited We have been working with predicates since the beginning of our studies in logic. Predicate symbols are used to express some property of objects or some relation between objects. Our discussion of predicates has been limited to those that refer to specific objects, e.g., Boy(Elliot) or Tet(b) where b is an actual entity in Tarski’s World. We will now expand this notion to allow variables to be used with predicates. So, we can now refer to Boy(x) or Apple(y) where the value of x and y may not be known. In general, we think of logic using predicates and variables (i.e., predicate logic) as a more “powerful” language. We can manipulate the variables within a predicate in a way that is just not possible with propositional logic. In fact, predicate logic is expressive enough to form the basis of a number of useful programming languages including Prolog, and SQL (structured-query language). Predicate logic is also used in reasoning systems or “expert” systems, such as medical-diagnosis programs and theorem-proving programs. When we work with predicate logic, the variables take on an important role. It is by defining a value for these variables that we can actually assign a true or false value to the predicate. For example, Apple(y) is true if and only if y is an apple. If y happens to a plum, then Apple(y) is false. The collection of values that can replace a variable in a predicate is called the universe or domain of discourse of the predicate. In an arity-one predicate, if a value from the universe can be substituted for the variable to make it true, we say the value satisfies the predicate, and the predicate is said to be satisfiable . The same ideas are extended to arity-n predicates. By itself, we can’t determine the truth value of a formula like Cube(x). But we can determine the value if we say what objects the formula applies to, in sentences like For some x in the world, Cube(x). For every x in the world, Cube(x).
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This note was uploaded on 10/01/2011 for the course CS 103A taught by Professor Plummer,r during the Winter '07 term at Stanford.

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18 Intro to Quantification - Handout #18 January 30, 2008...

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