# pred - Introduction to Predicate Logic Prepared by Kendra...

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1 Introduction to Predicate Logic Prepared by: Kendra Cooper Contents Introduction Predicates and Quantifiers Variables: bound and unbound Examples of Using the Existential and Universal Quantifiers Predicate Logic Definitions and Rules Universal and Existential Instantiation (aka Specification) and Generalization Negating Quantified Predicates Quantified Predicates With 2 Variables Translating English into Predicates Problems Introduction Predicate Logic -- Extending the Propositional Logic In propositional logic, there are only propositions (a statement that is either true or false) There are no variables allowed This restriction makes it difficult to express some conditions in propositional logic in a convenient way For example, consider the set of 2 nd year computer science students registered in this section of discrete mathematics: if we want to state that 26 computer science students are registered in this course using propositional logic, then we need to define these propositions: a: Elise is registered this course b: Abdullah is registered in this course a b c d e f g h i j k l m n o p q r s t u v w x y z

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2 c: Thuy is registered in this course x: Stacy is registered in this course y: Jasmin is registered in this course z: Greg is registered in this course a b c d e f g h i j k l m n o p q r s t u vv w x y z This does not scale well to large, finite sets or infinite sets (e.g., what would we need to do if we wanted to consider the set of integers?) Predicate logic allows us to introduce and use variables.
3 Predicates and Quantifiers Predicate (aka Open Statement) contains one or more variables. becomes a proposition when each variable is replaced by a choice from the variable’s universe. Notes: The universe for a variable is the set of all possible elements that are being considered (e.g., set of integers) The universe for each variable needs to be precisely defined (e.g., set of positive integers). Quantifiers There are two kinds of quantifiers: Existential and Universal Existential Quantifier: There exists one or more members in the universe Symbol is a backwards capital letter E Example: Let s be the set of all 2 nd year computer science students registered (s): the student is registered in this section of discrete mathematics s registered(s) If this statement is true, then at least one 2 nd year computer science student is registered in this section of discrete mathematics. Universal Quantifier: All of the members in the universe… Symbol is an upside down capital letter A Example: Let s be the set of all 2 nd year computer science students registered (s): the student is registered in this section of discrete mathematics s registered(s)

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4 If this statement is true, then every single 2 nd year computer science student is registered in this section of discrete mathematics.
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