Remark 1. 1. There is no relation between p and Q 2. The flaw is that"is/in"relationship can not be expressed by PL 3. Consider" there exists….","al…."," anong….",and"only 4. A richer language is needed. Solution(Continue). However, the statements can be abstracted further like 1. P: All As have property B 2. Q: C is one of Follow this way, we could find a solution 4 Predicates and quantifiers Example 3. Consider a simple sentence, every student is younger than some instructor. Solution. It is a declarative statement, which can be expressed by a proposition letter, say P. D Remark 2. 1. However, it means being a students, being a instructor and being younger than somebody else 2. P fails to reflect the finer logic structure of it Solution( Continue). Consider a special instance of this statement. Suppose Andy is a student and Let's introduce some predicates, which asserts something has some property Now we have 1. S(Andy ) Andy is a student. 3. Y(Andy, Paul): Andy is younger than Paul. Remark 3. 1. There are many instances. Too many symbols are needed 2. Introduce variable, which can represent any students or instructors. 3. How about"every/all"and"some"? (a)3 means"there is", which is called existential quantifier (b )v means "for all", which is called universal quantifiersRemark 1. 1. There is no relation between P and Q. 2. The flaw is that ”is/in” relationship can not be expressed by PL. 3. Consider ”there exists ...”, ”all ...”, ”among ...”, and ”only ...”. 4. A richer language is needed. Solution(Continue). However, the statements can be abstracted further like 1. P: All As have property B. 2. Q: C is one of As. Follow this way, we could find a solution. 4 Predicates and quantifiers Example 3. Consider a simple sentence, every student is younger than some instructor. Solution. It is a declarative statement, which can be expressed by a proposition letter, say P. Remark 2. 1. However, it means being a students, being a instructor and being younger than somebody else. 2. P fails to reflect the finer logic structure of it. Solution(Continue). Consider a special instance of this statement. Suppose Andy is a student and Paul is a instructor. Let’s introduce some predicates,which asserts something has some property. Now we have 1. S(Andy): Andy is a student. 2. I(P aul):Paul is an instructor. 3. Y (Andy, P aul): Andy is younger than Paul. Remark 3. 1. There are many instances. Too many symbols are needed. 2. Introduce variable, which can represent any students or instructors. 3. How about ”every/all” and ”some”? (a) ∃ means ”there is”, which is called existential quantifier. (b) ∀ means ”for all”, which is called universal quantifiers. 2