1. PHIL012 Class Notes 1/15/2001

2. Outline Announcements, web page Review Homework Problems (1-7) Set Theory Review & Problem 8 (if time) Assignment for Wednesday (1/17)

3. Announcements Notes are online. Syllabus has been updated. URL for web page: www.courses.psu.edu/phil/phil012_pam208 Note the URL’s are CASE SENSITIVE.

4. Last Time Atomic sentences make claims that have truth value. In other words, they are TRUE or FALSE. An atomic sentence consists of a predicate followed by a list of names, the number of which correspond to the predicate’s arity. Names refer to objects. Predicates refer to properties or relations of objects.

5. Homework Problems 1-7

6. 2.4 The Language of Set Theory Set Theory  First Order Logic (FOL) Set Theory is a formal language of mathematics, used to describe counting. Set Theory, unlike FOL, has only two symbols: –=, meaning “is the same number or set” – , meaning “is a member of”

7. The Domain of Set Theory In set theory, the domain of objects is the set of numbers, usually the whole numbers: - , …, -1, 0, 1, …,  In set theory, names can also (sometimes) refer to sets of numbers. A set is simply a collection of numbers, of other sets, or of a mix of numbers and sets.

8. Examples of Set Theory Sentences Good: a = { 1 } Good: b = { 2, 3, 4 } Good: c = { 1, 2, 3, 4 } which is the same as: c = { a, b } and also the same as c = { 1, b } and c = { a, 2, 3, 4 } and c = { 1, { 2, 3, 4 }} Bad: Cube(c)

9. The Identity Symbol “= ” “= ” means the same thing in both FOL and Set Theory. “a=b” means that “a” and “b” are names that refer to the same objects, which can denote numbers or sets. “a=b” also means that whatever claims are made of a must also be true of b (and vice versa) if “a=b” is true.

10. The Identity Symbol “= ” So, if “a = { 2 }” is true and if “a = b” is true, we know that “b = { 2 }” is true also.

11. The Membership Symbol “  ” The Membership symbol means “is a member of” “a  b” means that “a is a member of b” This means that if “a” and “b” are sets, all of the members of “a” appear at least once in “b” “b” may or may not have additional members, besides those in “a”

12. The Membership Symbol “  ” So, assuming: a = { 6, 10 } b = { 2, 4, 6, 10 } c = { 10, 12 } c  b b  a a  b false true

13. Truth Value & Reference in Set Theory In Set Theory, once the reference of a name is fixed, the truth value of all sentences containing that name is fixed once and for all. If a= {1} and b = { 1, 2 }, all statements about a and b will always The only way the truth value of these statements could change would be to change the reference of a or b.

14. Truth Value & Reference in FOL This is NOT the case in FOL. Suppose the name “Phillip” refers to me and the predicate “Indoors” means “is indoors”. Without changing the reference of “Phillip” the truth of the sentence “Indoors(Phillip)” will change the moment I go outside.

15. Homework Problem 8

16. Assignment for Wednesday (1/17) Read 2.5 and 2.6 if you haven’t already Work problems 9-13. The homework system should be set up by Wednesday so we’ll spend some time talking about it.