PHIL 012 Class Notes 1152001 Outline Announcements web

PHIL 012 Class Notes 1/15/2001

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

Announcements • Notes are online. • Syllabus has been updated. • URL for web page: www. courses. psu. edu/phil 012_pam 208 Note the URL’s are CASE SENSITIVE.

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.

Homework Problems 1 -7

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”

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.

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)

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.

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

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”

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

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.

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.

Homework Problem 8

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.