PHIL012 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/phil012_pam208

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 }

c bb aa b

falsefalsetrue

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.