PHIL012 Class Notes 1/15/2001. Outline Announcements, web page Review Homework Problems (1-7) Set...
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Transcript of PHIL012 Class Notes 1/15/2001. Outline Announcements, web page Review Homework Problems (1-7) Set...
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.
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.