Download PHIL012 Class Notes

PHIL012 Class Notes
1/15/2001
Outline
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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 “”
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So, assuming:
a = { 6, 10 }
b = { 2, 4, 6, 10 }
c = { 10, 12 }
ab
ba
cb
true
false
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.