Transcript Philosophy 115 Lecture 8b The Square of Opposition

Critical Thinking
Lecture 9
The Square of Opposition
By David Kelsey
The Square of Opposition
• The square of opposition:
represents the logical
relationships that can hold
between any two corresponding
standard form categorical
claims.
• For any two standard form
claims to correspond to
each other:
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•
•
•
•
•
•
•
•
A……….Contraries…………….E
.
.(cannot both be ____) .
.
.
.
.
.
.
Contradictories
.
.
(__________________)
.
.
.
.
.
.
.
.
.
. .
. .
I…………Subcontraries……….O
(cannot both be _____)
Contraries &
Subcontraries
•
Corresponding A and E claims
are contraries:
•
Corresponding I and O claims are
subcontraries.
•
Contraries cannot both be true.
•
Subcontraries cannot both be
false.
–
For example
–
For example
Contradictory claims
•
Contradictories:
–
–
Corresponding A and O claims are contradictories.
And corresponding I and E claims are contradictories.
•
One true, the other false:
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Never the same T-value:
Using the square
•
Inferring truth values using the square:
– Using the square of opposition and given the truth value of any standard
form categorical claim one can always infer the truth value of at least one of
the other corresponding three standard form claims.
•
Say the A-claim All monkeys are mammals is true. The square tells us:
What can be inferred
from the square
•
•
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A true clam at the top of the
square of opposition: given this
one can infer the truth value of
•
A false claim at the bottom of the
square: from this one can infer the
truth value of
•
Given an I claim is false:
•
And similarly for a false O claim.
Given an A claim is true:
And similarly for a true E claim.
What can be inferred in
using the square #2
•
Given a false claim at the top of the square of opposition one can infer the
truth value of
–
•
Similarly for a true claim at the bottom of the square
Say the A-claim All sharks are monkeys is false:
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And similarly for false E claims.
–
And also similarly for true I or O claims.
Three Operations
•
Conversion, Obversion & Contraposition:
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–
•
3 operations that can be performed on any standard form categorical claim.
The operations help us find a new truth value for a new claim…
Conversion:
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Finding the converse: find the converse by simply switching the positions of the
subject and predicate term
• Example:
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For any E or I claim:
–
A and O claims:
Obversion
•
Obversion: guides one to finding the obverse of a claim.
•
Find the obverse by:
– 1) changing it from affirmative to negative or vice versa
• Remember that while ______ claims are affirmative, _______ claims are
negative.
– To change an A claim to negative:
– To change an E claim to affirmative:
– To change an I claim to negative:
– To change an O claim to affirmative:
– 2) replacing the predicate term with its complementary term.
Complementary terms
•
A complementary term:
–
•
refers to or picks out a complementary class of things.
Two complementary classes:
–
refers to two different classes of things which together pick out all and only the
members of some universe of discourse.
– A universe of discourse is the group of things that a claim is about.
• Example: Everyone got an A on the exam.
Universe of Discourse
•
A universe of discourse (UD): contains classes of things within it.
– Subsets of a UD: a group of members of the UD that all possess some
property in common.
• Example:
•
Every subset has a complement: for every subset of a UD there is a
complement to that class.
– Example:
•
Complementary classes: (for some UD) pick out all and only the members of
that UD.
Complementary terms: refer to complementary classes.
•
Some examples
of complementary terms
•
Replacing a term with its complement:
–
–
•
But replacing a term with its complement is sometimes tougher:
–
•
Non-: Is often as easy as putting ‘non-’ in front of it.
Examples:
Example:
Looking back at obversion: Find the obverse of All Presbyterians are
Christians.
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–
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1) Change it from Affirmative to negative:
2) Replacing the predicate term with its complement:
Thus the obverse is:
Obversion recap
•
•
So to find the obverse of a claim:
–
change it from affirmative to negative or vice versa and then replace the predicate term
with its complement.
–
No fish are mammals:
• 1) Change it to affirmative: ______________
• 2) Replace the predicate term with its complement: __________
• So we get ____________
–
Other examples:
• All Catholics are Christians
• Some contestants are not winners:
• Some citizens are voters:
Logically Equivalent:
–
for any standard form categorical claim, it and its obverse are logically equivalent
Contraposition
•
•
Contraposition:
To find the contrapositive:
– 1) switch the places of the
subject and predicate terms
• just as in conversion
– 2) replace both the subject and
predicate term with their
complements.
• Examples:
– All Mongolians are Muslims
– Some citizens are not voters
•
Truth values?
–
–
A and O claims
E and I claims