Critical Thinking
Lecture 9
The Square of Opposition
By David Kelsey
The Square of Opposition
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The square of opposition:
represents the logical relationships
that can hold between any two
corresponding standard form
categorical claims.
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For any two standard form
claims to correspond to each
other:
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The square of opposition:
depicts a variety of logical
relationships that hold between
corresponding A, E, I and O claims.
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•
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•
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A……….Contraries…………….E
.
.(cannot both be true) .
.
.
.
.
.
.
Contradictories
.
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(Never the same t-value) .
.
.
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.
.
.
.
.
. .
. .
I…………Subcontraries……….O
(cannot both be false)
Contraries &
Subcontraries
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Corresponding A and E claims
are contraries:
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–
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For example
Corresponding I and O claims are
subcontraries.
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Both claims can be false or
One can be false while the other is
true, or vice versa.
Contraries cannot both be true.
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Both claims can be true or
One can be false while the other is
true, or vice versa.
Subcontraries cannot both be
false.
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For example
Contradictory claims
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Contradictories:
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Corresponding A and O claims are contradictories.
And corresponding I and E claims are contradictories.
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One true, the other false: For any two contradictory claims, one must be true
and the other false.
•
Never the same T-value:
Using the square
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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.
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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 all
three of the remaining
corresponding claims.
•
A false claim at the bottom of the
square: from this one can infer the
truth value of all three of the
remaining corresponding claims.
•
Given an I claim is false:
Given an A claim is true:
– And similarly for a true E claim.
– And similarly for a false O
claim.
What can be inferred in
using the square #2
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Given a false claim at the top of the square of opposition one can infer the
truth value of only one of the three corresponding claims: its contradictory.
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Similarly, Given a true claim at the bottom of the square one can infer the truth value of
only its contradictory.
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
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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 (of any standard form cat. claim).
• Example:
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For any E or I claim, it’s converse is logically equivalent to it.
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A and O claims are not logically equivalent to their converses.
Obversion
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Obversion: guides one to finding the obverse of a claim.
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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
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A complementary term:
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refers to or picks out a complementary class of things.
Two complementary classes:
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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 passed the exam.
Universe of Discourse
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A universe of discourse (UD): contains classes of things within it.
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Subsets of a UD: a grouping of the members of the UD that all ___________
• Example: ‘A’ students
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Every subset has a complement: for every subset of a UD there is a complement to that
class.
– Complements to subsets: all the members of the UD that aren’t members of the
___________.
–
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‘A’ students again:
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
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Replacing a term with its complement:
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But replacing a term with its complement is sometimes tougher:
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People who took the exam:
Looking back at obversion:
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Non-: Is often as easy as putting ‘non-’ in front of it.
Students and Surfers:
The second step to obversion: replace the predicate term with its complement.
Example: Find the obverse of All Presbyterians are Christians.
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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
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So to find the obverse of a claim:
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change it from affirmative to negative or vice versa and then replace the predicate term
with its complement.
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No fish are mammals:
• 1) Change it to affirmative: ______________
• 2) Replace the predicate term with its complement: __________
• So we get ____________
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All Catholics are Christians: its obverse is
Some contestants are not winners:
Some citizens are voters:
Logically Equivalent:
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for any standard form categorical claim, it and its obverse are logically equivalent.
Same T-Value:
Contraposition
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Contraposition:
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All Mongolians are Muslims:
its contrapositive is
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To find the contrapositive:
– 1) switch the places of the
subject and predicate terms
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Some citizens are not voters:
its contrapositive is
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All A and O claims are logically
equivalent to their
contrapositives.
– A and O claims and their
contrapositives always have the
same __________
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E and I claims are not logically
equivalent to their
contrapositives.
• just as in conversion
– 2) replace both the subject and
predicate term with their
complements.