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: A……….Contraries…………….E . .(cannot both be ____) . . . . . . . Contradictories . . (__________________) . . . . . . . . . . . . . I…………Subcontraries……….O (cannot both be _____)

Contraries & Subcontraries Corresponding A and E claims are contraries: Contraries cannot both be true. For example Corresponding I and O claims are subcontraries. Subcontraries cannot both be false. For example

Contradictory claims Contradictories: One true, the other false: Corresponding A and O claims are contradictories. And corresponding I and E claims are contradictories. One true, the other false: 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 A true clam at the top of the square of opposition: given this one can infer the truth value of Given an A claim is true: And similarly for a true E claim. 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.

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: And similarly for false E claims. And also similarly for true I or O claims.

Three Operations Conversion, Obversion & Contraposition: Conversion: 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: Finding the converse: find the converse by simply switching the positions of the subject and predicate term Example: 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: Two complementary classes: 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. 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: Non-: Is often as easy as putting ‘non-’ in front of it. Examples: But replacing a term with its complement is sometimes tougher: Example: Looking back at obversion: Find the obverse of All Presbyterians are Christians. 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 Examples: Truth values? 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