Examples 4.5
The Traditional Square of Opposition
Key:
Contradictory = opposite truth value
Contrary = at least one is false (not both true)
Subcontrary = at least one is true (not both false)
Subalternation = truth flows downward, falsity flows upward
Contradictory
– if a given A proposition is true, then the O proposition must be false.
Conversely, if a given O proposition is true, then the A proposition must be false.
All computers are things that are annoying. (assuming T)
Some computers are not things that are annoying. (therefore has to be F)
Some cats are not male (T)
All cats are male (F)
(Thus, we see that both statements can’t be true or false at the same time.
A given
statement will always tell us the truth value of the other.
Contrary
– if a given A proposition is true, then the corresponding E statement must be
false (because at least one has to be false).
Also, if an E statement is true, then we know
the A statement must be false, for the same reason.
However, if, say, a given A statement
is false,
then the E statement’s truth value is undetermined
.
All computers are things that are annoying (assuming T)
No computers are things that are annoying (therefore has to be F)
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All animals are cats (F)
No animals are cats (F)
(the point being, here, that they
can
be both false)
All cats are dogs (F)
(thus the falsity of the first guarantees nothing of the truth value of
No cats are dogs (T)
the other . . .)
Subcontrary
– Here, if a given I proposition is false, then the corresponding O statement
must be true—because both statements can’t be false at the same time.
Also, if a given O
statement is false, then we know the I statement has to be true.
If one of them is given as
true, however,
we won’t know the truth value of the other
.
The rule only states they can’t
both be false—so they could both be true.
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 Spring '08
 Barrett
 Logic, Truth value, Subalternation

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