The Power of Logic: 5.2 The Traditional Square of Opposition
Subcontrary
"some lunches are free" is false, "some lunches are not free" must be true.
A and O; E and I
Contradictory
A and E propositions. Cannot both be true, but they can both be false.
Contrary
An A proposition, e.g., "all giraffes have long necks" cannot be true at the same time as the corresponding E proposition: "no giraffes have long necks."
Contrary
Subaltern and Falsity
False flows upward
I and O propositions. cannot both be false, but they can both be true.
Subcontrary
Subaltern and truth
Truth flows downward
Contradictories
Two statements that cannot both be true and they cannot both be false.
An inference is immediate when
a conclusion is drawn from only one premise
Corresponding Statements
categorical statements having the same subject term and the same predicate term.
Sub alternation
is the logical relationship between a universal statement and its corresponding particular statement.
Subaltern
is the particular.
Superaltern
is the universal statement
Propositions are subcontrary when
it is impossible for both to be false. Because "some lunches are free" is false, "some lunches are not free" must be true.
Subcontrary
that it is possible for corresponding I and O propositions both to be true but they cannot both be false, as with "some nations are democracies," and "some nations are not democracies."
Contradictory
the truth of a proposition of the form All S are P implies the falsity of the corresponding proposition of the form Some S are not P.
Statements are Contradictory when
the truth of one implies the falsity of the other, and conversely.
Subcontraries
two statements that can both be true but they cannot both be false.
Contraries
two statements that cannot both be true but they can both be false.
Necessary falsehood
A statement that is false and cannot be true in any possible circumstances.
Necessary Truth
A statement that is true and cannot be false under any possible circumstances.
corresponding A and E propositions, while contrary, are not contradictory.
While they cannot both be true, they can both be false, as with the examples of "all planets are gas giants" and "no planets are gas giants."
Sub alternation
when the truth of the first ("the superaltern") implies the truth of the second ("the subaltern"), but not conversely.
