Square of Opposition

The square of opposition is a chart that was introduced within
classical (categorical) logic to represent the logical relationships
holding between certain propositions in virtue of their form. The
square, traditionally conceived, looks like this:

square-of-opposition

The four corners of this chart represent the four basic forms of
propositions recognized in classical logic:

A propositions, or universal affirmatives take the form: All S are P.
E propositions, or universal negations take the form: No S are P.
I propositions, or particular affirmatives take the form: Some S are P.
O propositions, or particular negations take the form: Some S are not P.

Given the assumption made within classical (Aristotelian) categorical
logic, that every category contains at least one member, the following
relationships, depicted on the square, hold:

Firstly, A and O propositions are contradictory, as are E and I
propositions. Propositions are contradictory when the truth of one
implies the falsity of the other, and conversely. Here we see that 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. For
example, if the proposition "all industrialists are capitalists" (A)
is true, then the proposition "some industrialists are not
capitalists" (O) must be false. Similarly, if "no mammals are aquatic"
(E) is false, then the proposition "some mammals are aquatic" must be
true.

Secondly, A and E propositions are contrary. Propositions are contrary
when they cannot both be true. 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." Note, however, that
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."

Next, I and O propositions are subcontrary. 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.
Note, however, that it is possible for corresponding I and O
propositions both to be true, as with "some nations are democracies,"
and "some nations are not democracies." Again, I and O propositions
are subcontrary, but not contrary or contradictory.

Lastly, two propositions are said to stand in the relation of
subalternation when the truth of the first ("the superaltern") implies
the truth of the second ("the subaltern"), but not conversely. A
propositions stand in the subalternation relation with the
corresponding I propositions. The truth of the A proposition "all
plastics are synthetic," implies the truth of the proposition "some
plastics are synthetic." However, the truth of the O proposition "some
cars are not American-made products" does not imply the truth of the E
proposition "no cars are American-made products." In traditional
logic, the truth of an A or E proposition implies the truth of the
corresponding I or O proposition, respectively. Consequently, the
falsity of an I or O proposition implies the falsity of the
corresponding A or E proposition, respectively. However, the truth of
a particular proposition does not imply the truth of the corresponding
universal proposition, nor does the falsity of an universal
proposition carry downwards to the respective particular propositions.

The presupposition, mentioned above, that all categories contain at
least one thing, has been abandoned by most later logicians. Modern
logic deals with uninstantiated terms such as "unicorn" and "ether
flow" the same as it does other terms such as "apple" and "orangutan".
When dealing with "empty categories", the relations of being contrary,
being subcontrary and of subalternation no longer hold. Consider,
e.g., "all unicorns have horns" and "no unicorns have horns." Within
contemporary logic, these are both regarded as true, so strictly
speaking, they cannot be contrary, despite the former's status as an A
proposition and the latter's status as an E proposition. Similarly,
"some unicorns have horns" (I) and "some unicorns do not have horns"
(O) are both regarded as false, and so they are not subcontrary.
Obviously then, the truth of "all unicorns have horns" does not imply
the truth of "some unicorns have horns," and the subalternation
relation fails to hold as well. Without the traditional
presuppositions of "existential import", i.e., the supposition that
all categories have at least one member, then only the contradictory
relation holds. On what is sometimes called the "modern square of
opposition" (as opposed to the traditional square of opposition
sketched above) the lines for contraries, subcontraries and
subalternation are erased, leaving only the diagonal lines for the
contradictory relation.