The Proprietor wrote:The four corners of the chart shown above represent the four basic forms of propositions recognized in classical (Aristotelian) logic:
A propositions, or universal affirmatives, take the form: All S are P or Every S is a P.
E propositions, or universal negations, take the form: No S are P or No S is a P.
I propositions, or particular affirmatives, take the form: Some S are P or Some S is a P.
O propositions, or particular negations, take the form: Some S are not P or Some S is not P.
On the assumption made within classical (Aristotelian) categorical logic, that every category contains at least one member, the following relationships hold within the square:
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. Thus, 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 expert consensus opinions are trustworthy”
(A) is
true, then the proposition “Some expert consensus opinions are not trustworthy”
(O) must be
false. Similarly, if “No expert consensus views are trustworthy”
(E) is
false, then the proposition “Some expert consensus views are trustworthy” 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 experts are right” — cannot be true at the same time as the corresponding
E proposition: “No experts are right.” Please note, however, that
A and
E propositions, while they are contrary, are not contradictory. While they cannot both be
true, they
can both be
false; “All experts are right” and “No experts are right” don’t exhaust the logical possibilities: “Some experts are right” (a proposition of form
I), if it is the case, obviously falsifies the corresponding
A and
E propositions.
Next,
I and
O propositions are what classical Aristotelian logic terms
subcontrary. Propositions are subcontrary when it is impossible for both to be
false. Because “Some experts are infallible” is false, “Some experts are not infallible” must be
true. Please notice, however, that it is possible for corresponding
I and
O propositions both to be
true, as with “Some authorities are reliable,” and “Some authorities are not reliable.” Again,
I and
O propositions are subcontrary, but they are neither contrary nor contradictory.
Finally, two propositions are deemed 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 relation of subalternation with the corresponding
I propositions. The truth of the
A proposition “All experts are fallible” implies the truth of the proposition “Some experts are fallible.” However, the truth of the
O proposition “Some experts are not correct” does not imply the truth of the
E proposition “No experts are correct.” In traditional logic, the truth of an
A or an
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 a universal proposition carry downwards to the respective particular propositions.
