1
Four relations among four forms
A and O, and E and I, are contradictories: exactly one of each pair is true. A and E are contraries: not both true. I and O are subcontraries: not both false. A implies I and E implies O, which is subalternation.
Lesson · Arguments and Validity · Categorical logic
The Square of Opposition and Existential Import
The square of opposition links the four categorical forms by contradiction, contrariety, subcontrariety and subalternation. On the Boolean reading only the contradictions survive. The rest return when every term is assumed to be non-empty, which is the existential import the old square relied on.
A and O, E and I: contradictories
A to I, E to O: subalternation
Boolean reading: terms may be empty
import: every term has members
only contradictories survive without import
The square, with and without import
Core ideas
What this lesson establishes
The traditional square arranges the four forms so that each relation is a claim about truth values. Whether the claims hold depends on a background assumption that is easy to miss.
2
One inference carries the rest
A is true only if E is false, and E is false exactly when I is true. So the contrariety of A and E is the same claim as A implies I. In the same way the subcontrariety of I and O is E implies O. Only subalternation and the contradictions need testing.
3
Import is a switch
Existential import is the assumption that every term has at least one member. Without it, All unicorns are white is true and Some unicorns are white is not, and the square loses its sides. With it, the square returns. The lab makes the assumption a menu.
The material
What the square needs
The same four forms of the last lesson, now compared in pairs, and the one assumption that decides which comparisons hold.
The square arranges the four forms in a box. A at the top left, E at the top right, I at the bottom left and O at the bottom right. The relations are claims about what can be true or false together, and each can be checked as an inference.
pair relation claim
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A and O contradictories exactly one is true
E and I contradictories exactly one is true
A and E contraries not both true
I and O subcontraries not both false
A to I subalternation A true, so I true
E to O subalternation E true, so O true
On the Boolean reading of the last lesson the contradictions hold in every case, because A empties the region that O occupies and E empties the region that I occupies. The rest fail on one pattern. Let nothing be an S. Then All S are P is true, and so is No S are P: the contraries are both true. Some S are P and Some S are not P are both false: the subcontraries are both false. And A is true while I is false, so subalternation fails.
Example · All unicorns are white
Read as a region claim, it empties the unicorns that are not white, and that region is empty because no unicorn is anywhere. So the statement is true. Some unicorns are white needs a white unicorn, and there is none. The lab's unicorns preset draws the pattern with every region empty as the counterexample to the step from All to Some.
Existential import is the repair. The Aristotelian reading assumes every term names something: each circle has at least one occupied region. Under that assumption an A statement, which empties the outside of P, leaves only the overlap to hold the S, so the overlap is occupied and Some S are P follows. Every relation on the square returns with it.
So the reading is a choice with a price on each side. The Boolean reading lets “All trespassers will be prosecuted” be true when there are no trespassers, and a law is not shown false by a quiet year; the price is that the old square collapses to its two diagonals. The Aristotelian reading keeps the whole square; the price is that it cannot say anything universal about what may not exist without also saying that it does.
The lab shows the reading as a switch, with the Boolean one shipped. Its status line always reports the other verdict as well. Take the step from All S are P to Some P are S, traditionally called conversion per accidens. On the Boolean reading it has a counterexample; on the Aristotelian reading it has none, because S is non-empty and every S is a P.
Theorem · What the switch does not change
Contradictories hold on both readings: A with O, and E with I, never agree. An inference that never needed a term to have members, such as converting No S are P, keeps its verdict when the reading changes. Only inferences that move from a universal to a particular claim are affected.
The contradictory preset has All S are P as its premise and Some S are not P as its conclusion. The inference is invalid on both readings. That is what contradiction says: when A is true, O is false, so the counterexample is exactly a case where A holds and O fails. Switch the reading and only the counterexample moves.
Interactive
Categorical logic by regions
Every pattern of empty and occupied regions
Method
Deciding whether an inference needs import
Four steps, one of them a comparison.
1
Run it on the Boolean reading
Find whether any pattern satisfies the premise and fails the conclusion, with no assumption that a term has members.
2
Run it with every term non-empty
Require each circle to hold a cross somewhere, and test again. This is the only change the switch makes.
3
Compare the two verdicts
If they agree, the inference does not depend on import. If it is invalid on the first and valid on the second, import is the extra premise doing the work.
4
State the assumption in the conclusion
Say, “assuming there are S”, and mean it. An inference that needs import is sound only for terms that name something.
Reading the square
Score 0 / 0Choose the answer you can justify.
Trespassers and unicorns
“All unicorns are white. So some unicorns are white.” Take the premise as a region claim and run the step on both readings.
premise unicorns outside white: empty
Boolean every region empty satisfies it
conclusion unicorns inside white: occupied
Boolean fails there: invalid
Aristotelian unicorns exist: the overlap is occupied
On the Boolean reading the pattern with nothing anywhere is allowed by the premise and fails the conclusion. Require the unicorns to exist and that pattern is gone: the only place left for them is inside the white region. The step from All to Some is a good inference about trespassers and a bad one about unicorns, and the logic cannot tell which a term is. That is the cost of having a switch, and the lab's status line shows it.
Common mistakes
1
Thinking “All S are P” implies “Some S are P”It does only if some S exists. On the Boolean reading the lab finds the pattern with every region empty: the premise is true, the conclusion false. The step feels safe because the examples we reach for (dogs, squares) name things that exist, and the logic has no way to know that.
2
Treating the Boolean reading as the claim that there are no such thingsThe Boolean reading does not say that unicorns are absent. It declines to assume they are present. The empty pattern is one the premise allows among many, and an argument is valid only if the conclusion survives all of them.
3
Supposing the switch changes every verdictContradictories hold on both readings. Converting No S are P is valid on both. Only the steps from a universal claim to a particular one move. Check the status line, which always reports the verdict on the reading you are not using.
Check your understanding
✓
Finish when you can say which side of the switch an inference sits on.
Given an inference from a categorical premise, state its verdict on the Boolean and on the Aristotelian reading, say whether it needs existential import, and say which relations on the square hold on each reading.
Note: Modern logic took the Boolean reading and rebuilt the particular statements as explicit existence claims, which is the move made in “Quantifiers and Their Order”. The older reading is not an error; it is a theory of what ordinary general statements presuppose.