Square of Opposition

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The Traditional Square of Opposition

Above is the Square of Opposition!
— SAP, SIP, SEP, and SOP have the same subject (S) and the same predicate (P).
— These categorical propositions only differ in quality and/or quantity.

Contradictory Propositions…
One must be true, and the other false.

Contrary Propositions…
Both cannot be true, but both may be false.

Subcontrary Propositions…
Both cannot be false, but both may be true.

Subaltern Propositions…
If the universal is true, then the opposing particular is true.
If the particular is false, then the opposing universal is false.

The Square of Opposition is an elegant way to relate all four forms of a categorical proposition with a given subject and predicate. That is, as an A, I, E, and O proposition.

We can take a given proposition and then compare it with three different propositions that only differ with that given proposition by quality and/or quantity.

Maybe it’s best to jump into an example.

We’re given that “all men are mortal” is true.

It’s an A proposition.
We’ll symbolize it as HAM.

What’s the truth, falsehood, or indeterminacy of its “oppositional” propositions?

We can use our “common sense” to verify those results.
The laws of thought guard our “common sense.”

Contradictory Opposition. . .
The truth value of two contradictory propositions will always be opposite!

It’s the “strongest” opposition because there’s a difference in quality and quantity.
So, it’s the opposition between A and O or E and I.

If “all men are mortal” is true, then “some men are not mortal” must be false. Conversely, if “all men are mortal” is false, then “some men are not mortal” must be true. One being false guarantees the other one being true. One being true guarantees the other one being false.

The Law of Contradiction assures us that both propositions cannot be true. And the Law of Excluded Middle assures us that of two contradictory propositions one must be true.

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Contrary Opposition. . .
The truth values of two contrary propositions cannot both be true, but both may be false.
In fact at least one will be false.

It’s an opposition of quality with universal propositions.
So, it’s the opposition between A and E.

The propositions “all men are mortal” and “no men are mortal” cannot both be true. They can both be false, however, as then their contradictories will be true (namely, the O and I propositions: “some men are not mortal” and “some men are mortal”).

If “all men are mortal” is true, then “no men are mortal” must be false. Conversely, if “all men are mortal” is false, then “no men are mortal” is indeterminate. One being true guarantees the other one being false. But only knowing that one is false doesn’t tell us the truth value of the other proposition (since both can be false).

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Sub-contrary Opposition. . .
The truth values of two sub-contrary propositions cannot both be false, but both may be true. In fact at least one will be true.

It’s an opposition of quality with particular propositions.
So, it’s the opposition between I and O.

The propositions “some men are mortal” and “some men are not mortal” may both be true. They cannot both be false, however, as then both of their contradictories would be true (but this is logically impossible because two contrary propositions cannot both be true).

If “some men are not mortal” is false, then “some men are mortal” must be true. Conversely, if “some men are mortal” is true, “some men are not mortal” is indeterminate. One being false guarantees the other one being true. But only knowing that one is true doesn’t tell us the truth value of the other proposition (since both can be true).

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Subaltern Opposition. . .
The truth of a universal implies the truth of its corresponding particular.
The falsity of a particular implies the falsity of its corresponding universal.

It’s the opposition of the universal to the particular of the same quality.
So, it’s the opposition between A and I or E and O.

Truth moves downward (from A to I or from E to O). Falsity moves upward (from I to A or from O to E). However, truth doesn’t move upward and falsity doesn’t move downward.

If “all men are mortal” is true, then “some men are mortal” must be true. Conversely, if “all men are mortal” is false, then “some men are mortal” is indeterminate. Yet if “some men are mortal” is false, then “all men are mortal” must be false.

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Additional Examples. . .

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Oppositional “Inference”

As there are four types of categorical propositions (A, I, E, and O) and each of them can either be true or false, there are a total of eight possible Squares of Opposition. We can consider each one. Our rules can derive the results, and each rule ultimately can be proven with the Laws of Thought (which are self-evidently true).

1) If A is true; then O is false, E is false, and I is true.
2) If A is false; then O is true, E is indeterminate, and I is indeterminate.
3) If E is true; then I is false, A is false, and O is true.
4) If E is false; then I is true, A is indeterminate, and O is indeterminate.
5) If I is true; then E is false, A is indeterminate, and O is indeterminate.
6) If I is false; then E is true, O is true, and A is false.
7) If O is true; then A is false, E is indeterminate, and I is indeterminate.
8) If O is false; then A is true, I is true, and E is false.

Furthermore, we can take a proposition and reach another proposition with these rules.

Example Problem.
Premise: SEP
Conclusion: SOP (by oppositional “inference,” given the subaltern relation)
That is, given the premise SEP is true, it therefore follows SOP is true.

Contradictories and Singular Propositions

Remember, while we can sometimes treat a “singular proposition” as if it were a universal proposition, it is, strictly speaking, neither universal nor particular. A singular proposition concerns a single/specific individual, e.g., “Socrates is a man.” It has no place on the Square of Opposition. Yet it does have a contradictory, namely, “Socrates is not a man.”

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Versus the Boolean viewpoint

A Note on Modern Logic.

Modern logic defines “all” and “some” differently than Aristotelian-Scholastic logic. The result is that all forms of opposition, minus contradictory opposition between A vs. O and E vs. I, become logically undetermined.

Universal propositions are taken as hypothetical (a.k.a. conditional). Particular propositions are taken to claim that the subject really or actually exists.

Contraries, subalterns, and subcontaries are logically undetermined.

Unlike traditional logic, contraries can both be true because A and E are both hypotheticals. They don’t claim the existence of their subject. An “if P, then Q” statement is said to be true even when both its antecedent (P) and consequent (Q) are false. “All S are P” is symbolically translated as “(∀x)(Sx → Px).” This means that “for all x, if x is an S, then x is a P.” “No S are P” is symbolically translated as “(∀x)(Sx → -Px).” This means “for all x, if x is an S, then x is not a P.”

I and O are treated as having “existential import.” The subject exists. “Some S are P” is translated as “(∃x)(Sx & Px).” This means “there exists an x such that x is an S and x is a P.” “Some S are not P” is translated as “(∃x)(Sx & -Px).” This means “there exists an x such that x is an S and x is not a P.”

Subaltern opposition is undetermined because we cannot move from a proposition that doesn’t claim existence to a proposition that does claim existence. Nor can the falsity of the particular necessarily bring falsity to the universal. Although no unicorns exist, e.g., it doesn’t follow that it is false to claim that a unicorn, if it exists, is a being with a horn on its forehead.

Subcontrary opposition is also undetermined because an I and O can both be false. Once again, this is due to the differing definitions. For example, “some unicorn is white” and “some unicorn is not white” can both be false because there are no unicorns that really exist.