Dating back *at least* as far as Aristotle in the 4th Century BC, the **Square of Opposition** is a visual framework for representing the logical connections and relationships between four basic categorical propositions:

**A:**All S are P**E:**No S is P**I:**Some S are P**O**: Some S are not P

These propositions express information about the classes or categories of objects or individuals. The square of opposition below represents how these propositions relate to each other:

Note: This is the *traditional* square of opposition. These relationships don’t necessarily hold in the *modern* square of opposition.

## 1. Contradictory Pairs:

**A proposition** (All S are P) and **O proposition** (Some S are not P) are *contradictories*, as are **E proposition** and **I proposition**.

When propositions are contradictory, the truth of one proposition implies the falsity of the other **and vice versa**. In other words, if one is true the other must be false, and if one is false the other must be true.

For example:

**A:**All cats are mammals**O:**Some cats are not mammals

If all cats are mammals (A proposition) is true, then it must be false that some cats are not mammals (O proposition). If one proposition is true, the other must be false.

And vice versa: If all cats are mammals (A proposition) *were* false, then it would be true that some cats are not mammals (O proposition).

Another example:

**E:**No mammals can fly**I:**Some mammals can fly

Again, these propositions directly contradict each other: If no mammals can fly (E proposition), then it must be false that some mammals can fly (I proposition) and vice versa.

## 2. Contrary Pairs:

**A proposition** (All S are P) and **E proposition** (No S are P) are *contraries*.

When propositions are contrary, they **cannot both be true simultaneously**. In other words, if one is true, the other must be false.

However, although it’s impossible for contraries to both be true, **it is possible for both contraries to be false**.

For example:

**A:**All birds can fly**E:**No birds can fly

In this example, *both* statements are false. A claims that all birds can fly, but we know there are flightless birds like penguins and ostriches, which makes this statement false. On the other hand, E asserts that no birds can fly, which is also false.

However, if A proposition *were* true (all birds could fly), then it would be impossible for E proposition to also be true (no birds can fly).

## 3. Subcontrary Pairs:

**I proposition** (Some S are P) and **O proposition** (Some S are not P) are *subcontraries*.

When propositions are subcontrary, they **cannot both be false simultaneously**. In other words, if one is false, the other must be true.

However, although it’s impossible for contraries to both be true, **it is possible for both contraries to be true**.

For example:

**I:**Some cats are friendly**E:**Some cats are not friendly

In this example, *both* statements are true. I claims that some cats are friendly, and some cats are indeed friendly. E claims that some cats are not friendly, and this is also true.

## 4. Subalternation Pairs:

**A proposition** (All S are P) implies the corresponding **I proposition** (Some S are P). Similarly, **E proposition** (No S are P) implies the corresponding **O proposition** (Some S are not P). These pairs are known as *subalterns*.

When propositions are subalterns, the truth of the universal proposition **implies the truth of the specific proposition**.

However, notice that the arrow on the diagram only goes *one way:* The truth of the specific proposition (I or O) **does not** imply the corresponding universal proposition (A or E).

For example:

**A:**All mammals have hair**I:**Some mammals have hair

In this example, if we accept the A proposition that all mammals have hair, the I proposition that some mammals have hair logically follows. So, the truth of the universal A proposition implies the corresponding particular I proposition.

However, all mammals have hair (A proposition) does not logically follow from some mammals have hair (I proposition).

Another example:

**E:**No birds can swim**O:**Some birds can not swim

Again, the truth of the universal E proposition implies the truth of the specific O proposition: If it’s true that no birds can swim, then it’s true that some birds can’t swim.

And again, it doesn’t work the other way though: Just because some birds can’t swim (O proposition), it doesn’t necessarily mean no birds can swim (E proposition).

## The modern vs. traditional square of opposition

The difference between the *traditional* and *modern* squares of opposition lies in the treatment of **existential import**. The traditional square *assumes* the existence of the subject class in universal statements, while the modern square does not.

#### The traditional square: Universal import

So, in the *traditional* square of opposition, when we say “All S are P,” it is implied **that there are actually some S that exist**. If no S exist, then the statement “All S are P” would be considered false.

For example:

**A**: All unicorns are white- This statement would be
*false,*because unicorns don’t exist

- This statement would be
**E**: No unicorns are white- This statement would also be
*false*, because unicorns don’t exist

- This statement would also be

The *problem* with the traditional square of opposition is that **the relationships it describes don’t hold when the S is empty**.

For example, the traditional square says that if A is true, then O must be false because they are contradictories. But this doesn’t work with unicorns: All unicorns are white (A) is false, and some unicorns are not white (O) is *also* false.

#### The modern square: *No* universal import

However, the *modern* square of opposition **removes the assumption of universal import**. Instead, it adopts a more abstract approach where universal statements do not imply the existence of members of the subject class. This means that “All S are P” can be true even if there are no S that exist, because it’s considered a *vacuously true* statement.

For example:

**A**: All unicorns are white- This statement is treated as
*true*– even though unicorns don’t exist – because there are no unicorns that disprove this claim

- This statement is treated as
**E**: No unicorns are white- This statement would also be treated as
*true*– even though unicorns don’t exist – because, again, there are no unicorns that disprove this claim

- This statement would also be treated as

However, although *universal* statements (A and E) do not have existential import in the modern square, *particular* statements (I and O) *do*.

A consequence of this is that the **I proposition** no longer follows from the **A proposition** according to the modern square of opposition.

For example:

**A:**All unicorns are white- This statement would be considered vacuously true, but it does not follow that:

**I:**Some unicorns are white- Because this statement implies that there is at least one unicorn
*that exists*and is white.

- Because this statement implies that there is at least one unicorn