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24.6 Aristotle’s A E I O

Since Aristotle’s day it’s been common practice to label Aristotle’s forms with the first four vowels of the Latin alphabet: A, E, I and O.

A == All Ps are Q.
I == Some Ps are Q.

A is the first form, “All Ps are Q”, and I is “Some Ps are Q”.

They are both affirmative (they don’t have a negation), and A and I are the two vowels in “affirm”.

A == All Ps are Q.

I == Some Ps are Q.

E == No Ps are Q.
O == Some Ps are not Q.

The other two forms are negative, and E and O are the two vowels in “nego”, Latin for negate.

E == No Ps are Q.

O == Some Ps are not Q.

Given that there are a finite number of Aristotelian forms, there are a finite number of categorical syllogisms we can make out of them. (We will assume that all syllogisms are in standard form.)

There are 256 different categorical syllogisms. Only 15 of them are valid.

As we will explain below, there are 256 different syllogisms we can make out of the Aristotelian forms. What Aristotle did was first identify all the valid ones (there are 15 of them). Then he tried to create rules to explain why they are valid.

Mood: the letters/forms of the sentences in the syllogism, such as AAA or EIO.
Figure: the pattern of the middle term in the premises (1, 2 3, or 4).

Identifying a syllogism has two parts, called the “mood” and “figure”.

  1. Mood: the forms of each premise and the conclusion.
  2. Figure: the pattern of the middle term in the premises.

The mood is a list of three letters, such as EIO. “EIO” means the first (major) premise is form E, the second (minor) premise is I, and the conclusion is O.

For example, let’s use “S” for the minor term (the subject of the conclusion), and “P” for the major term (the predicate of the conclusion. Then here’s the structure of an AAA syllogism:

1. All __ are __
2. All __ are __
Thus:
3. All S are P.

All those those sentences have form A, so this is an AAA syllogism.

Since the first premise is the major premise, it must have term P, the major term. It also needs the middle term, M.

But notice that there are still two ways we could finish filling in the blanks: “All Ps are M” and “All Ms are P”.

The same goes for the second premise: it could be “All S are M” or “All Ms are S”.

How we fill in the blanks with P, S and M determines the “figure” of the syllogism.

How we fill in the blanks with P, S and M determines the “figure” of the syllogism.

There are 4 possible figures. Figure 1 puts the middle terms first and last:

AAA-1
1. All M are P.
2. All S are M.
Thus:
3. All S are P.

Figure 2 puts the major term first and the middle term last:

AAA-2
1. All P are M.
2. All S are M.
Thus:
3. All S are P.

Figure 3 puts the middle term first and subject term last:

AAA-3
1. All M are P.
2. All M are S.
Thus:
3. All S are P.

Figure 4 puts the major term first and the subject term last:

AAA-4
1. All P are M.
2. All M are S.
Thus:
3. All S are P.

Notice that not all four of these syllogisms are valid.

Only once we specify both mood and figure do we have a syllogism that is valid or invalid.

Some moods, like AAA or OAO, are valid in one figure but not others. So only once we specify both mood and figure do we have a syllogism that is valid or invalid.

Now let’s practice these concepts.

There are 15 valid syllogisms.

Now that we can identify the mood and figure of any syllogism, we can go through the list of all possible syllogisms to catalogue which are valid.

There are 15 valid syllogisms:

Figure 1: AAA-1, EAE-1, AII-1, EIO-1

Figure 2: AEE-2, EAE-2, EIO-2, AOO-2

Figure 3: AII-3, IAI-3, OAO-3, EIO-3

Figure 4: AEE-4, IAI-4, EIO-4

You don’t need to memorize this list, but you should be able to figure out whether a syllogism is valid or not. And once you learn how to do formal proofs in FOL, you will be able to prove the validity of all of these!

Notice that the EIO mood is unique: it is only mood valid in all four figures.

Here’s some more practice.