Moods & Figures of Syllogisms

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Figures & Moods of Syllogisms

A categorical syllogism’s “structure” can be captured by its figure and mood.
— Its figure concerns the placement of the middle term in the two premises.
— Its mood concerns what type (A, I, E, or O) each of its propositions are.

The Medieval Mnemonic:
Barbara, Celarent, Dari, and Ferio.
Cesare, Camestres, Baroco, and Festino.
Darapti, Datisi, Disamis, Bocardo, Felapton, and Ferison.
Bramantip, Camenes, Dimaris, Fesapo, and Fresison.

Four Figures

Either the middle term will appear as the subject or the predicate (i.e., there are two possibilities here). There are two premises. Thus, there are four possible figures.

In the diagram above, e.g., notice that Figure 1 has it that the middle term appears as the subject in the first premise and then as the predicate in the second premise.

Aristotle believed that Figure 1 was the most “perfect” because it appeared to him most intuitively evident how a universal conclusion can be inferred in that figure. Figure 4 Aristotle didn’t treat! It’s often thought to be the most unintuitive of the figures.

Moods

Mood deals with the quantity and quality of the three propositions. When thinking in terms of the two premises, each premise can take on four possibilities (A, I, E, or O). Therefore, in this selection, there are 16 possibilities (or permutations).

Not all of those possibilities can produce a valid syllogism, however!

When we consider the conclusion, it too can take on four possibilities. Therefore, there are a total of 64 possible moods. (But, again, not all of them are valid.)

Mood – Figure

It’s customary to write a syllogism by its mood and then figure.
For example, AAA-1 means the syllogism has the mood AAA in figure 1.

“All men are mortal. Socrates is a man. Therefore, Socrates is mortal.”
— This is AAA-1.
“No Catholic is an atheist. Mike is an atheist. Therefore, Mike is not Catholic.”
— This is EAE-2.

Considering all possible arrangements of figure and mood, there is a total of 256 (since there are 64 possible moods and four possible figures). Most are actually invalid: 232 of those possibilities are invalid with only 24 being valid.

Valid Moods

Let’s consider the basic rules of the categorical syllogism. Some moods are guaranteed to produce only invalid conclusions. And, given the moods of two premises, only one or two valid conclusions are possible with its respective mood.

Consider quality. Two negative premises cannot produce any valid conclusion. Therefore, we can rule out: EE, EO, OE, and OO. Consider quantity. At least one universal premise is required. Therefore, we can rule out: II, IO, OI, and OO.

Valid moods will always (1) have a conclusion that is no more universal in scope than the weakest premise and (2) have a conclusion that is negative when there is a negative premise. For example, with premises AI, a valid conclusion must be an I (and so AII is valid). There’s no way AIA is valid!

What about AA as premises? The most natural conclusion that is valid will be A. We can technically get an I conclusion, too, but why weaken the conclusion? Likewise, AE can get us an E conclusion; though we can also get an O conclusion, O weakens the conclusion. Generally speaking, we want the conclusion to be as “strong” as possible. So, if we can derive a universal conclusion, derive that and not something particular.

The Medieval Mnemonic
And “Perfect” versus “Imperfect” Figures

The mathematician and logician Augustus De Morgan might have said: “They [these verses] are magic words, more full of meaning than any that were ever made.”

Within these mnemonics you’ll find (1) the valid moods of figure 1, figure 2, figure 3, and figure 4; and (2) guidelines to transform an “imperfect” syllogism to the first figure.

Vowels represent moods!

Barbara is in Figure 1. It is an AAA–1 syllogism.
Darapti is in Figure 3. It is an AAI–3 syllogism.

It’s often been thought that Barbara is a “perfect syllogism.” The middle term’s role is most evident in it. Its extension is greater than the minor term and less than the major term. It’s thereby a perfect “bridge” between the major and minor. Barbara is also “perfect” because we get a very strong conclusion, i.e., a universal conclusion that’s affirmative.

“Strong” versus “Weak” Conclusions . . .

“Strong” ≈ universal conclusion
“Weak” ≈ particular conclusion

“Strong” ≈ affirmative conclusion
“Weak” ≈ negative conclusion

This implies that the “weakest” Figure 1 syllogism is Ferio.

Figure 1 not only has a “perfect syllogism,” it can give us conclusions for all types of categorical propositions (A, I, E, and O). Thus, it has been called the most “perfect” figure.

Note that the mnemonic for Figure 1 only has four words. This doesn’t mean there are only four valid moods in this figure. That’s because the AAA-1 implies AAI-1. Similarly, EAE-1 implies EAO-1. But, remember, we generally want the conclusion to be as “strong” as possible. Why give an AAI-1 argument when you can give the stronger AAA-1?

Figure 2 is “weaker” than Figure 1 because it can only give us negative conclusions: Cesare, Camestres, Baroco, and Festino. The middle term, as Figure 2 implies, always shows up as the predicate in the premises. Note that EAE-2 implies EAO-2 and that AEE-2 implies AEO-2.

Figure 3 is also “weaker” than Figure 1 because it only yields particular conclusions: Darapti, Datisi, Disamis, Bocardo, Felapton, and Ferison.

Figure 4 is the black sheep! It’s not covered in Aristotle’s Organon. Some traditional logicians have gone so far as to reject using it in practice because it is so “awkward.” This is not to say these syllogisms are invalid, just that they are usually more convoluted.

Here are the valid ones: Bramantip, Camenes, Dimaris, Fesapo, and Fresison.
Also note, given Camenes, AEE-4 implies AEO-4.

Reduction of “Imperfect” Syllogisms to the First Figure

There’s another reason Figure 1 can be viewed as “perfect.” This is because, Aristotle said, “the first figure has no need of the others.” We generally can “reduce” a syllogism to Figure 1 (or can engage in a reduction ad absurdum proof).

The Medieval Mnemonic. . .
Consonants represent how to reduce!

Figure 1: Barbara, Celarent, Dari, and Ferio.
Figure 2: Cesare, Camestres, Baroco, and Festino.
Figure 3: Darapti, Datisi, Disamis, Bocardo, Felapton, and Ferison.
Figure 4: Bramantip, Camenes, Dimaris, Fesapo, and Fresison.

B – reduce form through contradiction to a Barbara syllogism
C – reduce form to a Celarent syllogism
D – reduce form to a Darii syllogism
F – reduce form to a Ferio syllogism

M – the vowel before M should be muted
{premise indicated by the vowel should be switched with the other premise}
P – the vowel before P should be converted accidentally (per accidens)
{premise indicated by the vowel should undergo conversion by limitation}
S – the vowel before S should be converted simply
{premise indicated by the vowel should undergo conversion}

C as it appears in Baroco and Bocardo – premise before C should be removed and replaced with the contradiction of the conclusion

Direct or Ostensive Reductions. . .

Cesare Syllogism Example. . .
The mood is EAE. It’s in Figure 2.
Premise 1: PEM (No P are M.)
Premise 2: SAM (All S are M.)
Conclusion: SEP (No S are P.)

Goal: reduce syllogism to Figure 1.
Notice the C and S in “Cesare .”
C indicates we will reduce this syllogism to a Celarent syllogism.
S follows the vowel e in “Cesare .” So, PEM should be simply converted.

Disamis Syllogism Example. . .
The mood is IAI. It’s in Figure 3.
Premise 1: MIP (Some M are P.)
Premise 2: MAS (All M are S.)
Conclusion: SIP (Some S are P.)

Goal: reduce syllogism to Figure 1.
Notice the D, S, M, and S in “Disamis .”
D indicates we will reduce this syllogism to a Darii syllogism.
S follows the vowel i in “Disamis .” So, MIP should be simply converted.
M follows the vowel a in “Disamis .” So, MAS should be muted.
S follows the vowel i in “Disamis .” So, SIP should be simply converted.

Indirect Reductions. . .

Aristotle employed an “indirect proof,” which is also known as a reductio ad absurdum proof, for some of the “imperfect” syllogisms. Mathematicians to this day often employ indirect proofs. They show something false by assuming it as true and then derive a contradiction from that assumption. Since the assumption leads to a logical absurdity, the assumption must be false.

A Baroco syllogism, unless we resort to obversion, cannot be directly reduced.
(1) All P are M.
(2) Some S are not M.
(3) Thus, some S are not P.

Following Aristotle, we can do an indirect proof.
First, contradict the conclusion.
This gives us “all S are P.”
Second, form a Barbara syllogism.
We’ll do so with line one and the contradiction of line three.

Accordingly, we now have. . .
(1′) All P are M.
(2′) All S are P.
(3′) All S are M.

A contradiction has been reached! Insofar as the original syllogism had true premises, it must be the case that it actually does have a true conclusion. Why? Because (3’) contradicts (2). However, by hypothesis, (2) is true and likewise (1) and (1’) are true; ergo, the premise (2’) must be false.

Additionally, insofar as we know a Barbara syllogism is valid, it must be, based on this indirect reasoning, that the conclusion of a Baroco syllogism is valid.

It turns out, we can resort to a direct reduction with a Baroco. . .

Aristotle deals with a Baroco syllogism and a Bocardo syllogism indirectly.
We can deal with them directly, with the mnemonics Faksoko and Doksamosk.

Notice the ks. This indicates that the premise must be obverted and then converted.
Notice the k. This indicates that the premise must be obverted.

With a Baroco, we have. . .
(1) All P are M.
(2) Some S are not M.
(3) Thus, some S are not P.

We’ll directly reduce to Figure 1 with Faksoko .

“aks” indicates to obvert and then convert (1).
— PAM is obverted to PEM’ and that is converted to M’EP.
“ok” indicates to obvert (2).
— SOM is obverted to SIM’.

So, now we have the Ferio syllogism. . .
(1) No non-M are P.
(2) Some S are non-M.
(3) Thus, some S are not P.