The Names of Syllogisms: A Guide
Part of what prompted me to write this was Lewis Powell being very cheeky about how historical philosophers named their syllogisms. To be fair, I was cheeky first to contemporary philosophers, but they deserve every bit of it. So this impudence will not stand. The names of syllogisms are no joke! They are mnemonically useful. If you know how to read them, they tell you how to construct all the valid syllogistic moods.
This isn’t how the names are normally used; normally they show you how to reduce moods to the first figure moods, which is part of how Aristotle proved them. They’re records of proofs. But the proofs can be turned around into recipes for constructing all the moods from the first-figure moods. Thus if you’re given the name of a mood, and you know the first-figure moods, you can work out what the precise form is.
The forms of propositions
Traditional syllogistic deals with arguments that have two premises and one conclusion. The premises are quantified propositions that come in four forms, two affirmative forms (one universal and one particular), and two negative forms (again one universal and one particular). A form of a proposition is something like ‘Every S is P’, where you can make propositions by replacing ‘S’ and ‘P’ with words. What ‘universal’, ‘particular’, ‘affirmative’, and ‘negative’ mean can be gathered from the examples at the end of the next section.
Propositional forms are assigned letters: the affirmative ones are called ‘a’ and ‘i’ (from the first two vowels of the Latin affirmo — I affirm), and the negative ones ‘e’ and ‘o’ (from the first two vowels of the Latin nego — I deny).
So there’s the first handy mnemonic: the universal form of a proposition is the first vowel of affirmo or nego, depending on whether you’re affirming or denying, and the particular form is the second vowel.
The mnemonic verses sometimes given in medieval logic textbooks to help students remember this seem superfluous. You just need to remember the two Latin words and the rule first vowel, universal; second vowel, particular. That said, once you get those verses in your head they stick, uselessly, so I won’t repeat them here.
The propositional forms are then these:
a: Every S is (a) P/ P belongs to all S (e.g. every animal is mortal).
e: No S is (a) P / P belongs to no S (e.g. no angel is mortal)
i: Some S is (a) P / P belongs to some S (e.g. some animal is a man)
o: Some S is not (a) P / Some S is not a P (e.g. some animal is not a man)
The moods of syllogisms
The mood of a syllogism is determined by the forms of its three propositions. The names given to the syllogisms (‘Barbara’, e.g.) contain three vowels, telling you the form of the two premises and the conclusion. So Barbara is a syllogism whose two premises and conclusion are all a-propositions, e.g. ‘Every swan is a bird; every bird is an animal; therefore every swan is an animal’.
(Many online ‘logic guides’ get no further than Barbara in explaining what a valid argument is; from this one example you’re supposed to intuit what syllogistic validity is. This takes ‘showing not saying’ to an unhealthy extreme.)
Notice that the name of a syllogistic mood doesn’t tell you the mood; it only tell you something about the mood, viz. the forms of the propositions it involves. The syllogism ‘every swan is a bird; every pigeon is a bird; therefore every swan is a pigeon’ is also composed of three a-propositions. But it isn’t Barbara; it isn’t a valid mood at all (you can see that the example-argument is fallacious).
The so-called ‘first figure’ moods are Barbara, Celarent, Darii, Ferio. Using ‘SaP’ for ‘Every S is P’, SiP for ‘Some S is P’, etc., these moods are:
Barbara: MaP, SaM ∴ SaP
Celarent: MeP, SaM ∴ SeP
Darii: MaP, SiM ∴ SiP
Ferio: MeP, SiM ∴ SoP
You can see that they all have the form M_P, S_M ∴ S_P, with the blanks to be filled by the three vowels of the mood’s name. So if you remember this and the names, you can form these four first-figure syllogisms.
The other moods
In the names of the other moods, the consonants help you. Take, for instance, the third-figure syllogism Disamis. We know that this has an i-proposition and an a-proposition for premises and an i-proposition for a conclusion.
Technically, the consonants are supposed to tell you how to convert the mood to a first-figure mood. But by reversing the rules, we can show how to get to the mood from a first-figure mood.
First let’s deal with the ‘m’. This is about swapping the premises around, but it can get a bit tricky because the other consonants tell us what to do with the premises. My suggestion is to apply the ‘m’ rule by swapping the first two syllables around in the word. Just keep the first letter as the first letter and eliminate the ‘m’ now that its purpose has been served. Thus Disamis becomes Daisis.
Next, the first letter tells us which first-figure mood we’re beginning with; in this case, Darii, which is the first-figure mood beginning with ‘D’.
Then the ‘s’ tells us to convert the premise of that syllable . In this case it ends the second syllable, so we convert the second premise. And ‘s’ tells us to convert it simpliciter, which means just reversing the terms. So from Darii — MaP, SiM ∴ SiP — we get MaP, MiS ∴ SiP.
We then apply the next ‘s’, which converts the conclusion. So now we have: MaP, MiS ∴ PiS.
All the traditional forms, however, have a conclusion of the form S_P. So we need to swap around the S and the P, resulting in: MaS, MiP ∴ SiP. This is the form of Disamis.
The order of the vowels doesn’t match the order in the name, but that’s because our rules have ended up swapping the premises around. It doesn’t affect validity, but if you like you can swap the premises back around: MiP, MaS ∴ SiP. For example: ‘Some syllogisms are fun, every syllogism is an inference, therefore some inferences are fun.’
A more complicated case
The consonant ‘p’ tells us that, in reducing a mood to a first-figure mood, we apply conversion ‘per accidens’ to that syllable’s proposition. This means moving from a universal to a particular and swapping the terms: from SaP to PiS or from SeP to PoS. Using the rules backwards to go from first-figure moods to others, as we’re doing here, you have to do the conversion in reverse.
So now take Felapto.
The first-figure mood beginning with ‘F’ is Ferio: MeP, SiM ∴ SoP
The ‘p’ tells us to reverse-convert the second premise, SiM, per accidens: thus SiM -> MaS.
And now we know that Felapto is MeP, MaS ∴ SoP, for instance ‘No syllogisms are fun, some inferences are syllogisms, therefore some inferences are not fun’.
The ‘c’ rule
There is a more complicated case again: the consonant ‘c’ (in some versions of the name it’s a ‘k’, or sometimes it’s a ‘ch’). The rule here, the right way around, is complex. Where you have, e.g., Bocardo, the rule means that the contradictory of the conclusion entails the contradictory of the premise in which the ‘c’ appears, when combined with the other premise. Entails it how? By way of a syllogism in the first-figure mood corresponding to the first letter; thus Barbara in this case.
I haven’t explained contradictories, but you can just think of them as negations. If it is not the case that every S is P, it must be that some is not P, thus the contradictory of SaP is SoP. And if it’s not the case that no S is P, it must be that some S is P, so SiP is the contradictory of SeP.
To explain the ‘c’ rule, let me first just tell you what Bocardo is: MoP, MaS ∴ SoP. Now the ‘c’ rule works like this to ‘reduce’ it to first-figure Barbara. First, suppose the contradictory of the conclusion, which is SaP. From this and the second premise, MaS, we have an instance of Barbara: SaP, MaS ∴ MaP (this is an instance of MaP, SaM ∴ SaP, since you can just put in ‘M’ for ‘S’ and ‘S’ for ‘M’). But MaP is the contradictory of the first premise of Bocardo. So the denial of Bocardo’s conclusion contradicts is premise, and so the conclusion must follow from the premise. It hasn’t been reduced to Barbara, but Barbara has been used to prove it.
Did you follow all that? Well the good news is, it doesn’t matter. When we’re running the rules the other way it’s much simpler. Where you have a ‘c’, the rule is simply: turn the premise corresponding to that syllable into its contradictory, turn the conclusion into its contradictory, and swap the terms in the other premise.
Thus to get from Barbara to Bocardo. Start with Barbara: MaP, SaM ∴ SaP. Turn the first premise into its contradictory, MoP. Turn the conclusion into its contradictory, SoP. Swap the terms in the other premise to get MaS. And voilà! You’ve got Bocardo: MoP, MaS ∴ SoP. To make Baroco from Barbara you just contradict the conclusion again, and this time the second premise, then reverse the terms of the first premise, so Baroco is PaM, SoM ∴ SoP.
The Indirect Forms
Some interesting exercises can be done with the so-called ‘indirect’ first-figure moods.
Fapesmo:
First apply the ‘m’, so we have Fesapo.
Then start with Ferio: MeP, SiM ∴ SoP.
To deal with the ‘s’, convert the first premise simpliciter to get PeM, SiM ∴ SoP
To deal with the ‘p’, reverse-convert per accidens the second premise to get PeM, MaS ∴ SoP.
And that’s Fapesmo, though again with the premises reversed so that the vowels don’t match.
Baralipton:
Start with Barbara: MaP, SaM ∴ SaP.
We only have one rule to apply — the ‘p’ rule, applied to the conclusion. Thus we have MaP, SaM ∴ PiS.
But again, to make this into the standard form, we have to reverse P and S, to get: MaS, PaM ∴ SiP.
Thus the names of the syllogisms work as mnemonic devices, not just for showing how to reduce other moods to the first-figure (and thus to take a step towards proving them), but also to get to other moods from the first-figure moods. I think we can admire the intricacy and ingenuity of this.
