Syllogism

In Aristotelian and scholastic logic, a syllogism is defined as a form of discourse in which certain things – called premisses – are postulated, and another thing – a conclusion – follows from them.

The definition is from Aristotle's Prior Analytics. It applies to forms of argument other than syllogisms, however Aristotle confines his discussion to so-called categorical syllogisms, i.e. argument consisting of three categorical propositions, the first two of which are the premisses, the third of which is the conclusion. The first is called the major premiss, the second the minor premiss.

It does not matter whether the premisses are true or false. The conclusion is said to follow from the premisses (or to follow 'by necessity') if the premisses can never be true when the conclusion is false.

Structure of the syllogism

A categorical syllogism is an argument consisting of three categorical propositions, the first two of which are called premisses, the third of which is called a conclusion. For example

Every animal is a creature

Every giraffe is an animal

Every giraffe is a creature

In scholastic logic, the first proposition is called the major premiss, the second, the minor premiss. Every syllogism has three terms, namely the major term (P), the minor term (S), and the middle term (M). The middle term is the one that appears in both premisses (e.g. 'an animal'). The major term is the one taken with the middle in the major premiss ('a creature '), the minor term is the one taken with the middle in the minor premiss ('a giraffe'). The syllogism is called direct if the major term is predicated of the minor term in the conclusion. If the other way round, it is called indirect. Thus the middle term never occurs in the conclusion.

The sixth century Greek commentator John Philoponus had another definition of 'major term' as the one that is the predicate in the conclusion, and of the minor term as the one that is the subject in the conclusion. This definition was adopted after the medieval period, alongside the adoption of a fourth figure (see below).

There are other forms of argument which conform to Aristotle's definition, but which are not technically syllogisms. The following argument has only one premiss, but is valid, since the premiss cannot be true with the conclusion false.

It is day

Therefore, it is day

And the following argument, although it has two premisses and a conclusion, does not consist of categorical propositions:

If the sun is out, then it is day

The sun is out

Therefore, it is day

Figure and mood

A syllogism has figures and moods. The figure is defined by the arrangement of terms in the propositions. If the middle term is the subject in one premiss and the predicate in the other, then the syllogism is in the first figure. If the middle term is predicate in both premisses, the syllogism is in the second figure. If it is subject in both premisses, the syllogism is in the third figure. There cannot be any other figures, because three terms in two premisses cannot be varied in any other way, using the definitions of major and minor premiss given in the introduction (but see below on the 'fourth figure').

The mood of a syllogism is the arrangement of the categorical forms (A, E, I, O) of the three propositions. Given that there are four possible forms of each of the three propositions, it follows by elementary mathematics that there are 64 possible moods. However, few of these are valid (see below).

The medieval writers encoded the moods using the codes for the forms of each of the three propositions. For example, the first figure syllogism given in the introduction

Every animal is a creature (universal affirmative = A)

Every giraffe is an animal (universal affirmative = A)

Every giraffe is a creature (universal affirmative = A)

consists of three universal affirmative propositions. The form of the universal affirmative has the code 'A', thus the mood is encoded as 'AAA'. Similarly the second figure syllogism

No stone is an animal (universal negative = E)

A giraffe is an animal (universal affirmative = A)

No giraffe is a stone (universal negative = E)

has the code 'EAE'. The scholastic textbook writers gave mnemonic names to the moods, whose vowels were the codes. For example, the AAA first figure mood was called Barbara, the second figure EAE Cesare (see the verses below).

Moods of the first figure

The first figure has nine moods, as follows.

(1) Barbara consists of two universal affirmatives and a universal affirmative conclusion. Example: "every animal is a substance, every man is an animal, therefore every man is a substance".

(2) Celarent consists of a universal negative, a universal affirmative, and a universal negative conclusion. Thus "no animal is a stone, every man is an animal, no man is a stone".

(3) Darii consists of a universal affirmative, a particular affirmative, and a particular affirmative conclusion. Thus "every man is a substance, some animal is a man, some animal is a substance".

(4) Ferio consists of a universal negative, a particular affirmative, and a particular negative conclusion. Thus "no man is an ass, some animal is a man, some animal is not an ass".

(5) Baralipton consists of two universal affirmatives and a particular affirmative conclusion. Thus "every animal is a substance, every man is an animal, some substance is a man". It can be reduced to the first mood (Barbara), because its premisses lead to a conclusion ("every man is a substance" which implies the stated conclusion of this mood ("some substance is a man").

(6) Celantes consists of a universal negative, a universal affirmative, and a universal negative conclusion. Thus "no animal is a stone, every man is an animal, no stone is a man". This mood can be reduced to the second (Celaraent) by simple conversion of the conclusion to " no man is a stone".

(7) Dabitis consists of a universal affirmative, a particular affirmative, and a particular affirmative conclusion. Thus "every man is an animal, some substance is a man, some animal is a substance". This reduces to the third mood (Darii) by simple conversion of the conclusion.

(8) Fapesmo consists of a universal affirmative, a universal negative, and a particular negative conclusion. Thus "every man is an animal, no stone is a man, some animal is not a stone". This reduces to the fourth (Ferio) by conversion of the major by limitation, the simple conversion of the minor, and the transposition of the premisses.

(9) Frisesomorum consists of a particular affirmative, a universal negative, and a particular negative conclusion. Thus "some lion is an animal, no man is a lion, some animal is not a man". This reduces to the fourth (Ferio) by simple conversion of the premisses (some animal is a lion, no lion is a man), then transposing them (no lion is a man, some animal is a lion, ergo etc)

Moods of the second figure

The second figure has four moods.

(1) Cesare consists of a universal negative, a universal affirmative, and a universal negative conclusion. Thus "no man is a stone, every pearl is a stone, no pearl is a man". It reduces to the second mood of the first figure (Celarent) by conversion of the major.

(2) Camestres consists of a universal affirmative, a universal negative, and a universal negative conclusion. Thus "every pearl is a stone, no man is a stone, no man is a pearl". It reduces to the second mood of the first figure (Celarent) by conversion of the conclusion and of the minor proposition and transposition of the premisses.

(3) Festino consists of a universal negative, a particular affirmative, and a particular negative conclusion. Thus "no man is a stone, some pearl is a stone, some pearl is not a man". It reduces to the fourth mood of the first figure (Ferio) by conversion of the major.

(4) Baroco consists of a universal affirmative, a particular negative, and a particular negative conclusion. Thus "every pearl is a stone, some man is not a stone, some man is not a pearl". It reduces to the first mood of the first figure (Barbara) per impossibile, i.e. by taking one of the premisses with the (contradictory) opposite of the conclusion and conclude the opposite of the other premiss. I.e. take the opposite of the conclusion ("every man is a pearl") with the major ("every pearl is a stone"), to give the syllogism "every pearl is a stone, every man is a pearl, every man is a stone".

Moods of the third figure

The third figure has six moods.

(1) Darapti consists of two universal affirmatives, and a particular affirmative conclusion. Thus "every man is an animal, every man is a substance, some substance is an animal". It reduces to the third mood of the first figure (Darii) by conversion of the minor.

(2) Felapton consists of a universal negative, a universal affirmative, and a particular negative conclusion. Thus "no man is an ass, every man is an animal, some animal is not an ass". It reduces to the fourth mood of the first figure (Ferio) by conversion of the minor.

(3) Disamis consists of a particular affirmative, a universal affirmative, and a particular affirmative conclusion. Thus "some man is an animal, every man is a substance, some substance is an animal". It reduces to the third mood of the first figure (Darii) by conversion of the major and of the conclusion, then the transposition of the premisses.

(4) Datisi consists of a universal affirmative, a particular affirmative, and a particular affirmative conclusion. Thus "every man is an animal, some man is a substance, some substance is an animal". It reduces to the third mood of the first figure (Darii) by conversion of the minor.

(5) Bocardo consists of a particular negative, a universal affirmative, and a particular negative conclusion. Thus "some animal is not a man, every animal is a substance, some substance is not a man". It reduces to the first mood of the first figure (Barbara) per impossibile, i.e. taking the contradictory opposite of the major, where the opposite of the conclusion is taken in place of the major.

(6) Ferison consists of a universal negative, a particular affirmative, and a a particular negative conclusion. Thus "no man is an ass, some man is an animal, some animal is not an ass". It reduces to the fourth mood of the first figure (Ferio) by conversion of the minor.

Fourth figure

Following Aristotle, the scholastic philosophers recognised only three figures. As Ockham explains, the first is in which the middle term is the subject in the first premiss, and the predicate in the second. The second figure is when the middle term is predicated in both. The third figure is when the middle term is the subject in both. There is no fourth figure to be added, “for if the middle term is the predicate in the first proposition and the subject in the second, it will only be a transposition of the propositions given in the first figure, and therefore there will not follow any other conclusion than that which follows from the premisses arranged in the first figure”.

However, this is only true on the scholastic definition of major and minor. Writers from the seventeenth century followed the definition of John Philoponus, who argued that the major term should be defined as the predicate of the conclusion.

It is possible to define this [the major term] both in a way common to all the figures and in a way peculiar to the first figure. By the definition which is peculiar to the first figure, the major term is that which is predicated of the middle, and the minor that which is subject to the middle. But since in neither of the other two figures have the extremes a different position in relation to the middle, it is clear that this definition will not apply to them. We must therefore apply to the figures the common rule that the major term is the predicate of the conclusion’.

Under this definition there can be no indirect moods, which are instead capturing by distinguishing the ordering M-P/S-M/S-P as unique to the first figure, and P-M/M-S/S-P as unique to the fourth figure. This usage has not been traced back further than the Institutio logicae of John Wallis (Oxford 1687), and was popularised in the Artis logicae rudimenta of Henry Aldrich (Oxford 1691).

When the fourth figure moods were regarded as indirect moods of the first figure, they had different names: Baralipton (or Bramantip), Celantes, Dabitis, Fapesmo, Frisesomorum.

Subalternate moods

Apart from the 19 valid moods mentioned above, there are others which can be derived by 'subalternating' the conclusion. Thus, from the premisses of a valid mood with an A-form conclusion, we can also obtain the corresponding I-form conclusion, and so also from a valid mood with an E-form conclusion, we can derive an O-form conclusions. These gives a total of five so-called 'subalternate moods': two in the first figure (Barbari and Celaront), two in the second figure (Camestrop and Cesaro), and one in the fourth figure (Camenop).

Aristotle does not mention them. The first surviving discussion is by Ariston of Alexandria, c. 50 B.C.

Thus there are six valid moods in each of the four figures, giving twenty-four valid moods in all, as follows. Subalternate moods are in italics, indirect moods of the first figure in square brackets.

First figure direct: Barbara, Celarent, Darii, Ferio, Barbari, Celaront.

Second figure: Cesare, Camestres, Festino, Baroco, Cesaro, Camestrop.

Third figure: Darapti, Disamis, Datisi, Felapton, Bocardo, Ferison.

Fourth figure [first figure indirect]: Bramantip [Baralipton], Camenes [Celantes], Dimaris [Dabitis], Fesapo [Fapesmo], Fresison [Frisesomorum], Camenop.

Mnemonic verses

William of Sherwood is well known for a mnemonic poem to help students remember the names of the valid syllogistic forms. Note the different names of the fourth figure moods (in italics) when regarded as indirect moods of the first figure.

Barbara celarent darii ferio baralipton

Celantes dabitis fapesmo frisesomorum;

Cesare campestres festino baroco; darapti

Felapton disamis datisi bocardo ferison

This verse may not have originated with him, but it is the oldest known surviving version. Peter of Spain later gives an account of the verses which is more detailed, and also one which lacks mistakes in William's version. According to Kretzmann, this suggests their source is a single earlier version, now lost.

Aldrich’s version of the verse runs:

Bárbara, Célarént, Darií, Ferióque prióris.

Césare, Cámestrés, Festíno, Baróco secúndae.

Tértia Dáraptí, Disámis, Datísi, Felápton,

Bocárdo, Feríson habét. Quárta ínsuper áddit

Brámantíp, Camenés, Dimáris, Fesápo, Fresíson.

Quinque subalterni, totidem generalibus orti,

Nomen habet nullum, nec, si bene colligis, usum.

Links

• Spade's outline is very good.
• Lagerlund's discussion in SEP
• Britannica
• Aristotle’s Theory of the Assertoric Syllogism Stephen Read, 2017.
• [3] Sara Uckelman's article on the mnemonics.