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).

Fourth figure

Following Aristotle, the scholastic philosophers recognised only three figures. This is because the first figure is defined as 'the middle term is the subject in one premiss and the predicate in the other'. Using this definition, there are two possible orderings for the first figure,

M-P/S-M/S-P

P-M/M-S/S-P

The scholastic writers discussed these, but regarded the second as equivalent to the first, but with the terms of the conclusion transposed. For the premisses of the second ordering can be transposed without change of meaning to give M-S/P-M/S-P, and then relabelled according to the definition of the major term as the one occurring with the middle in the first premiss, as follows:

M-P/S-M/P-S

This is the schema for the indirect moods of the first figure, i.e. where the major term is the subject of the conclusion (not the predicate). As Buridan argues Aristotle did not bother about the fourth figure, because it merely differs by the transposition of premisses. "So it seems there is little to say about the fourth figure, as there are as many moods concluding directly in the first figure, as there are moods concluding indirectly in the fourth, differing only from those of the first by the transposition of premisses".

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.