Notes: Figure, Mood and the Possible Types of Syllogisms (Part I)

1.      Introduction

Arguments are complex. The syllogism has to be analysed in detailed to examine its complexities. In analyzing syllogisms, the premises are to be taken as true, notwithstanding whether they are actually true or not. 

Standard Form Syllogism 

Recall that:

·         An argument is a set of premises which support some conclusion.

·         Syllogism is an argument consisting of three statements: Two premises and One conclusion.

·         Categorical Syllogism is a syllogism consisting of three categorical propositions, and containing three distinct terms, each of which appears in exactly two of the three propositions.

·         There are three different terms in a categorical syllogism (besides the quantifiers and the copulas):

·         Take the following syllogism:

No politicians are professors

Some doctors are professors

Some doctors are not politicians

o   Major Term: predicate term of the conclusion (no politicians)

o   Minor Term: subject term of the conclusion (some doctors)

o   The conclusion only contains two of the three terms but one of the terms is found only in the premises.

·         Middle Term: term that does not appear in the conclusion (professors)

·         Major premise: premise containing the major term, i.e. predicate of the conclusion (occurs first in a standard syllogism)(no politicians)

·         Minor premise: Premise containing the minor term, i.e.,  subject of the conclusion (some doctors)

·         Standard premise order:

o   In a categorical syllogism, premise 1 usually contains the major term, while premise 2 contains the minor term.

o   Premise 1 is therefore called the major premise, while premise 2 is called the minor premise.

o   The standard form demands that the major premise (i.e., the one containing the major term) always be listed first.

Standard Form	Non-Standard Form
All humans are mortal. Joseph is a human. ∴ Joseph is mortal.	Joseph is a human. All humans are mortal. ∴ Joseph is mortal.

2.      Moods of Categorical Syllogism

Importance of Figure & Mood

·         Tools that will help us to determine when such syllogisms are valid or invalid.

·         Figure and mood together determine the structure of syllogism.

·         All categorical syllogisms have what is called a “mood” and a “figure.”

·         Understanding of the structure of a syllogism (or for that matter a deductive argument) is made much easier if ‘figures and moods’ of syllogism are considered.

·         An analysis of the structure of an argument in deductive inference is a pre-requisite to the classification of arguments into good (valid) and bad (invalid). Since the function of logic is to distinguish arguments in the aforesaid manner, a study of figure and mood occupies an important position in our study of syllogism.

Types of Categorical Propositions

Name	Type of Sentence	Relation Between Classes	English Form or Schema
A	Universal affirmative	Total Inclusion	All S are P
E	Universal negative	Total Exclusion	No S are P
I	Particular affirmative	Partial Inclusion	Some S are P
O	Particular negative	Partial Exclusion	Some S are not P

Mood

·         The mood of a categorical syllogism is a series of three letters corresponding to the type of proposition the major premise, the minor premise, and the conclusion are (A, E, I, or O).

·         When determining the mood of a categorical syllogism, you need to figure out which of the four forms of categorical proposition each line of the argument is (A, E, I, or O).

·         The order in which the three letters occur specifies the mood of the syllogism.

Examples

Mood: AAA
All mammals are creatures that have hair.	A
All dogs are mammals.	A
Therefore, all dogs are creatures that have hair.	A

For instance, in the argument above about dogs, ALL THREE statements are “A” propositions (of the form “All S are P”), so the mood of that argument would be “AAA”.

                                                                                   

Mood: EAO
No women named Deepti are outer island Yapese women.	E
All outer island Yapese women are weavers of the baskets.	A
∴ Some weavers of the baskets are not women named Deepti.	O

                                   

In the above syllogism the minor term (subject of the conclusion) is ‘weavers of the baskets’, the major term (predicate of the conclusion) is ‘women named Deepti’ and the middle term is ‘outer island Yapese women’. Therefore the first premise is the major, second is the minor and the third is the conclusion.

Mood: EIO
No S are P	E
Some S are P	I
Some S are not P	O

Mood:
All rocks are hard things.	A
No rocks are liquid.	E
∴ Some liquid things are not hard.	I

There are 64 Moods. (At this stage, let us not restrict ourselves to valid Moods).

But what is needed is to know how we arrive at this figure because the number is not fixed arbitrarily. There are four kinds of propositions which have to take three positions in such a manner that any proposition can occur in any one of the four different ways; 0, 1, 2 and 3. When we compute all possible arrangements, we arrive at 64.

There are two important aspects. First, we have discovered a certain number of structures in which syllogistic arguments can be constructed, and secondly, which we notice later, not all structures to which arguments subscribe are valid. It is in` this sense that the logical status of an argument is determined by the structure of that particular argument.

Figure

The figure of a categorical syllogism is a number which corresponds to the placement of the two middle terms.

Consider the below argument:

The middle term in the major premise is in the left and the middle term in the minor premise is in the right.

When this happens, argument has Figure 1.

Altogether, there are four possible figures:

·         Figure 1: The middle term is on the left in P1, and on the right in P2.

·         Figure 2: The middle term is on the right in both premises.

·         Figure 3: The middle term is on the left in both premises.

·         Figure 4: The middle term is on the right in P1, and on the left in P2.

In order to memorize the four kinds of figures, picture this “collar flap” image. From left to right, we see the layout of figures 1, 2, 3, and 4.

The Fourth Figure

From the above, it is clear that neither P nor S determines the figure of syllogism.

History has recorded that Aristotle accepted only the first three figures. The origin of the fourth figure is disputed. While Quine said that Theophrastus, a student of Aristotle, invented the fourth figure, Stebbing said that it was Gallen who invented the fourth figure. This dispute is not very significant.

Scientific Figures

Aristotle regarded the first figure as most ‘scientific’. It is likely that by ‘scientific’ he meant ‘satisfactory’. Reasons:

·         Nature of laws of mathematics and physical sciences establish laws in the form of the first figure.

·         Reasoned conclusion or reasoned fact is generally found in the first figure.

·         Aristotle believed that only universal affirmative conclusion can provide complete knowledge and universal affirmative conclusion is possible only in the first figure.

Fundamental Principle of Syllogism

Aristotle quotes the fundamental principle of syllogism: ‘One kind of syllogism serves to prove that A inheres in C by showing that A inheres in B and B in C’.

This principle can be expressed in this form:

Minor: A inheres in B

Major: B inheres in C

A inheres in C

Evidently, this argument satisfies transitive relation. This is made clear with the help of this diagram:

Figure 1

Premise	Particulars and Figure	Symbol
Major Premise	All artists are poets (M)                  (P)	AAP
Minor Premise	All musicians are artists (S)                    (M)	MAA
Conclusion	All musicians are poets (S)                   (P)	MAP

Figure 2

Premise	Particulars and Figure	Symbol
Major Premise	All saints are pious (P)                   (M)	SAP
Minor Premise	No criminals are pious (S)                    (M)	CEP
Conclusion	No criminals are saints (S)                   (P)	CES

Figure 3

Premise	Particulars and Figure	Symbol
Major Premise	All great works are worthy of study. (M)                  (P)	GAW
Minor Premise	All great works are epics. (M)                    (S)	GAE
Conclusion	Some epics are worthy of study. (S)                   (P)	EIW

Figure 4

Premise	Particulars and Figure	Symbol
Major Premise	No soldiers are traitors. (P)                   (M)	SET
Minor Premise	All traitors are sinners (M)                    (S)	TAS
Conclusion	Some sinners are not soldiers. (S)                   (P)	SOS

Figures and Moods: Interrelationship

Knowledge of the ‘figure of syllogism’ permits us to compute the total number of possible moods.

Mood is determined by quality and quantity of propositions, which constitute syllogism. Since there are four figures, in all 256 ways of arranging categorical propositions is possible.

However, out of 256, 245 are invalid by applying the rules and corollaries.

Only eleven valid moods. Even this is not sufficient to have a clear picture.

Six Valid Moods

Within the framework of traditional logic, in any given figure only six moods are valid.

I AAA, AAI, EAE, EAO, EIO and AII

II AEE, AEO, EAE, EAO, EIO and AOO

III AAI, AII, IAI, EAO, EIO and OAO

IV AAI, IAI, AEE, AEO, EAO, and EIO

In all these cases, first letter stands for the major premise, second for the minor and third for the conclusion.

Strengthened, Weakened & Normal Moods

Moods are represented above in three ways.

1.      Moods in italics and bold form are called strengthened moods,

2.      moods in mere italics are called weakened moods;

3.      All other moods are represented in normal form.

Figure	Strengthened Moods	Weakened Moods	Normal
I		AAI EAO	AAA EAE EIO AII
II		AEO EAO	AEE EAE EIO AOO
III	AAI EAO		AII IAI EIO OAO
IV	AAI EAO	AEO	IAI AEE EIO

Strengthened Moods: When the laws of syllogism permit two universal premises to yield logically only particular conclusion, then such moods are called strengthened moods.

e.g., Figure 3: AAI

Premise	Particulars and Figure	Symbol
Major Premise	All great works are worthy of study. (M)                  (P)	GAW
Minor Premise	All great works are epics. (M)                    (S)	GAE
Conclusion	Some epics are worthy of study. (S)                   (P)	EIW

Weakened Moods: If we deduce particular conclusion from two universal premises, even when the laws of syllogism permit two universal premises to yield logically a universal conclusion, then such moods are called weakened moods.

Logicians proved that from universal propositions alone particular proposition cannot be derived and vice versa. Accordingly, both strengthened and weakened moods become invalid. Thus, in the new scheme the number of valid moods reduces to FIFTEEN.

EIO & IEO

EIO is valid in all the figures and conversely IEO is invalid in all the figures.

Difference between EIO and IEO is that the minor and the major premises are only transposed which clearly shows that the position of premises, which is a part of the structure, determines the validity of argument.

Though EIO is valid in more than one figure it is one mood in one figure and some other in another figure.

AEE

Likewise, AEE is valid in the second and the fourth figures. But it is one mood in the second figure and a different mood in the fourth figure.

First Figure: The Perfect One

Since Aristotle argued that the first figure is the perfect figure, he felt the need to transmute all valid arguments in II and III figures to I figure so that if the transmuted mood is valid in I figure, then the corresponding mood in any figure other than the first is also valid.  

Transmutation from fourth figure to the first figure must have been evolved by the inventor of the former. Reduction is the tool to test the validity of arguments.

Mnemonics

In the thirteenth century, one logician by name Pope John XXI, devised a technique to remember the method of reducing arguments from other figures to the first figure. This technique is known as mnemonic verses.

Accordingly, each mood, excluding weakened moods, was given a special name:

                                   

Figure 1 AAA: BARBARA EAE:  CELARENT AII:    DARII EIO:   FERIO	Figure 2 EAE:  CESARE AEE:  CAMESTRES EIO:   FESTINO AOO: BAROCO
Figure 3 AAI:   DARAPTI IAI:    DISAMIS AII:    DATISI EAO: FELAPTON OAO: BOCARDO EIO:   FERISON	Figure 4 AAI: BRAMANTIP AEE: CAMENES IAI:   DIMARIS EAO:FESAPO EIO:  FRESISON

The method is like this.

If the names begin with C, then the syllogism has to be reduced to the first figure which begins with a C. For example, CESARE (a syllogism of the second figure) has to be reduced to CELARENT. Other consonants of the name have also their significance; ‘s’ (like in CESARE) signifies that the preceding ‘E’ needs to undergo simple conversion; ‘p’ signifies that the preceding proposition has to be converted by ‘limitation’; ‘t’ signifies that the order of the premises has to be changed; ‘st’ indicates that two operations, viz., simple conversion and transposition of the proposition represented by the preceding vowel are required to be carried out.

BAROCO and BOCARDO are reduced in a different manner. O propositions in both the moods have to be obverted first and then follow the relevant path to effect reduction However, the situation in modern logic is very different.