CHAPTER 3: THE LOGIC OF SYLLOGISM

OUTLINE OF CONTENTS
 
1.  STRUCTURES OF SYLLOGISTIC ARGUMENTS                   
2.  MOOD, FIGURE AND FORM OF AN “S.F.C.S”                  
3.  DIAGRAMMING THE FOUR CATEGORICAL PROPOSITIONS    
4.  SORITES                                                          
5.  ENTHYMEMES                                                    
6.  FORMAL FALLACIES AND RULES OF SYLLOGISM       

1. Structures Of Syllogistic Arguments
The structures of a syllogistic argument (syllogism) are;

     i.        Major term
   ii.        Minor term
  iii.        Middle term
  iv.        Major premiss
    v.        Minor premiss 

NOTE: A ‘Syllogism’ is a mediate inference, standard form categorical argument, which has two premises and one conclusion. For example;

All linguists are grammarians (premise 1)

Some grammarians are teachers (premise 2)

So, some teachers are linguists (conclusion)

     i.        The ‘MAJOR TERM’ of a syllogism is the predicate of its conclusion. E.g: ‘linguists’.

   ii.        The ‘MINOR TERM’ of a syllogism is the subject of its conclusion e.g: ‘teachers’.

  iii.        The ‘MIDDLE TERM’ of a syllogism is, the term which is absent in the conclusion but appears in the both premisses e.g: ‘grammarians’.

  iv.        The ‘MAJOR PREMISS’ is the premiss that contains the ‘major’ term e.g: All linguists are grammarians.

    v.        The ‘MINOR PREMISS’ is the premiss that contains the ‘minor’ term. E.g: Some grammarians are teachers.

PLEASE NOTE: A ‘standard form categorical syllogism’ is a syllogism that contains only standard form categorical propositions as premises and conclusion. It is normally stated in the following order;

§  Its major premiss comes first.

§  Its minor premiss comes second.

§  Its conclusion (which carries both the minor and major term) comes last.

2. Mood, figure and form of an ‘S.F.C.S’

The ‘FORM’ of a syllogism is mainly determined by its ‘MOOD’ and ‘FIGURE’.

§  MOOD

The ‘mood’ of a standard form categorical syllogism (S.F.C.S) is determined by the types of standard form categorical propositions (S.F.C.P) it contains. For example;

All lions are cats - A proposition

Some cats are beasts - I proposition

So, some beasts are lions - I proposition

The ‘mood’ of our syllogism is ‘AII’.

Another example is;

Some cats are not beasts - O prp

All cats are mammals - A prp

So, some mammals are not beasts - O prp

The ‘mood’ of our syllogism is ‘OAO’.  

§  FIGURE

The ‘figure’ of a standard form categorical syllogism (S.F.C.S) is determined by the positioning of the ‘middle term’ in its premisses. Accordingly, we have four types of figures, which are;

     i.        ‘Figure 1’ is when the middle term occurs as the subject of the major premiss (p1) and the predicate of the minor premiss (p2).

(Middle term) -------------- Predicate           (P1)

      Subject ----------------- (Middle term)    (P2)

   ii.        ‘Figure 2’ is when the middle term occurs as the predicate of both the major premiss (P1) and the minor premiss (P2).

      Subject ---------------- (Middle term)           (P1)

      Subject ---------------- (Middle term)      (P2)

  iii.        ‘Figure 3’ is when the middle term occurs as the subject of both the major premiss (P1) and the minor premiss (P2).

      (Middle term)   ---------------- Predicate      (P1)

      (Middle term)  ---------------- Predicate         (P2)

  iv.        ‘Figure 4’ is when the middle term occurs as the predicate of the major premiss (P1) and the subject of the minor premiss (p2).

       Subject ----------------- (Middle term)        (P1)

(Middle term)--------------- Predicate           (P2)

§  FORM

The concept of ‘form’ is difficult to define, as it signifies an abstract relationship which is better defined when instantiated. But loosely, we may say that, ‘the form of an argument, is the structure (figure) by which the propositions are arranged, as well as the nature (mood) of these same propositions. For example;

The form of a standard form categorical syllogism (S.F.C.S) which entails (or is a combination of) its ‘mood’ and ‘figure’ goes thus;

All Donatists are Pelagians (P1)

No Pelagians are Nestorians (P2)

So, No Nestorians are Donatists (C)

Its form is stated as ‘AEE – 4’. Mood is AEE; Figure is 4

3. Diagramming the four categorical propositions

In proving syllogisms valid or invalid, standard form categorical propositions must be diagrammed in a particular manner. Accordingly, each standard form categorical proposition has how it should be diagrammed, and this is highlighted thus;

     i.        UNIVERSAL AFFIRMATIVE (‘A’ Propositions)

All S is P

(No Existential Import)

ii.          UNIVERSAL NEGATIVE (‘E’ Propositions)

No S is P

(Existential Import)

iii.        PARTICULAR AFFIRMATIVE (‘I’ Propositions)

Some S is P

(No existential Import)

iV.        PARTICULAR NEGATIVE (‘O’ Propositions)

Some S is not P

(Existential Import)

4. SORITES

§  What is a ‘Sorites’?

A ‘Sorites’ is an elongated syllogism whose conclusion cannot be deduced from only a single set of premisses.

§  Other information

The conclusion of a ‘Sorites’ cannot be established by a single syllogism. Hence, at least two or more syllogisms are required to derive the conclusion.

The inferred conclusion of an initial syllogism would automatically serve as the major premiss of a latter one.

The formula ‘n – 1’ is used to known the number of ‘syllogisms’ a Sorites must produce. The letter ‘n’ stands for ‘number of premisses in a Sorites’. So, if n = 6, the number of syllogisms to be produced would be ‘6-1’ = 5.

Lastly, a ‘Sorites’ is valid, if and only if; all its syllogisms are also valid. Thus, if one syllogism is invalid and others valid, the Sorites (as a whole) is rendered invalid.

• 5. ENTHYMEMES

§  What is an ‘Enthymeme’?

An ‘Enthymeme’ is an incompletely articulated argument, of which either one of the premisses or the conclusion is unstated.

§  Types of Enthymemes

There are three (3) orders/types of Enthymemes;

A.  First order enthymeme

B.   Second order enthymeme

C.   Third order enthymeme

A.  First Order Enthymeme

Here, the ‘major premiss’ is unstated. For example;

No mammals are amphibians

So, No amphibians are dogs

NOTE: If an Enthymeme has a conclusion, and the major term (predicate of the conclusion) is not present in the given premiss; it is therefore a ‘first order enthymeme’.

B.   Second Order Enthymeme

Here, the ‘minor premiss’ is unstated. For example;

All dogs are mammals

So, No amphibians are dogs

NOTE: If an Enthymeme has a conclusion, and the minor term (subject of the conclusion) is not present in the given premiss; it is therefore a ‘Second order enthymeme’.

C.   Third Order Enthymeme

Here, the ‘conclusion’ is unstated. For example;

All dogs are mammals

No mammals are amphibians

NOTE: If an Enthymeme has ‘no’ conclusion, automatically, it is a ‘third order enthymeme’.

§  How To Solve An Enthymematic Argument

(i).    Recognize the order of the Enthymeme.

(ii).   Bring in the missing proposition.

(iii).  If necessary, restate the argument in standard order.

(iv).   Then, test the syllogistic argument for validity.

6. FORMAL FALLACIES AND RULES OF SYLLOGISM

These fallacies represent valid syllogistic rules, which when contravened, renders a syllogistic argument invalid. There are seven of them and they are highlighted thus;

   i.      Fallacy of ‘Equivocation’ or ‘Four terms’.

  ii.      Fallacy of ‘Exclusive premisses’

 iii.      Existential Fallacy.

 iv.      Fallacy of ‘Drawing an affirmative conclusion from a negative premiss.

  v.      Fallacy of ‘undistributed middle’.

 vi.      Fallacy of ‘illicit major’ or ‘illicit process of the major term’

vii.      Fallacy of ‘illicit minor’ or ‘illicit process of the minor term’.

   i.      Fallacy of ‘equivocation’ or ‘four terms’

RULE: A valid syllogism must have only three terms and they must be used without equivocation. For example;

All doctors are humanitarians

Some humanitarians are professionals

So, some professional are physicians

   i.      Fallacy of ‘exclusive premisses’

RULE:  A syllogism, whose both premisses have a negative quality, cannot be valid. For example;

No kings are paupers

Some paupers are not misers

So, some misers are not kings

  ii.      Existential Fallacy

RULE: In a valid syllogism, two ‘Universal’ premisses (has no existential import) cannot infer a particular’ conclusion (has existential import). For example;

All Fulanis are Nigerians

No Arabs are Nigerians

So, some Arabs are not Fulanis

 iii.      Fallacy of ‘drawing an affirmative conclusion from a negative premiss’

RULE: For an affirmative conclusion to occur in a valid syllogism, both premisses must be affirmative, and none negative. For example;

Some presidents are women

No women are men

So, some men are presidents

 iv.      Fallacy of ‘undistributed middle’

RULE: In a valid syllogism, the middle term must be distributed in at least one premiss. For example;

All popes are cardinals

Some cardinals are not vicars

So, some vicars are not popes

  v.      Fallacy of ‘illicit major’

RULE:  In a valid syllogism, if the major term is distributed in the conclusion, then it must also be distributed in the major premiss. For example;

All logicians are scholars

No dullards are logicians

So, No dullards are scholars

 vi.      Fallacy of ‘illicit minor’

RULE:  In a valid syllogism, if the minor term is distributed in the conclusion, then it must also be distributed in the minor premiss. For Example;

No pianos are drums

All drums are membranophones

So, No membranophones are pianos

7. TRANSLATING PROPOSITIONS INTO STANDARD ORDER

Some very important ideas are presented thus;

     i.        Only boys are cadets = All cadets are boys

   ii.        None but electronics are cookers = All cookers are electronics

  iii.        At least a thing called Moslem is Yoruba = Some Moslems are Yoruba people

  iv.        Talibans are deadly = All Talibans are deadly people

    v.        Some cars are expensive = Some cars are expensive things

  vi.        All snakes are non-mammals = No snakes are mammals (via obversion)

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