CHAPTER 2: THE LOGIC OF PROPOSITIONS

OUTLINE OF CONTENTS

1.  FOUR STANDARD FORM CATEGORICAL PROPOSITIONS       
2.  STRUCTURES OF AN “S.F.C.P.”                                 
3.  STRUCTURAL COMPONENTS OF AN “S.F.C.P”.                 
4.  SIMPLE AND COMPOUND PROPOSITIONS               
5.  CONDITIONS TO BE MET FOR A PROPOSITION TO BE A COMPONENT OF ANOTHER PROPOSITION
6.  MEDIATE AND IMMEDIATE INFERENCE          
7.  THREE TYPES OF IMMEDIATE INFERENCE       
8.  TRADITIONAL SQUARE OF OPPOSITION         
9.  TRUTH VALUES IN THE SQUARE OF OPPOSITION     
10.   EXISTENTIAL IMPORT OF CATEGORICALS

1. The four ‘standard form categorical propositions’

According to Traditional/Aristotelian Logic, we have four types of categorical proposition thus;
         i.      The ‘Universal Affirmative’ – ‘A’ proposition.
E.g. All animals are mammals
        ii.      The ‘Universal Negative’ – ‘E’ proposition.
E.g. No animals are mammals
       iii.      The ‘Particular Affirmative’ – ‘I’ proposition.
E.g. Some animals are mammals
       iv.      The ‘particular Negative’ – ‘O’ proposition.   
E.g. Some animals are not mammals

Examples of the four categorical propositions are;    

 ‘A proposition’ – All                   ‘I’ proposition - Some
‘E proposition’ - No                    ‘O proposition’ – some + not

2. ‘Structures’ Of Standard Form Categorical Propositions
The structures of an ‘S. F. C. P’ are;
(i) Quantity                     (ii) Quality        (iii) Distribution

   i.      Quantity
This simply refers to the nature of an ‘S.F.C.P.’ being either ‘Universal’ or ‘particular’.

  ii.      Quality
This simply refers to the nature of an ‘S.F.C.P.’ being either ‘Affirmative’ or ‘Negative’.

‘A’ proposition - ‘All men are human’
Quantity = Universal; Quality = Affirmative
‘E’ proposition – ‘No men are human’
Quantity = Universal; Quality = Negative
‘I’ proposition - ‘Some men are human’
Quantity = Particular; Quality = Affirmative
‘O’ proposition – ‘Some men are not human’
Quantity = particular; Quality = Negative.

       iii.      Distribution
This deals with ‘emphasis’ or ‘focus’, as regards the way subject or predicate terms occur in propositions.
§  What is ‘distribution’?
A term is said to be distributed in a proposition, if that proposition emphasizes or, makes reference to all ‘the members of the class’ designated by the term.
§  What determines the distribution of ‘terms’?
The distribution of ‘subject’ and ‘predicate’ terms depends mainly on the type (quantity and quality) of proposition, it is contained in. Therefore, as a matter of convention (fact), the following apply;

§  In ‘A’ Propositions: Universal Affirmative
Subject term – Distributed
Predicate term – Undistributed
E.g. All (men) are human
§  In ‘E’ Propositions: Universal Negative
Subject term – Distributed
Predicate term – Distributed
E.g. No (men) are (human)
§  In ‘I’ Propositions: Particular Affirmative
Subject term – Undistributed
Predicate term – Undistributed
E.g. Some men are human
§  In ‘O’ Propositions: Particular Negative
Subject term – Undistributed
Predicate term – Distributed
E.g. Some men are not (human)

3. ‘Structural components’ of an ‘S.F.C.P.’
The structural components of a standard form categorical proposition are;
§  The ‘Quantifier’ – All, No, some, and some + not.
§  The ‘Subject term’ – initial term
§  The ‘Copula’ – are or is (i.e. the joining word)
§  The ‘predicate term’ – Latter term

For example; All bags are articles
Quantifier = All
Subject term = bags
Copula = are
Predicate term = articles

4. Simple And Compound Propositions
A ‘Simple Proposition’ is one, which does not have or contain any component proposition. This, it has no extension. For example;
‘Richard graduated with a first class bachelor’.

A ‘Compound Proposition’ is a simple proposition, which contains another simple proposition as a component. Thus, it has an extension. For example;
‘Richard graduated with a first class bachelor, despite that, he had much pressures and troubles.

A ‘Complex Compound Proposition’ is a simple proposition, which has a compound proposition as a component. Thus, it has two (or even more) extensions. For example;
‘Richard graduated with a first class bachelor, despite that, he had much pressures and troubles; although, he was very much disciplined.

5. The Conditions To Be Met For A Proposition To Be A Component, Of Another Proposition
For a proposition to be a component, of another proposition, two (2) conditions must be met.
§  Firstly, the “component proposition” must be a proposition in its own right. i.e. it must be able to stand on its own. For example, Considering the propositions;
     i.        ‘A good number of saints are blessed people’.
   ii.        ‘Pastors are demi-gods for saints are blessed people’.

                The Statement/Component Proposition;
‘Saints are blessed people’, is a proposition in its own right (i.e.) it can stand on its own. Therefore, ‘condition one’ is met in both propositions ‘i’ and ‘ii’ 

§  Secondly, if the ‘component proposition’ is substituted by another proposition in the main one, it must always make sense. For example:- substituting, the component proposition; ‘Saints are blessed people’  with another one, like ‘Nowadays true love is scarce’; both main propositions would then become;
     i.        A good number of nowadays, true love is scarce.
   ii.        Pastors are demi-gods for nowadays, true love is scarce.

Note that proposition ‘i’ is meaningless but, proposition ‘ii’ is meaningful (maybe not really relevant).
Therefore, ‘Condition Two’ is not met in proposition ‘i’, but is met in proposition ‘ii’.

Finally, this implies that the statement ‘Saints are blessed people’ is not a component of the main proposition ‘i’;
‘A good number of saints are blessed people’.
... But it is a component of ‘ii’;
‘Pastors are demi-gods for saints are blessed people:

6. Mediate And Immediate Inference
A ‘Mediate Inference’ is a form of reasoning in an argument, by which the conclusion is not immediately derived from the initial premises, as it requires an intermediate premiss. An example of a ‘mediate inference argument is the syllogism (two premisses, one conclusion).
For example:
All furniture are made of wood
Settee is furniture
So, Settee is made of wood

An ‘Immediate Inference’ is a form of reasoning in an argument, by which the conclusion is immediately derived from an initial premiss without any intermediate premiss. An immediate inference argument has the form of an ‘atomic argument’ (one premiss, one conclusion), even though they are not actually the same.

Mediate and immediate inferences apply only to ‘standard form categorical propositions’.  

7. The three types of ‘immediate inference’
The three (3) main types of immediate inferences are;
(a)    Conversion         (b)    Obversion         (c) Contraposition

a.  CONVERSION
For ‘conversion’, the premiss is called ‘convertend’ while the conclusion is called ‘converse’.

Conversion: Convertend – Converse

Basically, conversion is done by, ‘switching or interchanging the subject and predicate terms of a standard form categorical proposition (S.F.C.P.)
§  Rules for applying ‘conversion’ on the four S.F.C.P^s.
     i.        In ‘A propositions’, conversion functions by LIMITATION. This implies that an ‘A’ prp would become an ‘I prp.
Hence, ‘A’ propositions undergo conversion by, interchanging the subject and predicate terms, and then changing the quantifier from ‘All’ to ‘Some’. For example;
(A prp): All logicians are scholars – Convertend (Converts to)
(I prp): Some Scholars are Logicians – Converse

   ii.        In ‘E propositions’  as well as ‘I propositions’,
Conversion is both ‘logically equivalent’ (i.e.) they are tautologous. So, ‘E’ prp remains an ‘E’ while ‘I’ prp also remains an ‘I’. Hence, ‘E’ and ‘I’ propositions undergo conversion by simply and only interchanging their subject and predicate terms.
For example;
(E prp): No logicians are dullards - Convertend
(Converts to)
(E prp): No dullards are logicians - Converse
Likewise;
(I prp): Some Logicians are Lawyers – Convertend
(Converts to)
(I prp): Some Lawyers are Logicians – Converse

  iii.        ‘O propositions’ cannot undergo conversion. Hence, conversion is generally ‘INVALID’ for any ‘O proposition’.
Note:  Although very rare, but for the sake of test. If one is given an ‘O’ proposition with the instruction ‘convert’, simply write – INVALID For ‘O’ prp.


b.  OBVERSION  
For ‘obversion’, the premise is called ‘obvertend’, while the conclusion is called ‘obverse’. It is only in obversion, that obvertend (premise) can become obverse (conclusion) and vice-versa.

OBVERSION: Obvertend - obverse

Generally, obversion is done by, ‘changing the ‘quality’ of the S.F.C.P. (i.e. affirmative to negative or vice-versa; and then replacing the “predicate” term with its complement.

Note: The complement of a ‘term’ or a ‘class’ is the combination of all things, that do not belong to it; not the opposite or contrary. E.g.  The complement of black is not ‘white’, but is ‘non-black’).

§  Rules for applying ‘obversion’ on the four S.F.C.P^s.
By convention, obversion applies and is valid for all the S.F.C.P.^s Hence, their application is similar.
     i.        ‘A propositions’ would obvert to ‘E propositions’. For example:
(A prp): All logicians are scholars - obvertend
(obverts to)
(E prp): No logicians are non-scholars - obverse

   ii.        ‘E propositions’ would obvert to ‘A propositions’. For example:
(E prp): No logicians are dullards - obvertend
(obverts to)
(A prp): All logicians are non-dullards – obverse

  iii.        ‘I propositions’ would obvert to ‘O propositions’. For example:
(I prp): Some logicians are lawyers - obvertend
(obverts to)
        (O prp): Some logicians are not non-lawyers - obverse 

iv.     ‘O propositions’ would obvert to ‘I propositions’. For example:
(O prp): Some logicians are not lecturers - obvertend
(obverts to)
(I prp): Some logicians are non-lecturers - obverse

c.  CONTRAPOSITION
For ‘contraposition’, the premiss is called ‘proposition’ while the conclusion is called ‘contrapositive

CONTRAPOSITION: Proposition - contrapositive

Generally, contraposition is done by, ‘substituting or replacing the subject term with the complement of the predicate term; and also substituting or replacing the predicate term with the complement of the subject term.

§  Rules for applying ‘contraposition’ on the four ‘S.F.C.P.^S
     i.        In ‘A propositions’ as well as ‘O propositions’, contraposition is both ‘logically equivalent’. So, ‘A’ prp remains an ‘A’ while ‘O’ prp remains an ‘O’. The only exception is that the general basic rule applies. For example;
(A prp): All logicians are scholars - proposition
(Contraposits to)
(A prp): All non-scholars are non-logicians - Contrapositive
Likewise;
(O prp): Some logicians are not lecturers - Proposition
(Contraposits to)
(O prp): Some non-lecturers are not non-logicians - C.positive

   ii.        In ‘In ‘E propositions’, contraposition functions by ‘LIMITATION’. This implies that an ‘E’ prp would become an ‘O’ prp. Hence, ‘E’ propositions undergo contraposition by, initially applying the basic rule, and then changing the quantifier from ‘No’ to ‘Some + not’.
For example;
(E prp): No logicians are dullards - proposition
(Contraposits to)
(O prp): Some non-dullards are not non-logicians - C.positive

  iii.        ‘I propositions’ cannot undergo contraposition. Hence, contraposition is generally ‘INVALID’ for any ‘I proposition’. (Note: Although very rare, but for the sake of test; If one is given an ‘I’ proposition with the instruction ‘apply contraposition’, simply write – INVALID for ‘I’ prp).

8. The Traditional Square Of Opposition
There are four (4) other forms/types of immediate inferences existing apart from the previous three earlier discussed. These immediate inferences are relationships that exist between the four S.F.C.P^S. These relationships are;
(A).   Contradictory                                (C).   Subcontrary
(B).   Contrary                                      (D).   Subalternation
These immediate inferences/relationships are normally displayed in a rectangular formation known as the ‘traditional square of opposition’.  It is shown below;

Note: Before there can be a relationship, all propositions in the traditional square of opposition must have the same subject and predicate terms.   

A. The relationship of ‘contradictories’

Contradictories are two S.F.C.P.^s that have different quality and different quantity. Accordingly;

§  ‘A’ and ‘O’ propositions are contradictories

(A prp): All robots are machines – universal affirmative

(O prp): Some robots are not machines – particular negative

Likewise;

§  ‘E’ and ‘I’ propositions are contradictories

(E prp): No robots are machines – universal negative

(I prp): Some robots are machines – particular affirmative

§  ‘RULE’ of Contradictories

If one is ‘true’, the other must be ‘false’ and vice-versa. Hence, they cannot both be true or false, as they both deny or contradict each other.

B. The relationship of ‘Contraries’

Contraries are two S.F.C.P^s that are universal propositions and have different quality. Accordingly;

§  Only, ‘A’ and ‘E’ propositions are contraries

(A prp): All robots are machines – universal affirmative

(E prp): No robots are machines – universal negative

§  ‘RULE’ of Contraries

If one is ‘false’, the other would be ‘undetermined’ but if one is ‘true’, the other would be ‘false’.

…. Put another way (in logic language)

Both ‘may’ (undetermined) be false, but both cannot be true simultaneously.

C. The relationship of ‘subcontraries’

Subcontraries are two S.F.C.P^s that are particular propositions and have different quality. Accordingly;

§  Only, ‘I’ and ‘O’ propositions are subcontraries

(I prp): Some robots are machines – particular affirmative

(O prp): Some robots are not machines – particular negative

§  ‘RULE’ of Subcontraries

If one is ‘true’, the other would be ‘undetermined’, but if one is ‘false’, the other would be ‘true’.

…… put another way (in logic language)

Both ‘may’ (undetermined) be true, but both cannot be false simultaneously.

D. The relationship of ‘Subalternation’

Subalternation exists between two S.F.C.P^s that have the same quality but different quantity.

Unlike other relationships in the traditional square of opposition; in ‘Subalternation’ one proposition is called ‘Superaltern’ while the other is called ‘Subaltern’. Accordingly;

§  ‘A’ and ‘I’ propositions share the subalternation relationship

(A prp):  All robots are machines – Superaltern

(I prp): Some robots are machines – Subaltern

Likewise;

§  ‘E’ and ‘O’ propositions share the subalternation relationship

(E prp): No robots are machines – Superaltern

(O prp): Some robots are not machines – Subaltern

§  ‘RULES of Subalternation

If Superaltern is ‘true’, Subaltern would be ‘true’.

If Superaltern is ‘false’, Subaltern would be ‘undetermined’.

...on the other hand

If Subaltern is ‘true’, Superaltern would be ‘undetermined’.

If Subaltern is ‘False’, Superaltern would be ‘false’.

9. ‘Truth values’ in the ‘square of opposition’

Due to the relationships exhibited in the ‘traditional square of opposition’, there are certain ‘truth value’ implications that follow for the four categorical propositions. These certain ‘truth value’ implications are based on the laws guiding these relationships. These are stated thus;

If > A = True            

E = False   (Contrary)                

I = True   (Subaltern)

O = False (Contradictory)   

If > A = False

E = Undetermined (Contrary)

I = Undetermined (Subaltern)

O = True (Contradictory)

If > E = True               

A = False (Contrary)        

I = False (Contradictory)  

O = True   (Subaltern)  

If > E = False

A = Undetermined (Contrary)

I = True (Contradictory)

O = Undetermined (Subaltern)

If > I = True               

A = Undetermined (Superaltern)

E = False (Contradictory)         

O = Undetermined (Subcontrary)                              

If > I = False

A = False    (Superaltern)

E = True     (Contradictory)

O = True    (Subcontrary)

If >O = True 

A = False (Contradictory)

E = Undetermined (Superaltern)

I = Undetermined (Subcontrary)

If > O = False

A = True (Contradictory)

E = False (Superaltern)

I = True (Subcontrary)

10. ‘Existential Import’ Of Categoricals

‘Existential import’ of propositions simply means that, ‘a proposition affirms the existence of something or another’. According to the existential import of categorical propositions, it is maintained that;

§     ‘UNIVERSAL PROPOSITIONS’ (A and E) do not have existential import.

§     ‘PARTICULAR PROPOSITIONS’ (I and O) do have existential import.

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