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TRADITIONAL SQUARE OF OPPOSITION

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TRADITIONAL
SQUARE OF OPPOSITION

I hope you have now been able to  construct syllogism  moods and figures such as :

a)EIO   b)AAA  c)III   d)AEE   e)IOI    f)AOO   g)AEE   h)OOO

As guided in our previuous Article: Critical Thinking and Logic Guide to Constructing Syllogism Mood and Figures

Welcome to this Article where we will continue to learn on Critical Thinking and Logic. This session will cover Constructing Traditional Square of Opposition. 

Traditional Square of Opposition is a square that describes the logical relationship of propositions

This square helps to answer questions such as:

Question 1:

If the Statement some sinners are thieves is false,

Determine the truth value of:

a)Sinners are thieves

b)No sinners are thieves

c)Some sinners are thieves

d)All sinners are thieves

Suppose the statement some sinners are thieves is True

  • Determine the truth value of the following propositions:

a)Sinners are thieves

b)No sinners are thieves

c)Some sinners are thieves

d)All sinners are thieves

Question 2:

Given that No lions are cats is false

  • Determine the truth value of the following

a)Lions are cats

b)Some lions are cats

c)Some lions are not cats

d)All lions are cats

Suppose No lions are Cats is True

  • Determine the truth value of the following

a)Lions are cats

b)Some lions are cats

c)Some lions are not cats

d)All lions are Cat

Image of the traditional square of opposition

PARTS OF TRADITIONAL SQUARE OF OPPOSITION

PARTS OF TRADITIONAL SQUARE OF OPPOSITION

1. CONTRARIETY

Contrariety- this is the relationship between A (Universal affirmative- All S are P) and E (Universal negative (No S are P) forms of categorical propositions.

Contrariety means that when one of the A or E proposition is given as true, the other one must be false. When one of the A or E proposition is given as false, the truth value of the other cannot be determined because it can either be true or false.

Example1 : Given that:

All student are rich is true

The statement: No student are rich is thus false

Example 2: Given that:

  • All student are rich is false

The statement: No students are rich is thus undetermined. ("All students are rich" being false simply means that not all students are rich. Some students might be rich, some might not, and the statement "No student are rich" could be true or

2. Subcontrariety

If I (Some S are P) is given as true then O (Some S are not P) has to be undetermined (true or false)

Example 

Given that the statement: some student are stingy is True

The proposition: Some student are not stingy  is undetermined.

This is because Proposition in the first proposition, at least one student is stingy and in the second proposition it means at least one student is not stingy and thus it is undetermined.

Similarly …

If O (Some S are not P) is given as true then I (Some S are P) is undetermined

Example:

  • Given that Some lions are not Cat is True
  • The proposition some Lions are Cats is undetermined.
  • This is because at least one lion is not a cat and also at least one lion is a cat. Therefore, it is undetermined (true or false).

When one subcontrariety is false the other one is true

I.e If I (Some S are P) is given as false then O (Some S are not P) has to be True.

Given that the statement: some student are stingy is False

The proposition: Some student are not stingy  is True

On the other hand

If O (Some S are not P) is given as true then I (Some S are P) is undetermined

Example:

Given that Some lions are not Cat is False

The proposition some Lions are Cats is True

Subcontrariety Summary

3. CONTRADICTIONS

  • Contradiction is the relationship between A and O and I and E
  • They are opposite of each other
  • When one is true the other is false
  • Example 1:
  • If A -(All S are P) is true Some S are not P is false (and vice versa)
  • Example 1: if all cats are lion is true, some cats are not lion is false (and vice versa)

Example 2:

  • IF Some s are p is True, No S are P is false (and vice versa)

e.g If some cats are lion is True, The statement No lions are Cats is False (and vice versa)

4. SUBALTERNATIONS

  • It is the relationship between A (all S are P) and I (some S are P) and E (No S are P) and O (Some S are not P)

A-I are called superalterns

E-O are called subalterns

  • Subalterns differ in quantity but they have the same quality
  • If A (all S are P) and I (some S are P) is true, the corresponding E (No S are P) and O (Some S are not P) is True.
  • This is because truth flows from Super-altern (A-I) to Sub alterns (E-O)

 

However, if propositions A-E are false, their corresponding I-O are undetermined (True or False) because falsity is not transmitted downward

For Example

 

On the other hand;

If any of the I-O subalterns is given as false, then the corresponding A-E is also false.

This is because, falsity is only transmitted upwards.

However, if subalterns I-O are true, the corresponding superalterns A-E _______________________

Example 

Given that Some students are not stingy is false, the statement No men is stingy is also false

Answers to the Questions on Truth value of Proposition using Traditional Square of opposition

Question 1:

Given that the Statement some sinners are thieves is false,

Determine the truth value of:

a)Sinners are thieves

b)No sinners are thieves

c)Some sinners are thieves

d)All sinners are thieves

Suppose the statement some sinners are thieves is True

Determine the truth value of the following propositions:

a)Sinners are thieves

b)No sinners are thieves

c)Some sinners are thieves

d)All sinners are thieves

Answers:

If the Statement some sinners are thieves is false,

Determine the truth value of:

a) Sinners are thieves – it is undetermined because the proposition has no quantifier

i.e has no measure of is it All, some, none etc

b) No sinners are thieves- is True—this is a contradiction

 

c) Some sinners are thieves- This is false because the first proposition had argued that some sinners are thieves is false

d) All sinners are thieves is False

In this we are moving from I-A subaltern I to Superaltern A: and if any subaltern I or O is given as false, then the superalterns A or E are also false because falsity is transmitted upwards.

Therefore, some sinners are thieves is false (I- is false) then the statement All sinners are thieves (A) is also False.

Question 2

Suppose the statement some sinners are thieves is True

Determine the truth value of the following propositions:

a. Sinners are thieves – this is undetermined/indefinite because the proposition has no quantifier

b.No sinners are thieves- this is false because if one contradiction is true the other is True

 

Suppose the statement some sinners are thieves is True
Determine the truth value of the following propositions:

c). Some sinners are thieves – this is True because the proposition itself have argued that the statement some sinners are thieves is True

d). All sinners are thieves this is _____________________________

 

Try working out the following we will give the answers in the next session

Question 2:

  1. i) Given that No lions are cats is false
  • Determine the truth value of the following

a)Lions are cats

b)Some lions are cats

c)Some lions are not cats

d)All lions are cats

 

  1. ii) Suppose No lions are Cats is True
  • Determine the truth value of the following

a)Lions are cats

b)Some lions are cats

c)Some lions are not cats

d)All lions are Cat

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