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Proposition

A proposition is a sentence that makes a statement and gives a relation between two terms. It is also called as a premise.

It consists of three parts:
  1. The subject.
  2. The predicate.
  3. The relation between the subject and the predicate is called ‘copula’.
All propositions either assert or deny something.

Some Examples of Propositions

  1. All tigers are wild.
  2. No school boy is disciplined.
  3. Some girls are beautiful.
  4. Some pups are not cute.
A subject is that part of the proposition about which something is being said. A predicate, on the other hand, is that term of the proposition which tells something about the subject.
 
Thus, in the four propositions given above, tigers, school boy, girl and pups are subjects. The words wild, disciplined, beautiful and cute are predicates. The words ‘are’, ‘is’ and ‘are not’ are called copula.

Types of Propositions

Categorical Proposition: Categorical proposition means the relationship between the subjective and predicative terms is categorical. In other words, the predicative terms affirm or deny the subjective term.
 
For example, ‘All foxes are cunning’, ‘No cat is beautiful’, ‘Some watches are good’, ‘Some dogs are not fatty’. All of these are categorical propositions because in each case the relationship between the subject and predicate is categorical.
‘All S are P.’
‘No S are P.’
‘Some S are P.’
‘Some S are not P.’

Hypothetical Proposition: A hypothetical proposition is a type of conditional sentence in which the ‘if’ clause is in present indefinite tense and the consequent clause in future tense.
 
For example, ‘If he studies hard, he will get rank.’
If ‘S’, then P.

Disjunctive Proposition: Disjunctive propositions are those in which there is an alternation between the first part and the second part of the statement.
 
For example, ‘Either India will win or lose against Pakistan in cricket.’

Relational Proposition: Relational proposition may establish either blood relationship between subject and predicate or any other type of relationship. Mr. Jack is the father of Mr. John. We are establishing a relationship between the subjective and predicative terms. Similarly, if we say ‘A is taller than B’ or ‘A is as tall as B’, we are referring a particular type of relational proposition.

Types of Categorical Propositions

Universal Proposition: A proposition where the subject either includes or excludes all is called universal proposition.
 
Examples:

All girls are naughty.
No box is square shaped.

Normally, universal proposition begins with ‘all’, ‘every’, ‘any’, etc., or ‘no’, ‘none of these’, ‘not a single’, etc.

Universal proposition is again sub-divided into two categories.
  1. Universal Affirmative: If the subject of a categorical proposition refers to all for which it stands and if the proposition is affirmative, it is called universal affirmative proposition.
     
    ‘All S are P’ is called universal affirmative.
     
    A universal affirmative proposition is generally denoted by the letter ‘A’.
  2. Universal Negative: If the subject of a categorical proposition refers to all for which it stands and if the proposition is negative, it is called universal negative proposition.
     
    For example, ‘No man is four-footed’. In this statement also, we are referring to ‘All men’ but from the negative point of view.
     
    If there is a negative term like ‘not or no’ in the statement, it is called negative statement.
     
    Universal negative proposition is generally denoted by the letter ‘E’.
Particular Proposition: If the subject does not refer to ‘all’, it is called particular proposition. The subject in this type of proposition can be ‘some’, ‘almost all’, ‘many’, ‘quite a few’, etc. Anything less than ‘all’ is considered as particular in logic.
 
Example:

Some girls are naughty.
Some boxes are not square-shaped.

Particular proposition is again sub-divided into two categories.
  1. Particular Affirmative: If the subject is not referring to ‘all’ and if the proposition is affirmative it is called particular affirmative proposition.
     
    For example, ‘Some birds are intelligent.’
     
    Particular affirmative is generally denoted by the letter ‘I’.
  2. Particular Negative: If the subject is not referring to ‘all’ and if the proposition is negative, it is called particular negative proposition.
     
    For example, ‘Some students are not honest’. Here, the subject does not refer to ‘all’ and the statement is negative.
     
    A particular negative proposition is generally denoted by the letter ‘O’.
The four types of propositions can be summarised in a tabular form as follows:
 
The definition of the A, E, I and O propositions are very important and the students must be in a position to recognise these types quickly.
 
Type of Propositions Universal Particular
Affirmative ‘A’ format
All S are P
‘I’ format
Some S are P
Negative ‘E’ format
No S are P
‘O’ format
Some S are not P

Classify the following propositions into their respective type of A, E, I or O.
 
1. All watches are expensive. [ ]
2. Some students are players. [ ]
3. All frogs are amphibians. [ ]
4. All mammals give milk. [ ]
5. Some vegetables are green. [ ]
6. Some fruits are not sweet. [ ]
7. Many of the books are spoiled. [ ]
8. Quite a few of the politicians are honest. [ ]
9. Some toothbrushes are not soft. [ ]
10. Children are generally honest. [ ]

Confusing Categorical Propositions

Consider the statement, ‘All students are not intelligent’. This statement cannot be considered as universal negative proposition. In fact, it is an ‘O’ proposition. The statement implies that ‘Some students are intelligent’ and ‘some students are not intelligent’. As the statement is negative, it is considered to be an ‘O’ proposition.

Exclusive Propositions

While working out the inference, these propositions are changed into ‘A’ proposition, by making the predicate as subject and the subject as predicate.
 
For example, the proposition ‘Only males are eligible’ should be converted into ‘All who are eligible are males’ before we start working out the arguments.

Exceptive Propositions

If a proposition contains the word ‘except’, it is called exceptive proposition. Exceptive propositions are considered to be particular propositions.

‘All persons except one are bachelors’ is an ‘I’ proposition and for reasoning purposes, it can be rewritten as ‘Some persons are bachelors.’ But if exception is with a particular name, it will be considered a universal proposition.
 
For example, ‘All except Japan are leaders’. We will have to consider it as ‘A’ proposition.




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