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Categorical Propositions


Disciplines Argument > Categorical Propositions

Definitions | Four types | Opposites | See also


Classical logic makes great use of the principle of putting things into categories, or classes. Categorical propositions tell you things about these categories.


Categorical term

A categorical term is something that will be categorized, such as 'dog' and 'cat'. It is usually a collective statement such as 'all dogs' or 'some dogs'.

Categorical proposition

A categorical proposition is simply a statements about the relationship between categories. It states whether one category or categorical term is fully contained with another, is partially contained within another or is completely separate.

A dog is an animal

Some dogs are friendly

No dog is a cat

Propositions may have quality: either affirmative or negative.

They may also have quantity: such as 'a', 'some', 'most' or 'all'. The 'all' quantity is also described as being universal and other quantities particular.

Predicate and subject

The first term in the proposition is the subject. The second term is the predicate.

Some dogs (subject) are friendly (predicate)


A categorical term is said to be distributed if the categorical proposition that contains it says something about all members of that categorical term. It is undistributed if the categorical proposition that contains it says does not something about all members of that categorical term.

Four types

There are four types of categorical proposition, each of which is given a vowel letter A, E, I and O. A way of remembering these is: Affirmative universal, nEgative universal, affIrmative particular and nOgative particular. To be more correct, A and I letters came from the Latin affirmo, and E and O from the Latin nego.


Form Type Quality Quantity Distribution of X Distribution of Y
All X is Y A Affirmative Universal Distributed Undistributed
No X is Y E Negative Universal Distributed Distributed
Some X is Y I Affirmative Particular Undistributed Undistributed
Some X is not Y O Negative Particular Undistributed Distributed


In this classification, 'some X is some Y' is I and 'some X is not some Y' is O, although it can be argued that these may be treated as an additional two variants.


There are several types of opposition used in categorical propositions. These can be traditionally placed in the Square of Opposition.




<-- Contraries --> E












<-- Subcontraries --> O


  • Contraries cannot both be true, but both can be false.
  • Subcontraries cannot both be false, but both can be true.
  • Subaltern pairs can both be true or both be false.
  • Contradictories cannot both be true and cannot both be false.

Opposites are also described in the converse, obverse and contrapositive.


The converse of a categorical proposition is categorical proposition where the predicate and subject of the original proposition are exchanged. Note that the quantity does not move with the subject or predicate.

No dogs are cats  -->  No cats are dogs

Some dogs are friendly creatures  -->  Some friendly creatures are dogs

All dogs are animals  -->  All animals are dogs

The converse of any true E or I proposition is also true (making it a useful test). A and O converses are seldom true.


The obverse of a categorical proposition has predicate term replaced with its complement and quality of the proposition reverse.

All dogs are animals  -->  No dogs are not animals

No dogs are not dangerous  -->  All dogs are dangerous

The obverse of all types of true categorical proposition are also true.


The contrapositive of a categorical proposition is formed by taking the complement of both subject and predicate and then reversing them.

All dogs are animals  -->  All non-animals are not dogs

Some dogs are friendly  -->  Some non-friendly creatures are not dogs

The contrapositive of any true A or O proposition is also true (making it a useful test). Contrapositives of E and I propositions are seldom true.

See also

Categorical syllogism, Set Theory

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