CATEGORICAL PROPOSITIONS

Categorical propositions divide the world into two distinct classes and make an assertion about members of those classes. Every categorical proposition is a statement about the members of two classes and their relationship to one another. For example,

All geraniums are flowering plants.

Some logicians are not rational thinkers.

are both categorical propositions.  For all practical purposes, until the nineteenth century the study of logic was limited to the study of categorical propositions and of arguments composed exclusively of categorical propositions.   Aristotle (384-322 B.C.) was the first to study ways of arguing and to formalize logic as a discipline. The form of argument that he identified and systematized uses subject-predicate statements in a syllogism (two premises and a conclusion), so it is sometimes called 'syllogistic logic'. Because it was first worked out by Aristotle, it is also known as 'Aristotelian logic'. And, finally, because it deals with categorical statements in a syllogistic form, it is sometimes known as the 'logic of the categorical syllogism' or simply 'categorical logic'.

Although modern logic has moved well beyond traditional categorical logic, categorical logic is worthy of study for at least three reasons. One, traditional logic has played a major role in the history of western thought. Indeed, it is the logic most people recognize as logic. Two, the categorical syllogism is a relatively accessible deductive system. It employs a limited number of propositional forms and its syllogisms can be tested for validity without too much technical difficulty. Three, categorical logic provides a useful bridge from the propositional logic we have been studying to modern predicate logic. 

The Four Kinds of Categorical Propositions

A categorical proposition is a statement that relates two classes, or categories in a subject-predicate relationship. Something is predicated, or said about, some subject. What is said is that the first class (the subject term) is either included in or excluded from the second class (the predicate term).   For any two classes S and P, there are only 4 distinct ways in which the members of those classes can be related to one another.  Each of those 4 ways is represented by a type of categorical proposition. Using "S" and "P" as symbols (to stand for "subject" and "predicate"), they are :

  1. All S's are P's -- (Type A)
  2. No S's are P's -- (Type E)
  3. Some S's are P's -- (Type I)
  4. Some S's are not P's -- (Type O)

Each categorical proposition is composed of 4 distinct parts: a quantifier, a subject term, a copula (linking verb), and a predicate term.  Logicians recognize only 2 quantifiers in English:  "all" and "some" ("a few", "most", "many", "several" and the like all mean 'some).  The words "all" and "some" are called "quantifiers" because they indicate the quantity of the subject. That is, they specify how many member of the subject class are included within the predicate class. "No" indicates that zero members are included, so "No S are P" means "All S are non-P", that is "All S are excluded from P". The copula  in a properly expressed categorical proposition is always some form of the verb "to be". Thus we get the following schema:

Type Quantifier Subject Copula Predicate
A All S are P
E All (No) S are not (are) P
I Some S are P
O Some S are not P

Properties of Categorical Propositions:

Quantity and Quality

Each categorical proposition has both a quantity and a quality.  The quantity of a categorical proposition is determined by the quantifier used.  If the quantifier is "All" the quantity is universal.   If the quantifier is "Some" the quantity is particular.   The quality of a categorical proposition is determined according to whether the proposition asserts of denies an overlap between the classes.  If a proposition asserts an overlap between the classes named, the quality of the proposition is affirmative.  It a proposition denies an overlap between the classes named, the quality is negative. 

Each term in a categorical proposition is either distributed or undistributed.  If the proposition refers to the entire class named by a term, that term is distributed.  If the proposition does not refer to the entire class named by a term, that term is undistributed.

To complete the chart begun above, we get:

Type Quantifier Subject Copula Predicate Quality Quantity Subject Term Predicate   Term
A All S are P Universal Affirmative Distributed Undistributed
E All (No) S are not (are) P Universal Negative Distributed Distributed
I Some S are P Particular Affirmative Undistributed Undistributed
O Some S are not P Particular Negative Undistributed Distributed

It is important to remember this chart because the properties of categorical propositions are used as one method of determining the validity of a categorical syllogism.

Historical Interlude

Why do we call the four basic categorical propositions A, E, I and O?  The letters "A" and "I" are the first two vowels in the Latin word, affirmo, "I affirm."  The letters "E" and "O" are the first two vowels in the Latin word, nego, "I deny".

Immediate Inference and the Square of Opposition

Immediate Inferences

An immediate inference is one that can be made solely on the basis of a single premise.  In propositional logic the inference rules simplification and addition are instances of immediate inference, as are all of the equivalence rules.  In categorical logic, immediate inferences are drawn based on our knowledge of the truth or falsity of one categorical proposition.  Three important immediate inferences deal with the logical relationships of conversion, obversion and contraposition (you may recall these terms from high school geometry class).

Conversion

Conversion switches the subject and predicate terms.  For example, the converse of "All S are P" is "All P are S".  Conversion is legitimate for E and I propositions, but not for A and O propositions.

Obversion

Obversion involves making  two changes: the quality is changed (either from affirmative to negative or negative to affirmative) and the compliment of the predicate class is substituted for the predicate term. The compliment of a class is the class of everything not in the original class. The class of non-S's is the compliment of the class S. So, the obverse of "All S are P" is "No S are non-P".  Obversion is always legitimate.

Contraposition

Contraposition also involves making two changes: both terms are replaced by their compliments, and the terms are switched (as in conversion). Thus, the contrapositive of "All S are P is "All non-P are non-S".  Contraposition may remind you of the equivalence rule transposition and some systems of logic call that rule 'contraposition'.   HOWEVER, contraposition in categorical (and full blown predicate) logic does not neatly match the rules for conditionals in propositional logic.  Contraposition is legitimate for A and O propositions, but not for E and I.

The following chart  gives the converse, obverse and contrapositive of each of our 4 categorical propositions and indicates which of those transformations are NOT VALID
 

 

A

E

I

O

Proposition All S are P No S are P Some S are P Some S are not P
Converse All P are S 
NOT VALID
No P are S Some P are S Some P are not S 
NOT VALID
Obverse No S are non-P All S are non-P Some S are not non-P Some S are non-P
Contrapositive All non-P are non-S No non-P are non-S 
NOT VALID
Some non-P are non-S 
NOT VALID
Some non-P are not non-S

 

The Square of Opposition

There is another way to analyze the four basic categorical propositions. Following the lead of Aristotle, we can arrange them in a square of opposition as follows:

 

The square of opposition relates pairs of categorical propositions according to the logical relations of contrariety (contraries) (A and E), sub-contrariety (sub-contraries) (I and O), contradiction (A and O, and E and I), sub-alternation (A and I, and E and O), and super-alternation (I and A, and O and E).

 

Contraries (A and E)

Two propositions are contraries if, but only if, the propositions cannot both be true but both can be false. A pair of  universal propositions exhibits this characteristic. Consider the A and E statements "All the students in logic are female" and "None of the students in logic are female".  It is logically impossible that both are true, but it is possible that both are false.  If we know that one of the universal propositions is true then we can be certain that its contrary is false.

 

Sub-Contraries (I and O)

Two propositions are sub-contraries if, but only if,  the propositions can both be true but cannot both be false. A pair of particular propositions exhibits this characteristic. Consider the I and O statements "Some of the students in logic are female" and "Some of the students in logic are not female".  It is logically possible that both are true (in fact, they are for our class), but it is impossible that both are false.  If we know that one of the particular propositions is false then we can be certain that its contrary is true.

 

Contradictories (A and O and E and I)

Two categorical propositions are contradictory if, but only if, exactly one of them is true and the other is false. In the traditional square of opposition, an A proposition and its corresponding O are contradictories, as are an E proposition and its corresponding I.  Knowing the truth value of one member of a pair of contradictories guarantees that the other member has the opposite truth value.

 

Sub-Alternation (A and I, and E and O)

Sub-alternation is a logical relation between a universal proposition (A or E) and its corresponding particular proposition (I or O).  In traditional Aristotelian logic, the truth of a universal proposition logically entails the truth of its corresponding sub-altern.  However, since the corresponding proposition requires the existence of members of the subject class for the proposition to be true (how, after all, could "Some logicians are rational thinkers" be true if there were no logicians at all?), the truth of a universal proposition in classical logic carries with it a belief that there are members of the subject class.  This belief is called existential import (because we are importing as assumption about the existence of certain objects).  If one assumes existential import, as Aristotle did, then truth flows down on the square of opposition, that is, the truth of a universal proposition logically entails the truth of its corresponding particular.  Modern logic rejects existential import for reasons that will soon become clear.

 

Super-Alternation (I and A, and O and E)

Super-alternation is a logical relation between a particular proposition (I or O) and its corresponding universal proposition (A or E).  In traditional Aristotelian logic, the falsity of a particular proposition logically entails the falsity of its corresponding super-altern.  However, since the corresponding proposition requires the existence of members of the subject class for the proposition to be false (how, after all, could "No logicians are rational thinkers" be false if there were no logicians at all?), the falsity of a particlar proposition in classical logic carries with it a belief that there are members of the subject class.  This belief is called existential import (because we are importing as assumption about the existence of certain objects).  If one assumes existential import, as Aristotle did, then falsity flows up on the square of opposition, that is, the falsity of a particular proposition logically entails the falsity of its corresponding universal.  Modern logic rejects existential import for reasons that will soon become clear.

Return to Tutorials Index           Go on to Venn Diagrams               Go on to Syllogisms