Abstract: The
most important properties of standard form categorical propositions are explained and
illustrated.
I. What is a categorical proposition?


A. The long range goal of our study of
traditional logic
is to give a theory of deduction, i.e., to explain the relationship
between the premisses and conclusion of a valid argument and provide
techniques for the appraisal of deductive arguments. Hence, we will
be distinguishing between valid and invalid arguments.



1. A deductive argument is defined as one
whose premisses are claimed to provide conclusive evidence for the truth of its
conclusion.



2. A valid deductive argument is one in
which it is impossible for the premisses to be true without the conclusion
being true also.


B. Our study of deduction, for the present, will
be about arguments stated in categorical
propositions.
Here is an example of this kind of argument:



No honest people are persons who embroider the
truth.
Some politicians are persons who embroider the truth.
Some politicians are not honest people.



1. A categorical proposition
is defined as any proposition that can be interpreted as asserting
a relation of inclusion or exclusion, complete or partial, between
two classes.



For example, the proposition or statement that “All human beings are
mortal” implies anything that is a human being is included into the
class of mortal things.



2. A class is defined as a collection of
all objects which have some specified characteristic in common. Understanding this
definiion is no more complicated than observing that the class of
“lightbulbs” all have the common characteristic of “being a
lightbulb.”




Thus, we can have four possible class relations in the
various kinds of categorical propositions:




Utilizing the classes, “people” and
“good beings” we can illustrate these possibilities:




a. complete inclusion ⟶ “All people
are good beings.”




b. complete exclusion ⟶ “No people
are good beings.”




c. partial inclusion ⟶ “Some people
are good beings.”




d. partial exclusion ⟶ “Some people
are not good beings.”




We can also describe these four kind of
statements respectively as




a. universal affirmative




b. universal negative




c. particular affirmative




d. particular negative



3. Often, it is convenient to look at the general
form of the statements given above. To these forms, special names are given:
A, E, I, and O.




A: All S is P.





E: No S is P.





I: Some S is
P.





O: Some S is
not P.





… where S and P stand for the
logical subject and the logical predicate of the statement respectively.




4. An oftused mnemonic device for remembering
the quality of statements is to remember the red underlined letters
in the Latin words:
Affirmo and Nego.
In this fashion, A and I statements are seen to be
affirmative, E and O are remembered as negative.



5. Note, in categorical propositions, the logical
subject S and the logical predicate P represent something different from
the grammatical subject and the grammatical predicate as taught
in English grammar classes.




For example, in the pretense statement,
“All (unfledged floithoisters) are (things apt
to become unflaggled)”
… the logical subject is composed of all of the words between the quantifier
“All” and the copula ”are,” and the logical predicate is
composed of all of the words after the copula “are” until the end of
the statement.
So the logical subject of this statement is “unfledged floithoisters”
and the logical predicate is “things apt to become
unflaggled.”



6. Also note that the word “Some” in
the particular affirmative O statement is taken to mean “at least
one.” This meaning differs somewhat from its ordinary language meaning.
For example, the statement “Some women are physicists at CERN is
considered true if there is at least one woman physicist at CERN. (Additionally
this statement would be true even if it turned out that all except one person
were women physicists at CERN.)



7. A model statement, then, can be represented in
general as




Quantifier
[subject term] copula [predicate term]. 
II. Analysis of the Categorical Proposition:
Quality, Quantity, and Distribution


A. The quantity of a categorical
proposition is determined by whether or not it refers to all members of
its subject class (That is, the statement is considered either universal or
particular in quantity.) To question “How many members of the
subject class are being discussed?” asks for quantity.


B. The quality of a categorical
proposition is determined by whether the asserted class relation is one of
inclusion or exclusion (That is, the statement or proposition is considered
either affirmative or negative in quality.)


C. Indicators of “how many” are called
quantity indicators (i.e., quantifiers) and specifically are
“All,” “No,” and “Some.”


D. Indicators of affirmative and negative are
quality indicators (i.e., qualifiers) and specifically are
“are,” “are not,” “is,” “is not,”
and “No.”



Note that “No” is both a quantifier and
a qualifier.


E. In sum, thorough
knowledge of the following table is absolutely essential to do well in
categorical logic:


Name 
Form 
Quantity 
Quality 
Distribution 
Subject 
Predicate 
A 
All S is P 
universal 
affirmative 
distributed 
undistributed 
E 
No S is P 
universal 
negative 
distributed 
distributed 
I 
Some S is P 
particular 
affirmative 
undistributed 
undistributed 
O 
Some S is not P 
particular 
negative 
undistributed 
distributed 


F. Distribution
of a term.



1. A distributed term is a term of a
categorical proposition that is used with reference to every member of a
class. If the term is not being used to refer to each and every member
of the class, it is said to be undistributed.



2. Consider the following propositions:




A: All birds are winged creatures.




E: No birds are wingless creatures.




I: Some birds are black things.




O: Some birds are not black things.

The kinds of distribution of subject and predicate
terms appearing in the above listed statements are explained here:


Subject

Predicate



A: refers to all
birds 
does not refer to every member,
e.g., bats, flying fish. 


E: refers to all birds
by indicating that they are not part of the predicate class 
refers to all wingless
creatures by indicating that they are not part of the subject class 


I: refers only to some
birds 
refers only to some black
things, viz., those which are birds 


O: refers only to some
birds, not all of them 
refers to all members of the
class! Viz., not one of them is in the class referred to by "some birds" 



3. For the predicate of the O
proposition, consider the following analogy. If we know that there is a
book not in a bookcase, then we know something about each and every shelf
in that bookcase—namely, we know that the book is not on that
shelf.



4. There are three ways to remember the
distribution status of subject and predicate for standard form categorical
propositions:




a. Memorize it. (Undoubtedly, this is most
recommended way.)




b. Figure it out from an example (as was done
above).




c. Remember the following rule:





The quantity of a standard form
categorical proposition determines the distribution of the subject (such
that if the quantity is universal, the subject is distributed and if the
quantity is particular, the subject is undistributed), and …





the quality of a standard form categorical
proposition determines the distribution status of the predicate (such that
if the quality is affirmative, the predicate is undistributed, and if the quality
is negative, the predicate is distributed).
