Lecture 7
Categorical Propositions
and
Immediate Inferences
Categorical
Propositions: Any
proposition which relates two or more classes of objects
All categorical propositions are divided into two distinct logical parts which are associated with the terms which make up the object classes being compared
I. Parts of Categorical Propositions:
A. Subject Term  first category or class
B. Predicate Term  second category or class
Examples:
Cats
are members of the mammal family.
Subject Term
Predicate Term
Asteroids
are astronomical objects.
Subject Term
Predicate Term
C. Copula
 that which links the terms of a C. Proposition together
Examples:
Cats
are members of the mammal class.
Copula
Asteroids are astronomical objects.
Copula
D. Quantifies  the words which specify the
quantity of the subject class which is related to the predicate class
1. Universal:
a) 'All'  inclusive of a whole class
b) 'No'  exclusive of a whole class
2. Particular: 'Some' 
inclusive of part of a class
Note: We
should keep in mind that the quantifiers ('all', 'no', and 'some') are implied
in all categorical propositions even though they are not explicitly
stated. If these quantifiers are not
explicitly used in the formulation of the proposition it is said to be in
nonstandard form. The standard
form for all categorical propositions is the following:
All S are P.
No
S are P.
Some
S are P.
Some
S are not P.
If our examples above were in
standard form they would look like this:
All cats
are members of the mammal family.
All
asteroids are astronomical objects.
E. Quality
 the kind of affirmation made by the proposition
1. Affirmative  asserts a quality to a class
2. Negative  denies a quality to a class
When we put the quality and
quantity of categorical proposition together we see that there are four and
only four possible arrangements:
Universal
Affirmative Particular
Affirmative
Universal
Negative Particular
Negative
Thus, there are only four
types of standard form categorical proposition possible. As a shortcut we may tag each with a
letter as follows:
A Universal
Affirmative All S are P.
E Universal Negative No S are P.
I Particular
Affirmative Some S are P.
O Particular Negative Some S are not P.
F. Distribution
 the quantity associated with either the subject or predicate term of a
categorical proposition
Definition: a term is 'distributed' if the proposition makes an assertion about every
member of the class denoted by the term.
1. Universal
Affirmative  All S are P.
2. Universal
Negative  No S are P.
Note: in
light of this distribution we might be tempted to formulate the E proposition
as "All S are not P."
But this formulation is ambiguous since it could mean one of two
different things: suppose we said
"All pit bulls are not mean."
This could be interpreted as
1. No pit bulls are mean, or
2. Some pit bulls are not mean
Without a specific context it
is impossible to determine which is the intended meaning. Thus, even though both terms are distributed
in the E claim, we always use the standard form "No s are P."
3. Particular
Affirmative  Some S are P.
4. Particular
Negative  Some S are not P.
II. Categorical Propositions and Immediate Inferences:
Aristotle first divided the
proposition horizontally by quantity and vertically by quality.
From this arrangement
Aristotle derived four logical relations
1. From A to E and E to A
2. From I to O and O to
I
3. From A to O and O to
A, and from E to I and I to E
4. From A to I and I to
A, and E to O and O to E
A. Aristotle's Square of Opposition and Immediate
Inference
1. Contrary  at
least one claim is false (they cannot both be true)
2. Subcontrary 
at least one is true (they cannot both be false)
3. Contradiction
Ð they have opposite truth values
4. Subalternation
 truth of the universal implies the truth of the particular, and falsity of
the particular implies the falsity of the universal.
B. Modern Square of Opposition and Immediate
Inference
1.
Universal propositions (Boolean)
All
S are P. = No members of S are outside P.
No
S are P. = No members of S are inside P.
2.
Particular propositions (Boolean)
Some
S are P. = Some S exists, and it is a P.
Some
S are not P. = Some S exists and it is not a P.
The Boolean interpretation of
the Categorical Propositions leaves only one possible relationship for
immediate inference: Contradiction.
3. Existential
Fallacy  occurs when the traditional
square is used in conjunction with nonexistent entities.
III. Categorical Propositions and Venn Diagrams:
The most useful device for
understanding the nature of modern Categorical Propositions is the circle
diagram of John Venn. A Venn
diagram is a model for some universe
of discourse.
Note that there are four
regions in the diagram each corresponding to some element of a categorical
proposition:
1. The region of S and only S (not P).
2. The region of S and P (both S and P).
3. The region of P and only P (not S).
4. The region of not S, and not P, and not S and P (i.e., what lies beyond both S and P (not S and
not P).
UA/A Ð All S is P. (None of S is outside of P.)
UN/E Ð No S is P. (None of S is inside of P.)
PA/I Ð Some S is P. (At least one S is inside of
P.)
PN/O Ð Some S is not P. (At least one S is not
inside of P.)
IV. Translating Ordinary Sentences to Categorical
Propositions:
A. The main rule to observe is that all standard form CPs must have a subject term, predicate
term, quantifier, and copula in the following form:
In the case of ambiguous sentences, isolate what you think is the primary meaning, then follow the guidelines below:
1. Where there is a Term without a noun: supply the missing noun
2.
Nonstandard Verbs: always replace
with 'are' and 'are not'
3. Singular Propositions: translate as 'all' with a parameter
4.
Adverbs and Pronouns: translate as follows 
a)
spatial adverbs  'places'
b)
temporal adverbs  'times'
c)
pronouns  'persons' or 'things'
5. Unexpressed Quantifiers: choose the probable quantity
6.
Nonstandard Quantifiers: translate as follows 
a)
few  'some'
b)
any or anyone  'all'
c)
not every or not all  'some ______ are not'
Note: a
statement beginning with 'few' must be translated as a compound I/O
proposition.
7.
Conditionals: translate as follows

a) if . . . then  'all' or 'no'
b) where 'if' occurs in the middle  move 'if
predicate' to beginning and translate 'all' or 'no'
c) transposition  negate both antecedent and
consequent and switch their order, then translate 'all' or 'no'
8. Exclusive Propositions: conditionalize then categorize
Note: 'only'
and 'none but' at the beginning require term reversal
9. 'The only'  translate as 'all'
10. Exceptive Propositions  must be translated as a
compound E/A proposition