Categorical Propositions
• To help us make sense of our experience, we humans
constantly group things into classes or categories. These
classifications are reflected in our everyday language. In
formal reasoning the statements that contain our premises
and conclusions have to be rendered in a strict form so that
we know exactly what is being claimed. These logical
forms were first formulated by Aristotle (384-322 B.C.).
They are four in number, carrying the designations A, E, I,
O, as follows:
– All S is P (A).
– No S is P (E).
– Some S is P (I).
– Some S is not P (O).
Categorical Propositions II
• The letter "S" stands for the class designated by
the subject term of the proposition. The letter "P"
stands for the class designated by the predicate
term. Substituting any class-defining words for S
and P generates actual categorical propositions.
• In classical theory, the four standard-form
categorical propositions were thought to be the
building blocks of all deductive arguments. Each
of the four has a conventional designation: A for
universal affirmative propositions; E for universal
negative propositions; I for particular affirmative
propositions; and O for particular negative
propositions.
Categorical Propositions III
•
These various relationships between classes are
affirmed or denied by categorical propositions.
The result is that there can be just four different
standard forms of categorical propositions. They
are illustrated by the four following propositions:
1.
2.
3.
4.
All politicians are liars.
No politicians are liars.
Some politicians are liars.
Some politicians are not liars.
Universal Affirmative
• The first is a universal affirmative proposition. It is about two
classes, the class of all politicians and the class of all liars,
saying that the first class is included or contained in the second
class. A universal affirmative proposition says that every
member of the first class is also a member of the second class.
In the present example, the subject term “politicians”
designates the class of all politicians, and the predicate term
“liars” designates the class of all liars. Any universal
affirmative proposition may be written schematically as
All S is P.
where the terms S and P represent the subject and predicate
terms, respectively.
Universal Affirmative II
• The name “universal affirmative” is appropriate
because the position affirms that the relationship
of class inclusion holds between the two classes
and says that the inclusion is complete or
universal: All members of S are said to be
members of P also.
Universal Negative Propositions
• The second example
– No politicians are liars.
Is a universal negative proposition. It denies of
politicians universally that they are liars.
Concerned with two classes, a universal negative
proposition says that the first class is wholly
excluded from the second, which is to say that
there is no member of the first class that is also a
member of the second. Any universal proposition
may be written schematically as
No S is P
Where, again, the letters S and P represent the
subject and predicate terms.
Universal Negative Propositions II
• The name “universal negative” is appropriate
because the proposition denies that the relation of
class inclusion holds between the two classes –
and denies it universally: No members at all of S
are members of P.
Particular affirmative propositions
• The third example
– Some Politicians are liars.
is a particular affirmative proposition. Clearly,
what the present example affirms is that some
members of the class of all politicians are (also)
members of the class of all liars. But it does not
affirm this of politicians universally: Not all
politicians universally, but, rather, some particular
politician or politicians, are said to be liars. This
proposition neither affirms nor denies that all
politicians are liars; it makes no pronouncement on
the matter.
Particular affirmative propositions II
• The word “some” is indefinite. Does it mean “at least one,”
or “at least two,” or “at least one hundred?” In this type of
proposition, it is customary to regard the word “some” as
meaning “at least one.” Thus a particular affirmative
proposition, written schematically as
– Some S is P.
says that at least one member of the class designated by the
subject term S is also a member of the class designated by
the predicate term P. The name “particular affirmative” is
appropriate because the proposition affirms that the
relationship of class inclusion holds, but does not affirm it
of the first class universally, but only partially, of some
particular member or members of the first class.
Particular negative propositions
• The fourth example
– Some politicians are not liars
is a particular negative proposition. This example, like the
one preceding it, does not refer to politicians universally but
only to some member or members of that class; it is
particular. But unlike the third example, it does not affirm
that the particular members of the first class referred to are
included in the second class; this is precisely what is denied.
A particular negative proposition, schematically written as
Some S is not P.
says that at least one member of the class designated by the
subject term S is excluded from the whole of the class
designated by the predicate term P.
Quality and Quantity
• Every categorical proposition has a quality, either
affirmative or negative. It is affirmative if the
proposition asserts some kind of class inclusion,
either complete or partial. It is negative if the
proposition denies any kind of class inclusion,
either complete or partial.
• Every categorical proposition also has a quantity,
either universal or particular. It is universal if the
proposition refers to all members of the class
designated by its subject term. It is particular if the
proposition refers only to some members of the
class designated by its subject term.
General Schema of Standard-Form
Categorical Propositions
• Standard-form categorical propositions consist of
four parts, as follows:
• Quantifer (subject term) copula (predicate term)
• The three standard-form quantifiers are "all," "no"
(universal), and "some" (particular). The copula is
a form of the verb "to be."
Sentence
Standard Form
Attribute
All apples are delicious.
A All S is P.
Universal affirmative
No apples are delicious.
E No S is P.
Universal negative
Some apples are
delicious.
I Some S is P.
Particular affirmative
Some apples are not
delicious.
O Some S is not P.
Particular negative
Distribution
• Distribution is an attribute of the terms (subject
and predicate) of propositions. A term is said to
be distributed if the proposition makes an
assertion about every member of the class denoted
by the term; otherwise, it is undistributed. In other
words, a term is distributed if and only if the
statement assigns (or distributes) an attribute to
every member of the class denoted by the term.
Thus, if a statement asserts something about every
member of the S class, then S is distributed;
otherwise S and P are undistributed.
All S are P
– Here is another way to look at All S are P.
The S circle is contained in the P circle, which represents
the fact that every member of S is a member of P.
Through reference to this diagram, it is clear that every
member of S is in the P class. But the statement does not
make a claim about every member of the P class, since
there may be some members of the P class that are outside
of S.
Distribution
• Thus, by the definition of “distributed term”, S is
distributed and P is not. In other words for any
(A) proposition, the subject term, whatever it may
be, is distributed and the predicate term is
undistributed.
No S are P
• “No S are P” states that the S and P class are separate, which
may be represented as follows:
This statement makes a claim about every member of S and
every member of P. It asserts that every member of S is
separate from every member of P, and also that every member
of P is separate from every member of S. Both the subject and
the predicate terms of universal negative (E) propositions are
distributed.
Some S are P
• The particular affirmative (I) proposition states that at least one
member of S is a member of P. If we represent this one member
of S that we are certain about by an asterisk, the resulting diagram
looks like this:
Since the asterisk is inside the P class, it represents something that
is simultaneously an S and a P; in other words, it represents a
member of the S class that is also a member of the P class. Thus,
the statement “Some S are P” makes a claim about one member (at
least) of S and also one member (at least) of P, but not about all
members of either class. Thus, neither S or P is distributed.
Some S are not P
• The particular negative (O) proposition asserts that at least one
member of S is not a member of P. If we once again represent
this one member of S by an asterisk, the resulting diagram is as
follows:
Since the other members of S may or may not be outside of P, it is
clear that the statement “Some S are not P” does not make a claim
about every member of S, so S is not distributed. But, as may be seen
from the diagram, the statement does assert that the entire P class is
separated from this one member of the S that is outside; that is, it does
make a claim about every member of P. Thus, in the particular
negative (O) proposition, P is distributed and S is undistributed.
Two mnemonic devices for
distribution
• “Unprepared Students Never Pass”
Universals distribute Subjects.
Negatives distribute Predicates.
• “Any Student Earning B’s Is Not On Probation”
– A distributes Subject.
– E distributes Both.
– I distributes Neither.
– O distributes Predicate.
Exercises
• Please turn to page 188 in your book.
The Traditional Square of
Opposition
• Quality, quantity, and distribution tell us what
standard-form categorical propositions assert about
their subject and predicate terms, not whether those
assertions are true. Taken together, however, A, E, I,
and O propositions with the same subject and
predicate terms have relationships of opposition that
do permit conclusions about truth and falsity. In
other words, if we know whether or not a proposition
in one form is true or false, we can draw certain valid
conclusions about the truth or falsity of propositions
with the same terms in other forms.
Traditional Square of Opposition II
• There are four ways in which propositions may be
opposed-as contradictories, contraries,
subcontraries, and subalterns.
Contradictories
• Two propositions are contradictories if one is the
denial or negation of the other; that is, if they
cannot both be true and cannot both be false at the
same time. If one is true, the other must be false. If
one is false, the other must be true.
• A propositions (All S is P) and O propositions
(Some S is not P), which differ in both quantity
and quality, are contradictories.
Contradictories II
• All logic books are interesting books.
Some logic books are not interesting books.
• Here we have two categorical propositions with the same
subject and predicate terms that differ in quantity and quality.
One is an A proposition (universal and affirmative). The
second is an O proposition (particular and negative).
• Can both of these propositions be true at the same time? The
answer is "no." If all logic books are interesting, than it can't
be true that some of them are not. Likewise, if some of them
are not interesting, then it can't be true that all of them are.
• Can both propositions be false at the same time? Again, the
answer is "no". If it's false that all logic books are interesting,
then it must be true that some of them are not interesting.
Likewise if it's false that some of them are not interesting, then
all of them must be interesting.
• Like this pair, all A and O propositions with the same subject
and predicate terms are contradictories. One is the denial of
the other. They can't both be true or false at the same time.
Contradictories III
• E propositions (No S is P) and I propositions (Some S is P)
likewise differ in quantity and quality and are
contradictories.
• Example: No presidential elections are contested elections.
Some presidential elections are contested elections.
• Here again we have two categorical propositions with the
same subject and predicate terms that differ in both
quantity and quality. In this case, the first is an E
proposition—universal and negative—and the second is an
I proposition—particular and positive.
• Can both be true at the same time? The answer is "no." If
no presidential elections are contested, then it can't be true
that some are. Likewise is some are contested, then it can't
be true that none are.
Contradictories IV
• Can both be false at the same time? Again the
answer is "no." If it's false that no presidential
elections are contested, then it must be true that
some of them are. Likewise if it's false that some
are contested, then it must be the case that none
are.
• Like this pair, all E and I propositions with the
same subject and predicate terms are
contradictories. One is the denial of the other.
They can't both be true or false at the same time.
Contraries
• Two propositions are contraries if they cannot both be
true; that is, if the truth of one entails the falsity of the
other. If one is true, the other must be false. But if one is
false, it does not follow that the other has to be true.
Both might be false.
• A (All S is P) and E (No S is P) propositions-which are
both universal but differ in quality-are contraries unless
one is necessarily (logically or mathematically) true.
• For example:
• All books are written by Stephen King.
• No books are written by Stephen King.
• Both are false.
Subcontraries
• Two propositions are subcontraries if they cannot
both be false, although they both may be true.
• I (Some S is P) and O (Some S is not P)
propositions-which are both particular but differ in
quality-are subcontraries unless one is necessarily
false.
• For example:
• Some dogs are cocker spaniels.
• Some dogs are not cocker spaniels.
Subalternation
• Subalternation is the relationship between a universal
proposition (the superaltern) and its corresponding
particular proposition (the subaltern).
• According to Aristotelian logic, whenever a universal
proposition is true, its corresponding particular must
be true. Thus if an A proposition (All S is P) is true,
the corresponding I proposition (Some S is P) is also
true. Likewise if an E proposition (No S is P) is true,
so too is its corresponding particular (Some S is not
P). The reverse, however, does not hold. That is, if a
particular proposition is true, its corresponding
universal might be true or it might be false.
Subalternation II
• For example: All bananas are fruit. Therefore,
some bananas are fruit.
• Or, no humans are reptiles. Therefore, some
humans are not reptiles.
• However, we can’t go in reverse. We can’t say
some animals are not dogs. Therefore, no animals
are dogs.
• Or, some guitar players are famous rock
musicians. Therefore, all guitar players are
famous rock musicians.
Table of Inferences
If true:
false
true
false
A
E
I
O
All men are wicked creatures.
No men are wicked creatures
Some men are wicked creatures.
Some men are not wicked creatures.
If false:
undetermined
undetermined
true
• If true:
false
false
true
E
A
I
O
No men are wicked creatures.
All men are wicked creatures
Some men are wicked creatures.
Some men are not wicked creatures.
If false:
undetermined
true
undetermined
Table of Inferences II
If true:
Undetermined
False
undetermined
I
A
E
O
Some men are wicked creatures.
If false:
All men are wicked creatures
false
No men are wicked creatures.
true
Some men are not wicked creatures. True
•If true:
false
undetermined
undetermined
O
A
E
I
Some men are not wicked creatures.
All men are wicked creatures
No men are wicked creatures.
Some men are wicked creatures.
If false:
true
false
true
Conversion
• The first kind of immediate inference, called
conversion, proceeds by simply interchanging the
subject and predicate terms of the proposition.
• Conversion is valid in the case of E and I
propositions. “No women are American
Presidents,” can be validly converted to “No
American Presidents are women.”
• An example of an I conversion: “Some politicians
are liars,” and “Some liars are politicians” are
logically equivalent, so by conversion either can
be validly inferred from the other.
Conversion II
• One standard-form proposition is said to be the
converse of another when it is formed by simply
interchanging the subject and predicate terms of
that other proposition. Thus, “No idealists are
politicians” is the converse of “No politicians are
idealists,” and each can validly be inferred from
the other by conversion. The term convertend is
used to refer to the premise of an immediate
inference by conversion, and the conclusion of the
inference is called the converse.
Conversion III
• Note that the converse of an A proposition is not generally
valid form that A proposition.
• For example: “All bananas are fruit,” does not imply the
converse, “All fruit are bananas.”
• A combination of subalternation and conversion does,
however, yield a valid immediate inference for A
propositions. If we know that "All S is P," then by
subalternation we can conclude that the corresponding I
proposition, "Some S is P," is true, and by conversion
(valid for I propositions) that some P is S. This process is
called conversion by limitation.
Conversion IV
• Convertend
A proposition: All IBM computers are things that use electricity.
Converse
A proposition: All things that use electricity are IBM computers.
• Convertend
A proposition: All IBM computers are things that use electricity.
Corresponding particular:
I proposition: Some IBM computers are things that use electricity.
Converse (by limitation)
I proposition: Some things that use electricity are IBM computers.
• The first part of this example indicates why conversion applied directly to
A propositions does not yield valid immediate inferences. It is certainly
true that all IBM computers use electricity, but it is certainly false that all
things that use electricity are IBM computers.
• Conversion by limitation, however, does yield a valid immediate inference
for A propositions according to Aristotelian logic. From "All IBM
computers are things that use electricity" we get, by subalternation, the I
proposition "Some IBM computers are things that use electricity." And
because conversion is valid for I propositions, we can conclude, finally,
that "Some things that use electricity are IBM computers."
Conversion V
• The converse of“Some S is not P,” does not yield
an valid immediate inference.
• Convertend
O proposition: Some dogs are not cocker spaniels.
Converse
O proposition: Some cocker spaniels are not dogs.
• This example indicates why conversion of O
prepositions does not yield a valid immediate
inference. The first proposition is true, but its
converse is false.
Conversion Table
Does not convert to
A
A
All men are wicked creatures.
All wicked creatures are men.
Does convert to
E
E
No men are wicked creatures.
No wicked creatures are men.
Does convert to
I
I
Some wicked men are creatures.
Some wicked creatures are men.
Does not convert to
O
O
Some men are not wicked creatures.
Some wicked creatures are not men.
Obversion
• Obversion - A valid form of immediate inference
for every standard-form categorical proposition.
To obvert a proposition we change its quality
(from affirmative to negative, or from negative to
affirmative) and replace the predicate term with its
complement. Thus, applied to the proposition "All
cocker spaniels are dogs," obversion yields "No
cockerspaniels are nondogs," which is called its
"obverse." The proposition obverted is called the
"obvertend."
Obversion II
• The obverse is logically equivalent to the obvertend.
Obversion is thus a valid immediate inference when
applied to any standard-form categorical proposition.
• The obverse of the A proposition "All S is P" is the E
proposition "No S is non-P."
• The obverse of the E proposition "No S is P" is the A
proposition "All S is non-P."
Obversion III
• The obverse of the I proposition "Some S is P" is the O
proposition "Some S is not non-P."
• The obverse of the O proposition "Some S is not P" is the I
proposition "Some S is non-P."
• Obvertend
A-proposition: All cartoon characters are fictional characters.
Obverse
E-proposition: No cartoon characters are non-fictional
characters.
• Obvertend
E-proposition: No current sitcoms are funny shows.
Obverse
A-proposition: All current sitcoms are non-funny shows.
Obversion IV
• Obvertend
I-proposition: Some rap songs are lullabies.
Obverse
O-proposition: Some rap songs are not nonlullabies.
• Obvertend
O-proposition: Some movie stars are not geniuses.
Obverse
I-proposition: Some movie stars are non-geniuses.
Obversion V
• As these examples indicate, obversion always yields a
valid immediate inference.
• If every cartoon character is a fictional character, then it
must be true that no cartoon character is a non-fictional
character.
• If no current sitcoms are funny, then all of them must be
something other than funny.
• If some rap songs are lullabies, then those particular rap
songs at least must not be things that aren't lullabies.
• If some movie stars are not geniuses, than they must be
something other than geniuses.
Contraposition
• Contraposition is a process that involves replacing
the subject term of a categorical proposition with
the complement of its predicate term and its
predicate term with the complement of its subject
term.
• Contraposition yields a valid immediate inference
for A propositions and O propositions. That is, if
the proposition
• All S is P is true, then its contrapositive
All non-P is non-S is also true.
Contraposition II
• For example:
• Premise
• A proposition: All logic books are interesting
things to read.
• Contrapositive
• A proposition: All non interesting things to read
are non logic books.
Contraposition III
• The contrapositive of an A proposition is a valid
immediate inference from its premise. If the first
proposition is true it places every logic book in the
class of interesting things to read. The
contrapositive claims that any non-interesting
things to read are also non-logic books—
something other than a logic book—and surely
this must be correct.
Contraposition IV
• Premise:
• I-proposition: Some humans are non-logic
teachers.
• Contrapositive
• I-proposition: Some logic teachers are not human.
• As this example suggests, contraposition does not
yield valid immediate inferences for I propositions.
The first proposition is true, but the second is
clearly false.
Contraposition V
• E premise:
• No dentists are non-graduates.
• The contrapositive is: No graduates are nondentists.
• Obviously this is not true.
Contraposition VI
• The contrapositive of an E proposition does not yield a
valid immediate inference. This is because the propositions
"No S is P" and "Some non-P is non-S" can both be true.
But in that case "No non-P is non-S," the contrapositive of
"No S is P," would have to be false.
•
A combination of subalternation and contraposition
does, however, yield a valid immediate inference for E
propositions. If we know that "No S is P" is true, then by
subalternation we can conclude that the corresponding O
proposition, "Some S is not P," is true, and by
contraposition (valid for O propositions) that "Some non-P
is not non-S" is also true. This process is called
contraposition by limitation.
Contraposition VII
•
•
•
•
•
•
•
•
•
Premise:
E-proposition: No Game Show Hosts are Brain Surgeons.
Contrapositive
E proposition: No non-Brain Surgeons are non-Game show
hosts.
Premise:
E proposition: No game show hosts are brain surgeons.
Corresponding particular O proposition: Some game show
hosts are not brain surgeons.
Contrapositive
O proposition: Some non-brain surgeons are not non-game
show hosts.
Contraposition VIII
• The first part of this example indicates why contraposition
applied directly to E propositions does not yield valid
immediate inferences. Even if the first proposition is true
then the second can still be false. This may be hard to see at
first, but if we take it apart slowly we can understand why.
The first proposition, if true, clearly separates the class of
game show hosts from the class of brain surgeons, allowing
no overlap between them. It does not, however, tell us
anything specific about what is outside those classes. But
the second proposition does refer to the areas outside the
classes and what it says might be false. It claims that there
is not even one thing outside the class of brain surgeons
that is, at the same time, a non-game show host. But wait a
minute. Most of us are neither brain surgeons nor game
show hosts. Clearly the contrapositive is false.
Contraposition IX
• Contraposition by limitation, however, does yield
a valid immediate inference for E propositions
according to Aristotelian logic. By subalternation
from the first proposition we get the O proposition
"Some game show hosts are not brain surgeons."
And then by contraposition, which is valid for O
propositions, we get the valid, if tongue-twisting
O proposition, "Some non-brain surgeons are not
non-game show hosts."
Contraposition X
•
•
•
•
•
O proposition.
Premise:
Some flowers are not roses.
Some non-roses are not non-flowers.
This is valid. Thus we can see that contraposition
is a valid form of inference only when applied to
A and O propositions. Contraposition is not valid
at all for I propositions and is valid for E
propositions only by limitation.
Contraposition XI
Table of Contraposition
Premise
Contrapositive
A: All S is P.
A: All non-P is non-S.
E: No S is P.
O: Some non-P is not
non-S. (by limitation)
I: Some S is P.
Contraposition not
valid.
Some non-P is not
non-S.
O: Some S is not P.
Valid immediate inferences (other than from the square of opposition)
Proposition
A All S is P.
Converse
Contrapositive
No S is non- P.
Some P is S.
All non-P is non-S.
{true}
{false}
{true, limited}
{indeterminate}
{true}
{false}
All S is non- P.
No P is S.
Some non-P is not
not S
{when true}
{when false}
{true}
{false}
{true}
{false}
{true, limited}
{indeterminate}
Some S is P.
Some S is not
non-P
Some P is S
None Valid
{when true}
{when false}
{true}
{false}
{true}
{false}
{when true}
{when false}
E No S is P.
I
Obverse
O Some S is not P.
{when true}
{when false}
Some S is non- None Valid
P
Some non-P is not
non-S
{true}
{false}
{true}
{false}
Existential Import and the Interpretation of
Categorical Propositions
• Aristotelian logic suffers from a dilemma that
undermines the validity of many relationships in
the traditional Square of Opposition.
Mathematician and logician George Boole
proposed a resolution to this dilemma in the late
nineteenth century. This Boolean interpretation of
categorical propositions has displaced the
Aristotelian interpretation in modern logic.
Existential Import and the Interpretation
of Categorical Propositions II
• The source of the dilemma is the problem of existential
import. A proposition is said to have existential import if it
asserts the existence of objects of some kind. I and O
propositions have existential import; they assert that the
classes designated by their subject terms are not empty.
But in Aristotelian logic, I and O propositions follow
validly from A and E propositions by subalternation. As a
result, Aristotelian logic requires A and E propositions to
have existential import, because a proposition with
existential import cannot be derived from a proposition
without existential import.
Existential Import and the Interpretation
of Categorical Propositions III
• A and O propositions with the same subject and
predicate terms are contradictories, and so cannot
both be false at the same time. But if A
propositions have existential import, then an A
proposition and its contradictory O proposition
would both be false when their subject class was
empty.
• For example:
– Unicorns have horns. If there are no unicorns,
then it is false that all unicorns have horns and
it is also false that some unicorns have horns.
Existential Import and the Interpretation
of Categorical Propositions IV
• The Boolean interpretation of categorical propositions
solves this dilemma by denying that universal propositions
have existential import. This has the following
consequences:
• I propositions and O propositions have existential import.
• A-O and E-I pairs with the same subject and predicate
terms retain their relationship as contradictories.
• Because A and E propositions have no existential import,
subalternation is generally not valid.
• Contraries are eliminated because A and E propositions
can now both be true when the subject class is empty.
Similarly, subcontraries are eliminated because I and O
propositions can now both be false when the subject class
is empty.
Existential Import and the Interpretation
of Categorical Propositions V
• Some immediate inferences are preserved:
conversion for E and I propositions, contraposition
for A and O propositions, and obversion for any
proposition. But conversion by limitation and
contraposition by limitation are no longer
generally valid.
• Any argument that relies on the mistaken
assumption of existence commits the existential
fallacy.
Existential Import and the Interpretation
of Categorical Propositions VI
• The result is to undo the relations along the sides
of the traditional Square of Opposition but to leave
the diagonal, contradictory relations in force.
Symbolism and Diagrams for
Categorical Propositions
The relationships among classes in the Boolean interpretation
of categorical propositions can be represented in symbolic
notation. We represent a class by a circle labeled with the
term that designates the class. Thus the class S is
diagrammed as shown below:
Symbolism and Diagrams for
Categorical Propositions II
• To diagram the proposition that S has no members, or that
there are no S’s, we shade all of the interior of the circle
representing S, indicating in this way that it contains nothing
and is empty. To diagram the proposition that there are S’s,
which we interpret as saying that there is at least one
member of S, we place an x anywhere in the interior of the
circle representing S, indicating in this way that there is
something inside it, that it is not empty.
Symbolism and Diagrams for
Categorical Propositions III
• To diagram a standard-form categorical proposition, not
one but two circles are required. The framework for
diagramming any standard-form proposition whose subject
and predicate terms are abbreviated by S and P is
constructed by drawing two intersecting circles: