Logic and critical thinking

1. Presented by
Syed Peer Muhammad Shah
Syed Bilal Haider Shah

2.  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).

3.  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.

4.  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. All politicians are liars.
2. No politicians are liars.
3. Some politicians are liars.
4. Some politicians are not liars.

5.  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.

6.  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.

7.  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.

8.  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.

9.  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.

10.  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.

11.  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.

12.  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.

13.  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."

14. 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

15.  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.

16.  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.

17.  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.

19.  “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.

20.  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.

21.  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.

22.  “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.

23.  Please turn to page 188 in your book.

24.  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.

25.  There are four ways in which propositions may be
opposed-as contradictories, contraries, subcontraries,
and subalterns.

27.  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.

28.  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.

29.  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.

30.  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.

31.  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.

32.  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.

33.  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.

34.  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.

35. If true: A All men are wicked creatures. If false:
false E No men are wicked creatures undetermined
true I Some men are wicked creatures.
undetermined
false O Some men are not wicked creatures. true
 If true: E No men are wicked creatures. If false:
false A All men are wicked creatures
undetermined
false I Some men are wicked creatures. true
true O Some men are not wicked creatures.
undetermined

36. If true: I Some men are wicked creatures. If false:
Undetermined A All men are wicked creatures false
False E No men are wicked creatures. true
undetermined O Some men are not wicked creatures. True
•If true: O Some men are not wicked creatures. If false:
false A All men are wicked creatures true
undetermined E No men are wicked creatures. false
undetermined I Some men are wicked creatures. true

37.  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.

38.  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.

39.  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.

40.  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."

41.  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.

42. 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.

43.  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."

44.  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."

45.  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.

46.  Obvertend
I-proposition: Some rap songs are lullabies.
Obverse
O-proposition: Some rap songs are not non-lullabies.
 Obvertend
O-proposition: Some movie stars are not geniuses.
Obverse
I-proposition: Some movie stars are non-geniuses.

47.  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.

48.  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.

49.  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.

50.  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.

51.  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.

52.  E premise:
 No dentists are non-graduates.
 The contrapositive is: No graduates are non-
dentists.
 Obviously this is not true.

53.  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.

54.  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.

55.  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.

56.  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."

57.  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.

58. 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.
O: Some S is not P. Some non-P is not
non-S.

59. Valid immediate inferences (other than from the square of opposition)
Proposition Obverse Converse Contrapositive
A All S is P. No S is non- P. Some P is S. All non-P is non-S.
{when true}
{when false}
{true}
{false}
{true, limited}
{indeterminate}
{true}
{false}
E No S is P. 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}
I Some S is P. Some S is not
non-P
Some P is S None Valid
{when true}
{when false}
{true}
{false}
{true}
{false}
O Some S is not P. Some S is non-
P
None Valid Some non-P is not
non-S
{when true}
{when false}
{true}
{false}
{true}
{false}

60.  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.

61.  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.

62.  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.

63.  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.

64.  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.

65.  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.

66. 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:

67.  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.

68.  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: