Transcript Ch 5 Categorical Propositions

Chapter 5
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Categorical Propositions
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1. A deductive argument is one that claims to establish its conclusion
conclusively.
A valid deductive argument is one in which, if all the premises are
true, the conclusion must be true.
The theory of deduction aims to explain the relationship between
premises and conclusion in valid arguments. It also aims to provide
methods for evaluating deductive arguments.
There are two major logical theories that have been developed to
accomplish these aims: Aristotelian (or Classical) logic and Modern
symbolic Logic.
Chapters 5, 6, and 7 will cover Aristotelian logic. Chapters 8, 9, and 10
will cover Modern symbolic logic.
Aristotle
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Classical deductive (or Aristotelian) logic is based on the idea of
categories, or classes.
A categorical proposition affirms or denies, in whole or in part, that one
class is included in another.
Since categorical propositions deal with two states of two classes, there
are four possible categorical propositions.
(1) The universal affirmative (A) proposition states that every member
of one class is also a member of the second class.
(2) The universal negative (E) states that no member of one class is a
member of the second.
(3) In a particular affirmative (I), some members of one class are
members of the second
(4) And in a particular negative (O), some members of one class are not
members of the second.
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
Categorical Propositions
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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 classdefining 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.
Universal Affirmative
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All politicians are liars.
All men are mortal.
All good Web pages are written in html.
All good men come to the aid of their party.
All men have what it takes to become a successful
salesman.
All dogs go to heaven
The universal affirmative as stated by Aristotle is
not two-way. Consider, for example, example 1
above. It doesn’t mean that all liars are politicians.
In example 2, we are not stating that all mortal
things are men. (This “reverse” of the orignial
statement is called the converse of the statement)
All S are P Diagrams
Universal Negatives
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No politician is intelligent.
No man is immortal.
No good Web pages contain Java or browser-specific tags.
No good men will betray their principles.
No men have what it takes to be a successful mother.
The universal negative is effectively two-way, unlike the
universal affirmative. In other words, a universal negative
statement does imply its converse. For instance, in Example
1, we propose that no politicians are intelligent, and
therefore imply that no intelligent people are politicians. In
Example 2, we not only say that no men are immortal, but
that no immortal beings are men. This is the most important
distinction between the universal affirmative and the
universal negative, functionally speaking.
No S are P Diagrams
Particular Affirmative
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Some man is mortal.
There is a woman who is a politician.
At least one computer runs Microsoft products.
There is a fun Web site.
The particular affirmative states that there is at least one member of one class that is
a member of a second.
It doesn't imply that all members of one class are members of the second.These
sentences sound strange: a more natural language might say that "This Web site is
fun." or "Socrates is mortal." However, at this stage of the development of our
logical language, we want to be able to distinguish between saying that there is at
least one fun Web site and that a specific Web site is fun. While it is true that if this
Web site is fun then there is at least one Web site that is fun, it doesn't necessarily
follow that if there is at least one Web site that is fun, that this one is. You might
think this Web site was lame and Yahoo was fun, for instance.
This takes us naturally to the first thing to remember about the particular
affirmative: It isn't exactly right to talk about the particular affirmative having a
converse in the same way that it is to say that a universal term has one.
However, there is an implication involved in certain natural language statements
that forms the basis for most proofs of particular affirmative statements. It's simple:
to prove that some A is B, all you need to do is find one example of when A is B, and
bingo! You're done.
Some S is P Diagrams
Particular Negative
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Some fictional creatures are not mortal.
Some Web sites are not fun.
Some philosophers don't make sense.
Some computers are not expensive.
The connection between the particular affirmative and negative is easy to see. In
fact, in our natural language, we often don't make much of a distinction
between the two: modern logic doesn't either.
For example, when you think of the negative particular statement "Some
woman is not beautiful." it seems equivalent to the affirmative particular
statement "Some woman is homely." On further examination, we see that this is
only true if every woman is either beautiful or homely.
Similarly to the particular affirmative, the particular negative can be proven by
finding a single example. For instance, if we want to prove that some politician
is corrupt, all we have to do is find one corrupt politician.
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Exercises P. 187
Some S is not P Diagrams
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Propositions are said to have quality—either affirmative or negative—
and quantity—either universal or particular.
Quality: A and I are affirmative (help AffIrmo – Latin for I affirm); E and
O are negative (help nEgO- Latin for I deny)
Quantity: A and E are universal; I and O are particular.
In categorical propositions we use variations of the verb “to be” to
connect the subject and predicate terms – called a copula
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Propositions may also be distributed or undistributed:
A proposition is said to distribute a term if it refers to all members of the
class designated by the term.
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All dogs go to heaven -
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In the A proposition above, for example, the subject term (dogs) is
distributed, but the predicate term (things that go to heaven) is not.
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No men are immortal
In the E proposition above, the subject term (men) is distributed
because the whole class of men is excluded from the class of
immortality; it also asserts that all immortals (predicate term) are
excluded from being men.
Some Mexicans come illegally
In the I proposition, no assertion is made about Mexicans (subject)
and no assertion is made about those who come illegally
(predicate). NOT DISTRIBUTED
Some male dancers are not strippers
In this O proposition, nothing is said about all male dancers
(subject), it says that only part of male dancers are excluded from
the class of strippers. But these male dancers are excluded from the
whole of the stripper class (predicate). Given the particular dancers
being referred to, the proposition says that each and every member
of the class of strippers is not one of these particular male dancers.
Therefore, the whole class is referred to and the predicate class is
DISTRIBUTED.
Exer. P 193
Two mnemonic devices for
distribution
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“Unprepared Students Never Pass”
Universals distribute Subjects.
Negatives distribute Predicates.
“Any Student Earning B’s Is Not On Probation”
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A distributes Subject.
E distributes Both.
I distributes Neither.
O distributes Predicate.
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The traditional square of opposition graphically displays the
relationships that exist between the four different standard form
categorical propositions: A,E,I, O.
Propositions can be contradictories, contraries, subcontraries,
subalterns, or superalterns.
Contradictories
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A and O propositions are contradictory, as are E and I propositions.
Propositions are contradictory when the truth of one implies the falsity of
the other, and conversely. Here we see that the truth of a proposition of
the form All S are P implies the falsity of the corresponding proposition of
the form Some S are not P. For example, if the proposition “all
industrialists are capitalists” (A) is true, then the proposition “some
industrialists are not capitalists” (O) must be false. Similarly, if “no
mammals are aquatic” (E) is false, then the proposition “some mammals
are aquatic” must be true.
They cannot both be true and cannot both be false.
Contraries
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A and E propositions are contrary. Propositions are contrary when they
cannot both be true; if one is true, then other must be false. They can
both be false. An A proposition, e.g., “all giraffes have long necks”
cannot be true at the same time as the corresponding E proposition: “no
giraffes have long necks.” Note, however, that corresponding A and E
propositions, while contrary, are not contradictory. While they cannot
both be true, they can both be false, as with the examples of “all planets
are gas giants” and “no planets are gas giants.”
By saying that ‘some’ rather than all or none, both statements would be
false
Subcontraries
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I and O propositions are subcontrary. Propositions are subcontrary when
it is impossible for both to be false; if one is false the other must be true.
They can both be true. Because “some lunches are free” is false, “some
lunches are not free” must be true. Note, however, that it is possible for
corresponding I and O propositions both to be true, as with “some
nations are democracies,” and “some nations are not democracies.”
Again, I and O propositions are subcontrary, but not contrary or
contradictory.
Subalternation
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Two propositions are said to stand in the relation of Subalternation when
the truth of the first (“the superaltern”) implies the truth of the second
(“the subaltern”), but not conversely. A propositions stand in the
Subalternation relation with the corresponding I propositions. The truth
of the A proposition “all plastics are synthetic,” implies the truth of the
proposition “some plastics are synthetic.” However, the truth of the O
proposition “some cars are not American-made products” does not imply
the truth of the E proposition “no cars are American-made products.” In
traditional logic, the truth of an A or E proposition implies the truth of
the corresponding I or O proposition, respectively. Consequently, the
falsity of an I or O proposition implies the falsity of the corresponding A
or E proposition, respectively. However, the truth of a particular
proposition does not imply the truth of the corresponding universal
proposition, nor does the falsity of an universal proposition carry
downwards to the respective particular propositions.
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Inferences from square of opposition:
A number of very useful immediate inferences may be readily
drawn from the information embedded in the traditional square of
opposition. Given in the truth, or the falsehood, of anyone of the
four standard form categorical propositions, it will be seen that the
truth or falsehood of some or all of the others can be inferred
immediately.
A being given as True: E is false; I is true; O is false.
E being given as True: A is false; I is false; O is true.
I being given as True: E is false; A and O are undetermined.
O being given as True: A is false; E and I are undetermined.
A being given as False : O is true , E and I are undetermined.
E being given as False: I is true; A and O are undetermined.
I being given as False: A is false; E is true; O is true.
O being given as False: A is true; E is false; I is true.
Table of Inferences
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If true:
A
false
E
undetermined
true
I
undetermined
false
O
All men are wicked creatures.
No men are wicked creatures
If true:
E
false
A
undetermined
false
I
true
O
undetermined
No men are wicked creatures.
All men are wicked creatures
If false:
Some men are wicked creatures.
Some men are not wicked creatures.
Some men are wicked creatures.
Some men are not wicked creatures.
true
If false:
true
Table of Inferences II
If true:
I
Undetermined A
False
E
undetermined 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
Some men are not wicked creatures. If false:
All men are wicked creatures
true
No men are wicked creatures.
false
Some men are wicked creatures.
true
O
A
E
I
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The square of opposition has three important kinds of immediate
inference that are not directly associated with the square of opposition:
(1) Conversion
(2) Obversion
(3) Contraposition
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Conversion
Conversion
An inference formed by interchanging the subject and predicate terms of a
categorical proposition. Not all conversions are valid.
Conversion grounds an immediate inference for both E and I propositions That is,
the converse of any E or I proposition is true if and only if the original proposition
was true. Thus, in each of the pairs noted as examples either both propositions are
true or both are false.
Steps for Conversion: Reversing the subject and the predicate terms in the
premise.
Valid Conversions
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Convertend
A: All S is P.
E: No S is P
I : Some S is P
O: Some S is not P
Converse
I: Some P is S (by
limitation)
E: No P is S
I : Some P is S
(conversion not valid)
Conversion Example
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Example:
All bags are mangoes.-A
Some mangoes are bags.-I
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No men are intelligent.-E
No intelligent are men.-E
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Some cows are tables.-I
Some tables are cows.-I
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Some students are not cats.
(not valid)
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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.
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.
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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
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The converse of“Some S is not P,” does not yield an valid
immediate inference.
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Convertend
O proposition: Some dogs are not cocker spaniels.
Converse
O proposition: Some cocker spaniels are not dogs.
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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.
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.
Conversion Table
Obversion
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Obversion
An inference formed by changing the quality of a proposition and replacing the
predicate term by its complement. Obversion is valid for any standard form
Categorical proposition.
Obversion is the only immediate inference that is valid for categorical propositions of
every form. In each of the instances, the original proposition and its obverse must have
exactly the same truth-value, whether it turns out to be true or false.
Steps for Obversion:
Replace the quality of the given statements. That is, if affirmative, change it into
negative, and if negative, change it into affirmative.
Replace the predicate term by its complementary term.
Valid Obversions
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Obverted
A: All S is P.
E: No S is P
I : Some S is P
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O: Some S is not P
Obverse
E: No S is non-P.
A: All S is non-P.
O : Some S is not non-P
I: Some S is non-P
Obversion
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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 Example
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Example:
All females are perfect beings.-A
No females are non-perfect beings.-E
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No female are perfect beings.-E
All female are non-perfect beings.-A
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Some female are perfect beings.-I
Some females are not non-perfect beings.-O
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Some female are not perfect beings.-O
Some female are non-perfect beings.-I
Obversion II
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The obverse is logically equivalent to the obvertend. Obversion is
thus a valid immediate inference when applied to any standardform categorical proposition.
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The obverse of the A proposition "All S is P" is the E proposition
"No S is non-P."
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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.
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Obvertend
I-proposition: Some rap songs are lullabies.
Obverse
O-proposition: Some rap songs are not nonlullabies.
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Obvertend
O-proposition: Some movie stars are not
geniuses.
Obverse
I-proposition: Some movie stars are nongeniuses.
Obversion IV
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 nonfictional 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.
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Contraposition
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An inference formed by replacing the subject term of a proposition with the
complement of its predicate term, and replacing the predicate term by the
complement of its subject term. Not all contrapositions are valid.
Contraposition is a reliable immediate inference for both A and O propositions; that is,
the contrapositive of any A or O proposition is true if and only if the original proposition
was true. Thus, in each of the pairs, both propositions have exactly the same truth-value.
Note: In contraposition the subject of the conclusion is contradictory of the
predicate of the premise and predicate of the conclusion is contradictory of the
subject of the premise.
Steps for Contraposition:
a. Convert the statement: reverse the subject and the predicate terms.
b. Replace both terms by their complementary terms.
Valid Contrapositions
Premises
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
O: Some non-P is not non-S.
Contraposition
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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
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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
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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
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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.
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E premise:
No dentists are non-graduates.
The contrapositive is: No graduates are non-dentists.
Obviously this is not true.
Contraposition V
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.
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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.
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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 nongame show hosts.
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Contraposition VIII
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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.
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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 tonguetwisting O proposition, "Some non-brain
surgeons are not non-game show hosts."
Contraposition IX
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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 X
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)
Contraposition not
valid.
I: Some S is P.
O: Some S is not P.
Some non-P is not
non-S.
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Contraposition Example
Example:
All citizens are voter.-A
All non-voters are non-citizens.-A
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No politicians are honest.-E
Some-non-honest are not non-politicians.-O
(by limitation)
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Some applicants are graduate. -I
(cannot be contraposited)
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Some students are not scholarship holders.-O
Some non-scholarship holders are not non-students.-O
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Exer. P. 205-207
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}



The problem of existential import presents some problems for the
relationships suggested by the traditional square of opposition.
As a result, most modern logicians adopt a different interpretation of
the square, called Boolean.
Under this interpretation, particular propositions (I and O) have
existential import; but universal propositions (A and E) do not.
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.

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 II


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 III
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.



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 V

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.
Existential Import and the Interpretation
of Categorical Propositions VI


Diagrams and symbolizing techniques are useful in helping to visualize
the relationships of categorical propositions.
Venn diagrams are especially effective at exhibiting the relationships
between classes by marking and shading overlapping circles.
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:




If Tweety is a canary, then
Tweety is a bird.
Tweety is a canary.
Tweety is a bird.
Valid/sound





If Tweety is a bird, then
Tweety is a canary.
Tweety is a bird.
Tweety is a canary.
Valid/unsound





If Tweety is a canary, then
Tweety is a bird.
Tweety is a bird.
Tweety is a canary.


Using if p then q


Euler Diagram (circles)
Invalid





1. What are the properties of A, E, I, and O propositions? Come up with
examples of propositions for each of these types of standard form
categorical propositions.
2. What do affirmative propositions have in common? What do
particular propositions have in common? What about universal and
negative propositions? How do the terms quality and quantity come
into play in these considerations?
3. What is the difference between contraries and contradictories?
Between contraries and subcontraries?
4. When does conversion result in valid inferences? Why does it work
then, but not in other cases? Consider the same question with
contraposition and obversion as well.
5. Why is existential import so problematic for Aristotelian logic? What
changes does it require to the square of opposition?
DISCUSSION





1. What are the options for dealing with the question of existential
import? Why should we adopt one option over the other?
2. What is the meaning of the traditional square of opposition? How does
the position of each proposition exhibit the relationships between them?
What inferences does it illustrate?
3. When using Venn diagrams, what do shading, overlapping, and “x”
mean? How is each of the standard-form categorical propositions
diagrammed using this method?
4. What does existential import entail? Which propositions have
existential import? Why is this a problem for interpreting the traditional
square of opposition?
5. What changes to the square of opposition result from the Boolean
interpretation of existential import?
ESSAYS