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. All politicians are liars.

2. No politicians are liars.

3. Some politicians are liars.

4. 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 PWhere, 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: 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

Table of Inferences II

If true: I Some men are wicked creatures. If false:Undetermined A All men are wicked creatures falseFalse E No men are wicked creatures. trueundetermined 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

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 non-lullabies.

• 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 non-

dentists.• 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.

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

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}

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: