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H OME TA BL E OF CO NT ENT S

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2 Categorical Logic

In Categorical Logic, we learn how to use the Venn Diagram to determine the validity of a special

type of deductive arguments called categorical syllogisms. For example, to decide whether the

following argument is valid

Some fair people are apathetic and some apathetic people are not kind. So some fair

people are not kind.

we translate the argument into its categorical standard form

Some A are not K.

Some Fare A.

Some Fare not K.

The letters A, K, and F stand for the three terms in the arguments

A: apathetic people

K: kind people

F: fair people

Then, using a formal procedure called the Venn Diagram method, we can find out that the form is

invalid. The argument is therefore not a good argument.

2.1 Categorical Statements

A categorical statement expresses how two sets of things relate to each other. For example, the

statement All ravens are black birds tells us that every member of the set ravens is also a member

of the set black birds.

A categorical statement is made up of four components:

I L O G I C

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The subject term and the predicate term stand for the two sets whose logical relation is

characterized by the statement.

2.1.1 Sets

Aset is a group or class of things that share certain properties in common. For example, the set of

ravens is a group of large birds that have black feathers and a croaking cry. Each of such a bird is

said to be a member of the set.

A general term such as ravens or dinosaurs refers to, or denotes, a set. The set of all dinosaurs

that ever existed is called the referent of the term dinosaurs.

A set with no member is called an empty set, symbolized as . For example, since there is no pink

elephant, the term pink elephants denotes an empty set. Other terms that refer to empty sets

include centaurs and elves because they are depicted in mythologies and do not exist in the real

world.

The universe of discourse is the set of all things being discussed. If we are talking about

animals, the universe of discourse would be the set of animals. If we are discussing people, the

universe of discourse would be the set of people.

Visually, the blue circle represents the setS and a rectangle stands for the universe of discourse.

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The complement of the set S is the set non-S (or SC). It contains everything in the universe of

discourse except the members of the set S. Represented visually, non-S is the blue area outside the

circle.

The union of a set and its complement is the universe of discourse. That is, SSC

=U.

The visual way of representing a set with a circle may give rise to the impression that a set has a

clear border and we can always clearly and precisely determine whether something belongs to the

set. However, it should be noted that while some sets are crisp, many are fuzzy sets. A fuzzy set

has a grey area for whether something is its member. For example, there are borderline cases of

whether a man is bald. So the set of bald men is a fuzzy set. It is more accurate to represent a fuzzy

set as a circle with a blur boundary.

2.1.2 Quantifiers and Copula

The first word in a categorical statement is the quantifier. The quantifier determines the quantity

of a categorical statement. There are two quantifiers in Categorical Logic: All and Some. The

word All is the universal quantifier. It is used to say something about every member of the set

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denoted by the subject term. Some is the existential quantifier. It is sometimes called the

particular quantifier. It is used to say something about at least one member of the set denoted by

the subject term.

A copula is a verb that links the subject and predicate terms together. In the standard categorical

statement form, the only copula that can be used is the verb are. If a categorical statement is

written with other verbs, it has to be rephrased using the standard copula are. For example,

Some birds cannot fly.

is not a standard categorical statement. It has to be rewritten as

Some birds are not flyers.

In doing so, we then have two terms birds and flyers that designate two sets. The copula is used

to affirm or deny membership, that is, to indicate whether members of the first set are also

members of the second set. If it affirms membership, then we say the categorical statement is

affirmative; if it denies membership, then the statement is negative. Being affirmative or negative is

called the qualityof a categorical statement.

2.1.3 The A E I O Statements

Given two sets, it is possible that every member of the first set is also a member of the second set. It

is also possible that none of the member of the first set is a member of the second set. The third

possibility is that there is at least one member of the first set that is also a member of the second

set. The fourth is that at least one member of the first set is not a member of the second set. As a

result, there are four types of categorical statements.

TheAStatement

The A statement is the universal affirmative statement. It asserts that every member of the set S is

also a member of the set P. It has the statement form that

All S are P.

Here S is the variable used for the subject term and Pis the variable for the predicate term. The

statement

All dolphins are mammals.

asserts that every member of the set dolphins is also a member of the set mammals.

TheEStatement

The E statement is the universal negative statement. It says that none of the member of the set S is

also a member of the set P. It has the statement form that

No S are P.

The statement

No bats are birds.

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denies any bat the membership of the set birds. In other words, the two sets do not share any

member in common.

You may wonder why we do not use

All S are not P.

as the statement form for the E statement. The reason is that the statement form is ambiguous.

Given the context and our background knowledge about the world, we would sometimes read All S

are not P as meaning No S are P, but on other occasions, we would take it to mean Some S are

not P. For example, given what we know about emeralds, we would take All emeralds are not

blue to mean that no emeralds are blue. However, given what we know about apples, we would not

misinterpret All apples are not green as meaning that no apples are green. Rather, we would take

it as an emphatic way of saying that some apples are not green. To avoid confusion, logicians

choose to use the form No S are P for the E statement.

TheIStatement

The I statement is the particular affirmative statement. It asserts that at least one member of the

set S is also a member of the set P. It has the statement form that

Some S are P.

The statement

Some dinosaurs are carnivores.

asserts that at least there is one dinosaur that is also a carnivore.

TheOStatement

The O statement is the particular negative statement. It denies at least one member of the set S the

membership of the set P. That is, it claims that at least one member of the set S is not a member of

the set P. It has the statement form that

Some S are not P.

The statement

Some reptiles are not animals with legs.

asserts that there is at least one reptile that is not an animal with leg.

2.1.4 Venn Diagrams for Statements

A categorical statement has a subject term and a predicate term. We use two overlapping circles to

represent the two sets denoted by the two terms.

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The two overlapping circles form three areas: area , and . Everything in the area is a member

ofS, but not a member ofP. Everything in the area is a member ofS, and a member ofP.

Everything in the area is a member ofP, but not a member ofS.

The Venn Diagram for theAStatement

In the Venn Diagram for the A statement All dolphins are mammals, the area is shaded. This

means that the area is empty. Therefore, all members of the set D (the set of dolphins) must be inthe area. But if they are in the area, then they must also be members of the set M(the set of

mammals).

The Venn Diagram for theEStatement

In the Venn Diagram for the E statement No bats are birds, the area is shaded. This means that

the area is empty. Therefore, all members of the set B (the set of bats) must be in the area . But if

they are in the area , then they cannot be members of the set D (the set of birds).

The Venn Diagram for theIStatement

In the Venn Diagram for the I statement Some dinosaurs are carnivores, there is an X in the

area. This means that the area is not empty, i.e., there is at least one member of the set D (the set

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of dinosaurs) in the area. But if it is in the area , then it must also be a member of the set C(the

set of carnivores).

The Venn Diagram for theOStatement

In the Venn Diagram for the O statement Some reptiles are not animals with legs, there is an X

in the area . This means that the area is not empty, i.e., there is at least one member of the setR

(the set of reptiles) in the area . But if it is in the area , then it cannot be a member of the set A

(the set of animals with legs).

Exercise 2.1

Decide which of the following terms refer to empty sets.I.

dragons mammoths electrons

pirates ghosts soulspsychics magicians witches

geometric shapes music black holes

demons space aliens imaginary numbers

Decide which of the following terms refer to fuzzy sets.II.

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tables dogs books

white flowers trees skyscrapers

friends gadgets toddlers

cars smart people prime numbers

managers heroes planets

valid arguments middle class games

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