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 F are A. | ||
| Some F are 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:
The subject term and the predicate term stand for the two sets whose logical relation is characterized by the statement.
2.1.1 Sets
A set 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 set S and a rectangle stands for the universe of discourse.
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, S ∪ SC = 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 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 quality of 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.
The A Statement
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 P is 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.
The E Statement
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.
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.
The I Statement
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.
The O Statement
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.
The two overlapping circles form three areas: area α, β and γ. Everything in the area α is a member of S, but not a member of P. Everything in the area β is a member of S, and a member of P. Everything in the area γ is a member of P, but not a member of S.
The Venn Diagram for the A Statement
In the Venn Diagram for the A statement “All dolphins are mammals,” the area α is shaded. This means that the area is empty. Therefore, all members of the set D (the set of dolphins) must be in the area β. But if they are in the area β, then they must also be members of the set M (the set of mammals).
The Venn Diagram for the E Statement
In the Venn Diagram for the E statement “No bats are birds,” the area β 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 α. But if they are in the area α, then they cannot be members of the set D (the set of birds).
The Venn Diagram for the I Statement
In the Venn Diagram for the I statement “Some dinosaurs are carnivores,” there is an “X” in the area β. This means that the area is not empty, i.e., there is at least one member of the set D (the set of dinosaurs) in the area β. But if it is in the area β, then it must also be a member of the set C (the set of carnivores).
The Venn Diagram for the O Statement
In the Venn Diagram for the O statement “Some reptiles are not animals with legs,” there is an “X” in the area α. This means that the area is not empty, i.e., there is at least one member of the set R (the set of reptiles) in the area α. But if it is in the area α, 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.
| dragons | mammoths | electrons |
| pirates | ghosts | souls |
| psychics | magicians | witches |
| geometric shapes | music | black holes |
| demons | space aliens | imaginary numbers |
- Decide which of the following terms refer to fuzzy sets.
| 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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