diagramming categorical propositions

Logic

In Part 2 Modules 1 through 5, our topic is symbolic logic.

We will be studying the basic elements and forms that provide the structural foundations for critical reasoning.

Symbolic logic is a topic that unites the sciences and the humanities.

Researchers in logic may come from philosophy, mathematics, linguistics, or computer science, among other fields.

Statements in logic

In logic, a statement or proposition is a declarative sentence that has truth value.

When we say that a sentence has truth value, we mean that it makes sense to ask whether the sentence is true or false.

“Today is Monday” is a statement.“1 + 1 = 3” is a statement.

Quantifiers and categorical statements

In logic, terms like “all,” “some,” or “none” are called quantifiers.

A statement based on a quantifier is called a quantified statement or categorical statement.

“All bad hair days are catastrophes.”“No slugs are speedy.”“Some owls are hooty.”are examples of quantified or categorical statements.

Categories

Quantified or categorical statements state a relationship between two or more classes of objects or categories.

In the previous examples, bad hair days catastrophesslugsspeedy (things)owlshooty (things)are all categories.

Existential statementsA statement of the form “Some A are B” or “Some A aren’t B”

asserts the existence of at least one element (in logic, “some” means “at least one”).

Categorical statements having those forms are called existential statements.

“Some owls are hooty” “Some wolverines are not cuddly”are examples of existential statements.

Existential statements“Some owls are hooty” asserts that there exists at least one thing

that is both an owl and hooty.That is, the intersection of the categories “owls” and “hooty

things” is not empty.

We can convey that information by making a mark on a Venn diagram. We place an “X” in a region of a Venn diagram to indicate that that region must contain at least one element.

Diagramming existential statementsowlshooty thingsAccording to the statement “Some owls are hooty,” there must be at least one element in this region of the diagram.

X

Diagramming existential statementsThe existential statement “Some wolverines are not cuddly” asserts

that there must be at least one element who is a wolverine (W) but is not cuddly (C ).

WCAccording to the statement “Some W are not C,” there must be at least one element in this region of the diagram.

X

Universal statements

“All bad hair days are catastrophes” “No slugs are speedy”are examples of universal statements.

Negative universal statements

A statement of the form “No A are B” is called negative universal.It asserts that there is no element in both category A and category B

at the same time.

In other words, “No A are B” asserts that categories A and B are disjoint, which means that the intersection of the two categories is empty.

“No slugs are speedy” is a negative universal statement.

Diagramming negative universal statements

In logic, we use shading to indicate that a certain region of a Venn diagram is empty (contains no elements).

The negative universal statement “No slugs are speedy” asserts that the region of the diagram where “Slugs” and “Speedy things” intersect must be empty.

Diagramming negative universal statements

According to the statement “No slugs are speedy,” this region of the diagram must be empty.

Positive universal statements

A statement of the form “All A are B” is called positive universal.It asserts that there is no element in category A that isn’t also in

category B.

“All bad hair days are catastrophes” is an example of a positive universal statement.

Diagramming positive universal statements

The positive universal statement “All bad hair days are catastrophes” asserts that it is impossible to be a bad hair day (B) without also being a catastrophe (C).

This means that the region of the diagram that is inside B but outside C must be empty.

Diagramming positive universal statements

According to the statement “All B are C,” this region of the diagram must be empty.

Interpreting Venn diagrams in logicWe will use Venn diagrams (typically three-circle diagrams) to

convey the information in propositions about relationships between various categories.

Shading means “nothing here…”In logic, when a region of a Venn diagram is shaded, this tells us that that region contains no

elements.That is, a shaded region is empty.Suppose that we are presented with the marked Venn diagram shown below and on the following

slides. We should be able to interpret the meaning of the marks on the diagram.

These two regions containno elements.

An “X” means “something is here…”In logic, when a region of a Venn diagram contains an “X”,

this tells us that that region contains at least one element.

XThis region contains at least one element.

An “X” means “something is here…”In logic, when an “X”, appears on the border between two regions,

this tells us that there is at least one element in the union of the two regions, but we are not certain whether the element(s) are in the first region, the second region, or both regions.

XXThere is at least one element in these two regions combined.

No marking means “uncertain…”In logic, when a region of the Venn diagram contains no

markings, it is uncertain as to whether or not that region contains any elements.

XXWe don’t know if these two regions contain any elements.

ExampleSuppose we will use a three-circle Venn diagram to convey information about the

relationships between these three categories: Angry apes (A); Blissful baboons (B); Churlish chimps (C).

Select the diagram whose markings correspond to “No blissful baboons are angry apes.”Assume that we do not know of any other relationships between categories.

XX

SolutionSelect the diagram whose markings correspond to “No B are A.”According to the proposition “No B are A,” it must be impossible for an element that

is in category A to also be in category B. This means that the intersection of circles B and A must be empty (that is, shaded). This is what is shown in choice B below. The correct choice is B.

XX

More exercisesFor tutorials on diagramming categorical propositions, see The DIAGRAMMER on our

home page.

http://www.math.fsu.edu/~wooland/diagrams/diagramming.html