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 catastrophes slugs speedy (things) owls hooty (things) are all categories.

Existential statements A 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 statements

Diagramming existential statements The 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 ).

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

Universal negative statements A statement of the form “No A are B” is called universal negative. 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 universal negative statement.

Diagramming universal negative statements In logic, we use shading to indicate that a certain region of a Venn diagram is empty (contains no elements). The universal negative statement “No slugs are speedy” asserts that the region of the diagram where “Slugs” and “Speedy things” intersect must be empty.

Diagramming universal negative statements

Universal positive statements A statement of the form “All A are B” is called universal positive . 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 universal positive statement.

Diagramming universal positive statements The universal positive 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 universal positive statements

Interpreting Venn diagrams in logic We 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.

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.

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.

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.

Example Suppose 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.

Solution Select 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.

More exercises For tutorials on diagramming categorical propositions, see The DIAGRAMMER on our home page. http://www.math.fsu.edu/~wooland/diagrams/diagramming.html