### Unit 4: Venn Diagrams

In addition to using predicate logic, the limitations of sentential logic can also be overcome by using Venn diagrams to illustrate statements and arguments. Statements that include general words like "some" or "few" as well as absolute words like "every" and "all" – so-called categorical statements – lend themselves to being represented on paper as circles that may or may not overlap.

Venn diagrams are especially helpful when dealing with the logical arguments called syllogisms. Syllogisms are a special type of three-step argument with two premises and a conclusion, which involve quantifying terms. In this unit, you will learn the basic principles of Venn diagrams, how to use them to represent statements, and how to use them to evaluate arguments.

**Completing this unit should take you approximately 6 hours.**

Upon successful completion of this unit, you will be able to:

- create Venn diagrams as a means to represent and reason about relationships among classes;
- evaluate the validity of arguments using Venn diagrams; and
- describe the limitations of Venn diagrams as assessment tools.

### 4.1: Introduction to Venn Diagrams

### 4.1.1: Venn Diagrams as Illustrations of Sets or Classes

Read sections 2.14 and 2.15 (p. 114-126) to learn and apply a visual method for determining the validity of categorical inferences: Venn diagrams. Note the four categorical forms and what they mean: universal affirmative, universal negative, particular affirmative, and particular negative. Get comfortable drawing Venn diagrams for categorical statements and shading in the area or drawing a star for the statements you are given.

Complete exercises 18 and 19, checking your answers against the answer keys on pages 224-225, translating the diagrams into statements and using the Venn test of validity to determine the validity of the given categorical inferences.

Read the introduction and tutorial for an introduction to Venn diagrams. In Venn diagrams, circles represent sets or classes. These tutorials demonstrate how to use shading and overlapping to illustrate empty sets, as well as relations of all, every, and nothing.

### 4.1.2: More Complicated Venn Diagrams

Working with Venn diagrams that involve three circles is almost exactly the same as working with ones that involve two circles. The only difference is that there are now eight distinct regions, each with a specific logical meaning.

Complete these exercises and check your answers.

### 4.1.3: Illustrating Experience with Venn Diagrams

Read section 2.16 (p. 126-128) to learn about a potentially counter-intuitive relationship between universal affirmatives and particular affirmatives – namely that the one does not imply the other. This is because universal affirmatives do not contain what is called an "existential commitment" that is, a statement that there is anything in the category the universal affirmative references.

Complete exercise 20, keeping in mind that universal affirmatives do not contain existential commitments. Check your validity answers against the key on page 225.

### 4.1.4: Review of Introduction to Venn Diagrams

Complete these exercises relating to two-circle Venn diagrams. For each question, you must choose the sentence that best represents what is shown in the given diagram.

### 4.2: Venn Diagrams and Arguments

### 4.2.1: Using Venn Diagrams to Evaluate Syllogisms

Read section 2.17 on Venn diagrams (p. 128-138), which will help you use Venn diagrams to test the validity of whole categorical syllogisms, rather than only categorical inferences. Read the section and identify the categories in every statement of the syllogisms as you go, making your own Venn diagrams to test the validity, as directed.

Complete exercise 21, checking your validity answers against the key on pages 225-226.

Read this tutorial on how to use Venn diagrams to evaluate arguments. You will be introduced to the concept of a syllogism, a special type of argument that cannot be evaluated in SL. Venn diagrams are ideal for evaluating this type of argument. Remember that a Venn diagram can only tell us if an argument is valid, not whether it is sound.

### 4.2.2: Understanding the Logic of Venn Diagrams

This page reviews how to set up Venn diagrams as well as the rules for using Venn diagrams in evaluating argument validity. It also introduces the notion of conditional validity and explains how to use Venn diagrams to evaluate the validity of categorical syllogisms.

### 4.2.3: The Limitations of Venn Diagrams

Read this tutorial about the limitations of Venn diagrams. Although Venn diagrams are a powerful tool for representing some types of statements, there are many statements that they cannot handle.

Complete the exercises for this tutorial, then check your answers.

### 4.2.4: Review of Venn Diagrams and Arguments

Complete these exercises in which you will determine whether these arguments are valid or not. Draw out the Venn diagrams with pencil and paper.

Consider how you might adapt Venn diagrams to evaluate the validity of the following arguments.

- Most cooks are men. Most men are idiots. So most cooks are idiots.
- Very few plants are purple. Very few purple things are edible. So very few plants are edible.

Share your thoughts on the discussion forum by clicking on the link above and creating a free account, if you have not already done so. Review and respond to at least one or two other students’ posts.

- Most cooks are men. Most men are idiots. So most cooks are idiots.