Part III: Visualization

To get a better sense of how Universal Affirmative and Negative propositions compare to one another, it can be helpful to try to visualize the meanings of these two kinds of propositions. For this, we can use what are called “Euler Circles.” This is a common way for logicians to represent what these types of propositions mean.

• Euler Circles:

In Euler Circles, a circle represents a category. Thus, in figure 2, we see that the category “Cats” is entirely inside the category “Animals.” In other words, all members of the category “Cats” are also members of the category “Animals.” This relation represents the Universal Affirmative proposition “All cats are animals.” For another example, in figure 3, we see that there is no overlapping part between the category “Snakes” and the category “Birds.” In other words, there is no member of the category “Snakes” that is also a member of the category “Birds.” This relation represents the Universal Negative proposition “No snakes are birds.” 

Finally, let’s take a look at figure 4. The term “this book” refers to a category with only one member. That is, if one points to a book and says “this book,” one is only referring to one book. Therefore, if one says “This book is red,” one is saying that all members of the category “this book” are also members of the category “Red”. Namely, the only member of the category “this book” is also a member of the category “Red”. Figure 5 shows that no members of the category “James” (just James himself) are also members of the category “Happy”.