Part III: Visualization

To get a better sense of how Particular Affirmative and Negative propositions compare to one another, we can once again turn to Euler Circles.

• Euler Circles:

Remember from the previous lesson that in the case of Euler Circles, we must look at circles representing categories to interpret these kinds of propositions.

We are supposed to recognize that a portion of the area in the circle labeled “Snake” is included in the circle labeled “Poisonous.” This spatial relation represents the Particular Affirmative proposition “some snakes are poisonous.” 

We can think of the left circle as containing all of the members of the category “snake” and the right circle containing all of the members of the category “poisonous”. Notice, however, that if all we know is that some snakes are poisonous, it could be the case that all snakes are poisonous. That is, if “all snakes are poisonous” is true, then it would also be true that “some snakes are poisonous”! In other words, if all members of the category “snake” are also members of the category “poisonous,” then it is also true that some of the members of the category “snake” are also members of the category “poisonous.” 

Therefore, because we do not know just from “some snakes are poisonous” if there are any snakes “outside” of the things that are poisonous, we draw a dotted line to finish the circle. The dotted line represents what is unknown, while the solid line represents what is known. Since we do know that there are at least some things that are both members of the category “snake” and the category “poisonous,” we draw the part of the “snake” circle that is inside the “poisonous” circle with a solid line.

Furthermore, we notice that “Some snakes are poisonous” is true if and only if “some poisonous things are snakes” is also true. In other words, if 

i. Some members of the category “snake” are also members of the category “poisonous,”

then we know that

ii. There are some members of the category “poisonous” that are also members of the category “snake” (those very same members!).

 Therefore, the diagram is symmetrical. In other words, the outside of the “Poisonous” circle is also dotted. This diagram can represent both that “some snakes are poisonous” and that “some poisonous things are snakes.” All Particular Affirmative propositions will look like this when diagrammed. Let’s look at one more example:

Now, for an example of a Particular Negative proposition:

We are supposed to recognize that a portion of the area in the circle labeled “Cat” is not included in the circle labeled “Albino.” This spatial relation represents the Particular Negative proposition “some cats are not albinos.” Thus, we see that the part of the “Cat” circle outside of the “Albino” circle is a solid line. However, just from “Some cats are not albinos,” we do not know if there are any cats which are albinos! There might be, but we are not sure from the information given. Therefore, we draw a dotted line for the part of the “Cat” circle that is inside the “Albino” circle. 

Likewise, it could be the case that “all albinos are cats” while still being the case that “some cats are not albinos.” All we know is that which is enclosed by a solid line on all sides: the space which represents that there are some things that are in the circle of cats that are outside of the circle of albinos. All Particular Negative propositions will look this way. Here’s one more example: