Chapter8The Syllogism

8.1 Transitivity in a Syllogism

Here our middle term is “Saskatchewan,” and the relationship is one of containment. Containment, like the relation in example 1 (“_____ is older than _____”) is a transitive relationship. Transitivity is a relationship of ordering. If we know how things are ordered, then we can draw conclusions about the sequence. The order of example 1 is the order it would take to move from one place (Saskatoon) through another (Saskatchewan) to another (Canada).

The transitivity of containment is demonstrated in figure 8.1. For containment, if A is inside of B and B is inside of C, then it stands to reason that A is inside of C. Through the middle term B, A gets the transitive property of being inside of C.

This relation also works with relative height. If we use “_____ is taller than _____,” we can construct a conduit for deduction about those who are being measured for height. You might want to draw your own visual about “_____ is taller than _____” for three people you know using a vertical line. So we can formulate a more specific definition of a transitive relation.

Let’s consider another example using the “_____ is greater than _____” relation. Is it also transitive?

This pattern should again demonstrate that properly formulated syllogisms with three terms and a transitive relationship are valid. To revisit that definition, if the premises are true, then the conclusion must be true—or there’s no possibility for the premises to be true and the conclusion false. “Is greater than” is a transitive relation. Notice that the two premises have the form “a is R to b” and “b is R to c,” and the conclusion has the form “a is R to c” (notice that “b” is the middle term).

Other relations that are transitive:

8.2 Intransitivity

Of course, not every relation is transitive, and not every syllogism using a transitive relation is valid because the terms may not be in the right position in the argument.

When we look at this relation, we can think of the connection through the middle term. Here, the middle term is “Sarah.” In this example, the problem is that the relation “_____ is the mother of _____” is not transitive: the parent of your parent is not your parent (they are your grandparent). You could also say the friend of my friend is not necessarily my friend: recall the phrase “the enemy of my enemy is my friend!” So be on the lookout for relationships that are not transitive. Similarly, If A loves B and B loves C, does it follow that A loves C? No, because “love” is a relation between two terms, or a binary relationship. Intransitive relationships block the validity of three-term syllogisms.

Sometimes a relationship is transitive, but it fails to produce a valid argument because of the location of the terms in the argument. Above, we discussed that “is taller than” was transitive, but it is only when the terms are in the right place. Let’s look at an example:

All we know here is that both houses are taller than my car. This doesn’t give us any information on the relative heights of each other’s houses. Given the position of the terms, B (my car) fails to be a proper middle term. Its location blocks transitivity. If you read over the syllogism, do the premises make the case that our houses have a height difference? Could the premises be true andour houses be the same exact height (counter-example)? Yes. So it is possible for the premises to be true and the conclusion false, making it invalid.

In Chapters 9–12, we will spend some time looking at the traditional logic of terms also known as categorical logic. The syllogism pattern figures prominently in that logic in the form of the categorical syllogism. Here are two examples that are easily seen to be valid.

8.3 Containment Revisited

Consider the following example:

In the first case, we can imagine drawing a circle around all the ducks and a larger circle around all the birds (fig. 8.2). If all ducks are birds, then the duck circle will be inside the bird circle. Similarly for all living creatures, the bird circle will be inside the circle around all the living creatures, and the duck circle will be inside that. When we draw a circle that is “living creatures,” we think of it as “the set of all things that qualify as living creatures.” So the size of “birds” within that set is not proportionate! But what the circles demonstrate is that birds are wholly contained within “living creatures.”

Another transitive relationship that was mentioned above is the “if _____, then _____” relationship. Thinking of this explicitly as a transitive relationship allows us to understand the syllogism better.

Looking back at figure 8.2, if you are in the smallest circle we demonstrated (ducks), then you must be in the other two circles (birds and living creatures). But if you are in the biggest circle (living creatures), it doesn’t necessarily follow that you are a duck (you could be a gorilla!). And if you are a bird, it doesn’t necessarily follow that you are a duck (you could be a warbler!). So you can go from the inside out with “if _____, then _____” but not from the outside in.

Here is another way to think about why this argument is valid: First of all, to say that all ducks are birds is tantamount to saying that if something is a duck, then it is a bird. And to say that all birds are living creatures is tantamount to saying that if something is a bird, then it is a living creature. In the case of the ducks argument, unless something changes and ducks are no longer birds and living creatures, the argument isn’t just valid, it is sound.