Enthymemes - Amateur Logician

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Enthymemes

In logic, an enthymeme is an abridged categorical syllogism which omits one of its propositions, i.e., either one of the premises or the conclusion itself.

It’s especially important to master enthymemes for two reasons!

First, most argumentation in ordinary life consists of enthymemes, not explicitly formed syllogisms. Second, when we master enthymemes, this is accompanied with a deeper intuition of how categorical syllogisms work. Recall, logic is both a “science” and an “art.” Enthymemes are, so to speak, in the intersection of that distinction.

Let’s take a look at the classic example. . .

Generally speaking, what is omitted is the more universal principle. We might say something like, “Socrates is mortal because he is a man.” The “because” part explains why he is mortal, and this references the middle term (i.e., man or mankind). But this is true because it implicitly works on a more universal principle, i.e., “all men are mortal.” What’s missing, therefore, in “Socrates is mortal because he is a man” is the major premise that “all men are mortal.”

In any case, since an enthymeme omits one proposition, there are therefore three enthymemes possible with any categorical syllogism. A “first order” enthymeme omits the major premise. A “second order” enthymeme omits the minor premise. And a “third order” enthymeme omits the conclusion. The third order can be effective rhetorically, by the way, since it forces an audience to draw the conclusion on their own.

Practice makes perfect! For example, “Thomas shouldn’t be elected given that he’s a thief.” It’s a good enthymeme. Let’s reconstruct it into a categorical syllogism. The conclusion is “Thomas shouldn’t be elected” and there’s a premise “he’s a thief.” What’s the implicit premise? In this case, it’s the major premise. It also happens to be the universal principle. What is that? Well, it must be “no thief should be elected.”

How about this? “All conductors allow electricity to pass through it. Copper will allow electricity to pass through it.” If we think about it, it is pretty obvious that “all conductors allow electricity to pass through it” is a premise. “Copper will allow electricity to pass through it” applies that premise, and so it must be the conclusion. Thus, we have a “second order” enthymeme. What’s the missing premise? It would have to be “copper is a conductor.”

Treat an enthymeme as a puzzle! What is the missing piece? Consider the rules of a categorical syllogism. There are only three terms allowed, for example, in a good syllogism. There must be a middle term to compare the major term and the minor term in the conclusion. While the middle term will appear in the premises, it will never appear in the conclusion. At least one premise must have a universal quantity. And at least one premise must have an affirmative quality.

Additional Examples. . .

Example 1: “Metals contract when cooling. Iron will contract when cooled.”

Example 2: “Some atheists are not very rational and some are not logicians.”

Example 3: “He should not be elected president. The guy is a liar.”

Example 4: “All who study philosophy are bookish. Some of my friends study it.”

Figure out what is implicit?

If so, great!
They take practice.

***

Example 1 has the minor premise implicit (namely, “helium is a gas”). Also, the quantifier on the major is implicit (namely, “all.”) Example 2 has the major premise as implicit (namely, “all logicians are very rational”). Note that “some atheists are not very rational” is the minor and “some are not logicians” is the conclusion,” even though they are formed as a conjunction with the “and.” Example 3 has an implicit major premise (namely, “nobody dishonest should be elected president”). The minor premise is “the guy is a liar” and the conclusion is “he should not be elected president.” Example 4 has an implicit conclusion (namely, “therefore, some of my friends are bookish”).

Epicheiremas

In logic, an epicheiremas is a syllogism where at least one premise has an enthymeme in it. That enthymeme attempts to justify the respective premise.

Complex chains of reasoning are often involved when considering philosophical argumentation. An epichirmea is an implicit polysyllogism. Indeed, we can make an epichirmea more explicit by expanding it out into a polysyllogism.

A “popular” example in Scholastic texts is as follows (cf. Joyce’s Principles of Logic, p. 256):

We can render this implicit proof to something explicit. Notice that the first premise is justified as an enthymeme. (It’s a first order enthymeme.) Rendering this explicit, we get: “Whatever incapable of corruption is immortal. What is spiritual is incapable of corruption. Therefore, whatever is spiritual is immortal.”

The full polysyllogism is thus: “Whatever incapable of corruption is immortal. What is spiritual is incapable of corruption. Therefore, whatever is spiritual is immortal. But the human soul is spiritual. Ergo, the human soul is immortal.”

It’s a valid argument! Is it sound? A skeptic will likely need more convincing. Thomist philosophers might attempt to expand this chain of reasoning to further justify the premises as all being true.

Sister Miriam Joseph tells us, in her classic book The Trivium, that Cicero (106 – 43 B.C.) particularly liked to use the so-called double epicheirema.

A double epicheirema has two enthymemes in it. So, it has the major premise with an enthymeme to justify it, a minor premise with an enthymeme to justify it, and then the conclusion.

We can take a sorties of five propositions, e.g., and make it a double epicheirema (or vice versa).

A circle is an ellipse.
An ellipse is a conic section.
A conic section is a quadratic curve.
A quadratic curve has a variable squared.
Therefore, a circle has a variable squared.

In other words, there’s at least one variable squared.
A circle can be defined as (x-h)^2 + (y-k)^2 = r^2 when placed on the xy-plane.

Given that, the double epicheirema is: “A circle is a conic section because it’s an ellipse. A conic section has a variable squared because it’s a quadratic curve. Therefore, a circle has a variable squared.”