Let’s use symbols from math to help us understand this. “Greater than” (>) will mean “is hotter than (or warmer than),” and “less than” (<) will mean “is cooler than (is less hot than).”

(1) The Sahara is hotter than the Alps: Sahara > Alps

(2) My teakettle is warmer than the Alps but cooler than my oven: oven > teakettle > Alps

Both of these tell us that the Alps are cooler than the Sahara and my oven; however, they do not tell us anything about the relative positions of my oven and the Sahara; therefore, we must mark this as being uncertain.

Let’s use symbols from math to help us understand this.  “Greater than” (>) will mean “is healthier than,” and “less than” (<) will mean “is less healthy than.”

(1) Oatmeal is healthier than chili but less healthy than oysters: oysters > oatmeal > chili

(2) Oysters are less healthy than ham: oysters < ham

This could be rewritten: ham > oysters

By combining our statements, this likewise lets us write: 

ham > oysters > oatmeal > chili

Therefore, according to the logic of these statements, it is true to say that ham is healthier than chili.

Let’s use symbols from math to help us understand this. “Greater than” (>) will mean “is louder than,” and “less than” (<) will mean “is softer than.”

(1) Frederick is louder than Bonaventure but softer than Aaron: Aaron > Frederick > Bonaventure

(2) Aaron is louder than Helga who is louder than Bonaventure: Aaron > Helga > Bonaventure

Now, these two statements do tell us that Frederick and Helga both are softer than Aaron and louder than Bonaventure.  However, they do not provide enough information to say which is louder than the other.

Let’s use symbols from math to help us understand this. “Greater than” (>) will mean “is more famous than,” and “less than” (<) will mean “is less famous than.”

(1) Geoffrey is less famous than William but more famous than Reuel: William > Geoffrey > Reuel

(2) Reuel is more famous than Clives: Reuel > Clives

We can combine these two statements to get:

William > Geoffrey > Reuel > Clives

Therefore, it is indeed true to say that Geoffrey is more famous than Clives.

Let’s use symbols from math to help us understand this. “Greater than” (>) will mean “is more difficult to fabricate,” and “less than” (<) will mean “is less difficult to fabricate.”

(1) It is simpler to fabricate bricks than it is to fabricate cut stone: bricks < cut stones

Notice that we can rewrite this as: cut stones > bricks

(2) Cut stone is more difficult to fabricate than mud roofs but less complicated to fabricate than plutonium rods: plutonium rods > cut stone > mud roofs

Now, we combine the second form of (1) with (2) and get:

plutonium rods > cut stone > bricks

Therefore, it is false to say that plutonium rods are simpler to fabricate than bricks.

Let’s use symbols from math to help us understand this. “Greater than” (>) will mean “is more mystifying than,” and “less than” (<) will mean “is less mystifying than.”

(1) Porphyry is less mystifying than Plotinus, who is more mystifying than Ibn Gabirol. This gives us two statements, really:

(a) Porphyry < Plotinus, which could also be read: Plotinus > Porphyry

(b) Plotinus > Ibn Gabirol

(2) Avicenna is less mystifying than Ibn Gabirol: Avicenna < Ibn Gabirol

This could be rewritten: Ibn Gabirol > Avicenna

Now, this problem is meant to overwhelm you with data. Only pay attention to the relevant facts. We only need information from (1). Porphyry is less mystifying than Plotinus (see 1.a). The same holds for Ibn Gabirol (see 1.b). This does not, however, tell us anything about their relation to each other.

Let’s use symbols from math to help us understand this.  “Greater than” (>) will mean “is more frightening than,” and “less than” (<) will mean “is less frightening than.”

(1) Hobbes is more frightening than Machiavelli, who is less frightening than Ockham. This gives us two statements:

(a) Hobbes > Machiavelli

(b) Machiavelli < Ockham, which could also be written: Ockham > Machiavelli

(2) Ockham is more frightening than Scotus but less frightening than Billuart:

Billuart > Ockham > Scotus

Now, since Billuart is more frightening than Ockham, who is more frightening than Machiavelli, he (Billuart) is more frightening than Machiavelli.

Let’s use symbols from math to help us understand this. “Greater than” (>) will mean “is stricter than,” and “less than” (<) will mean “is less strict than.”

(1) Reginald is stricter than Jacques, who is less strict than Joseph. This gives us two statements:

(a) Reginald > Jacques

(b) Jacques < Joseph, which also could be rewritten: Joseph > Jacques

(2) Joseph is stricter than Peter: Joseph > Peter

Now, we know that Peter is less strict than Joseph; however, let's note the relationship between Joseph and Reginald. We know that both of these men are stricter than Jacques. Still, it is possible that Joseph is very strict, meaning that Peter could even be stricter than Reginald. Likewise, perhaps Joseph is less strict than Reginald. In this case, Peter would be less strict than Reginald.  Therefore, the conclusion is uncertain.

Let’s use symbols from math to help us understand this. “Greater than” (>) will mean “is wilder than,” and “less than” (<) will mean “is less wild (calmer) than.”

(1) Gerbert is wilder than Hilbert, who is wilder than Egbert: Gerbert > Hilbert > Egbert

(2) Egbert is calmer than Filbert: Egbert < Filbert, which also could be written Filbert > Egbert

Now, we know that Filbert is wilder than Egbert. This does not sufficiently tell us anything about the relationship between Filbert and either Hilbert or Gerbert.

Let’s use symbols from math to help us understand this. “Greater than” (>) will mean “is more popular than,” and “less than” (<) will mean “is less popular than.”

(1) Fantasy is more popular than biographies but less popular than self-help. This gives us:

self-help > fantasy > biographies

(2) Self-help is less popular than Thomism: self-help < Thomism

This is the same thing as: Thomism > self-help

That means that we can say:

Thomism > self-help > fantasy > biographies

Therefore, it is true to say that Thomism is more popular than biographies.