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.

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

(1) Nominalism is more problematic than conceptualism but less problematic than direct realism. This gives us:

direct realism > nominalism > conceptualism

(2) Conceptualism is more problematic than Poinsotiannism: conceptualism > Poinsotiannism

We can combine (1) and (2) to get:

direct realism > nominalism > conceptualism > Poinsotiannism

This means that it is false to say that Poinsotiannism is more problematic than direct realism. The complete oposite is the truth based on these premises.

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

(1) Mathematics is more abstract than natural philosophy but less abstract than metaphysics. This gives us:

Metaphysics > mathematics > natural philosophy

(2) Optics is more abstract than natural philosophy but less abstract than mathematics. This gives us:

Mathematics > optics > natural philosophy

This tells us that we can say:

Metaphysics > mathematics > optics > natural philosophy

It is not true to say that optics is more abstract than metaphysics.

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

(1) Hamburgers are more enjoyable than yogurt but less enjoyable than tuna steak. This gives us:

Tuna steak > hamburgers > yogurt

(2) Tuna steak is more enjoyable than bacon: tuna stead > bacon

This does not, however, tell us anything about the relative positions of bacon and hamburgers; therefore the conclusion is uncertain.

Let’s use symbols from math to help us understand this. “Greater than” (>) will mean “have oilier smoke than,” and “less than” (<) will mean “have less oily smoke than.”

(1) Cigars have oilier smoke than cigarettes but less oily smoke than pipes. This can be written:

Pipes > cigars > cigarettes

(2) Cornhusks have oilier smoke than cigars: cornhusks > cigar

Therefore, we can combine these to get:

Cornhusks > pipes > cigars > cigarettes

This means that cigarettes do indeed have less oily smoke than cornhusks.

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) Milk is healthier than apple juice and less healthy than carrot juice. This can be rewritten:

Carrot juice > milk > apple juice

(2) Apple juice is healthier than syrup water: apple juice > syrup water

We can combine all of these to get:

Carrot juice > milk > apple juice > syrup water

It is thus true to say that syrup water is less healthy than milk.

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

(1) Cajetan is more logical than Sylvester: Cajetan > Sylvester

(2) John is less logical than Cajetan but more logical than Francisco.  This can be rewritten:

Cajetan > John > Francisco

Notice that all we need to pay attention to is the second statement.  Here, it states that Fracisco is indeed less logical than Cajetan.