Formalizing a logical argument

Question

I want to formalize this reasonning

Many students will be either in Hegel’s or in Schopenhauer’s lectures, if they are scheduled at the same time. And of course Schopenhauer will schedule them at the same time as Hegel’s. If Hegel’s lectures are entertaining, then many students will go to them. That means of course many students will go to Hegel’s but not many will go to Schopenhauer’s lectures. For if Schopenhauer’s lectures are entertaining, Hegel’s must be entertaining as well; and of course many students will only come to Schopenhauer’s lectures if they are entertaining.

Let's highlight some premisses and the conclusion:

P: Many students will be either in Hegel’s or in Schopenhauer’s lectures, if they are scheduled at the same time.

Q: If Hegel’s lectures are entertaining, then many students will go to them.

R: Many students will only come to Schopenhauer’s lectures if they are entertaining.

Conclusion: Many students will go to Hegel’s but not many will go to Schopenhauer’s lectures

First question: Am I authorized to split premisses into parts ? e.g. :

Q: Q1 = Hegel's lectures are entertaining; Q2 = Many students will go to Hegel's courses.

R: R1 = Schopenhauer's lectures are entertaining; R2 = Many students will ONLY go to Schop's lectures.

So Q is Q1 ⇒ Q2 and R is R1 ⇒ (R2 ∧ ¬ Q2). Is that right ?

How can I use P, Q1, Q2, R1, R2 to prove the conclusion ?

The text is taken from The Logic Manual - V. Halbach

Answer 1

First, yes, you are allowed the break down the premises and the conclusion into smaller statements, and in fact this is usually what you have to do in order for the imrtant logical operators to appear so that statements can be combined.

Second, when symbolizing, it is usually a good idea to use letters that are indicative of what they symbolize. So, for example, I would use $H$ for 'Many students go to Hegel's lectures', $S$ for 'Many students go to Schopenhauer's lectures', etc.

Third, the last sentence of the passage is an 'only if' sentence. Those are always a bit tricky, but the basic idea is this: if we have '$P$ only if $Q$', then that is not $Q \rightarrow P$ (despite the 'if' going with the $Q$), because the 'only if' signals that $Q$ is a necessary condition for $P$, rather than a sufficient condition. Thus, $Q$ being true may not be enough to make $P$ true, as maybe there are further conditions that need to hold true. So again, $Q \rightarrow P$ does not work. However, you can say that $P \rightarrow Q$: if $Q$ is a necessary condition for $P$, then knowing that $P$ is true tells us that $Q$ must have been satisfied as well. So, long story short: '$P$ only if $Q$' is symbolized as $P \rightarrow Q$.

In your case, we have that students go to Schopenhauer's lectures ($S$) only if they are entertaining ($SE$), and thus we have $S \rightarrow SE$, rather than $SE \rightarrow S$, and since this sentence says nothing about Hegel, you should not make that part of your symbolization at all. So, your $R1 \rightarrow (R2 \land \neg Q2)$ (which with my notation would amount to $SE \rightarrow (S \land \neg H)$ is wrong for two reasons.

Fourth, and finally, the validity of this argument does indeed rely on a bit of rhetorical interpretation. When the first sentence declares that many students go to either Hegel's lectures or Schopenhauer's lecture, it is clear that this is meant as an exclusive or, especially given that the lectures are at the same time. However, it is not clear that this means that you cannot have many students at Hegel's lectures, and still have many students at Schopenhauer's lectures as well.

If, however, they mean to say that either you have many students at Hegel's lectures, or have many students at Schopenhauer's lectures, then the argument will turn out to be valid. Here would be its symbolization:

$T \rightarrow ((H \land \neg S) \lor (S \land \neg H))$

$T$

$HE \rightarrow H$

$SE \rightarrow HE$

$S \rightarrow SE$

$\therefore H \land \neg S$

Do you see why this is valid?