Another very basic Discrete Mathematics homework problem. I don't want the answer as much as I want to understand the question:
Problem 7
For each of the following sets of premises, what relevant conclusion(s) can be reached? Explain which rules of inference are used.
a) "If I play hockey, then I am sore the next day", "I use the whirlpool if I am sore", "I did not use the whirlpool"
b) "I am dreaming or hallucinating", "I am not dreaming", "If I am hallucinating, I see elephants smoking"
Okay, now my problem is with b, which ENDS with a conditional. I'm pretty confident that I already got a) correct, so let's look at b):
- $p$: I am dreaming
- $q$: I am hallucinating
- $r$: I see elephants smoking
According to the question, we have:
- $p$ V $q$
- ~$p$
- $q\rightarrow r$
The top two premises can be shortened to simply $q$ via "disjunctive syllogism":
- $q$
- $q \rightarrow r$
So…which rule can you use to draw any conclusions from the above, and what is the conclusion?
Using a truth table, if we look at the row where $q$ AND $q\rightarrow r$ are true, this means that $r$ must be true. So…is the conclusion $r$? But what rule is that?
Best Answer
You are correct in your application of the Disjunctive Syllogism in part (b). That gives you the derived premise $q$. Now, you can use Modus Ponens and note that from $q$ together with $q\rightarrow r$, we derive that $r$ holds.
Modus Ponens:
$$\begin{align} &\text{Modus Ponens }\\ \hline \\ & q \rightarrow r & q\\ & q & q\rightarrow r\\ \hline \\ \therefore & r &\therefore r\end{align}$$
The argument can be written as $\;q, \;(q\rightarrow r) \models r\;\;$ or as $\;(q\rightarrow r),\;q \models r$