# “Rules of inference” when the last premise is a conditional?

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?

• Try modus ponens – Hagen von Eitzen Jun 9 '13 at 22:03

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 $\;(q\rightarrow r),\;q \models r$

• Note that the order in which the two premises occur: $q$ and $q \rightarrow r$, does not matter. Modus Ponens needs both premises, in whatever order, to derive $r$. – Namaste Jun 9 '13 at 22:26
• Wow I feel really stupid. So you're saying the order of the premises can be changed...I had no idea! Thanks! – CptSupermrkt Jun 9 '13 at 22:33
• Don't feel stupid! Rules of inference are often presented rigidly; all that has to occur is that the needed premises for a given rule occur prior to "invoking"/applying the rule. Order of given premises doesn't matter either. – Namaste Jun 9 '13 at 22:40
• I just started this class, and I really enjoy it, but I've never been very math-minded, so it always takes me an extra explanation or two to really get the concepts. So far you're 2 for 2 (thanks for the other answer!). I'll be seeing you around here for sure :) – CptSupermrkt Jun 9 '13 at 23:02
• If the argument can get written as you put above, then you have a single formula (q∧(q→r) or ((q→r)∧q), and just one premise. Modus ponens requires two premises, and can get put q, (q→r) |= r, or {q, (q→r)}|= r, or (q→r), q|=r, or {(q→r), q}|=r, because you have two premises or a set with two premises in it. – Doug Spoonwood Jun 10 '13 at 3:43