Modus Ponens - Implication vs Disjunction

Question

The Modus Ponens inference rule is generally expressed as:

$$ \begin{array}{rl} & P\rightarrow Q \\ & P \\ \hline \therefore & Q\end{array} $$

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Is the below rule also considered to be Modus Ponens?

$$ \begin{array}{rl} & P \lor \lnot Q \\ & Q \\ \hline \therefore & P\end{array} $$

Answer 1

Yes.

Thanks to @Bernard Massé for pointing me in the right direction.

Here's the proof:

1. $(P \lor \lnot Q)$ can be written as $ (\lnot Q \lor P) $ - Commutative Property

2. $ (\lnot Q \lor P) $ can be written as $ (Q \rightarrow P) $ - Material Implication

3. By Modus Ponens :

$$ \begin{array}{rl} & Q\rightarrow P \\ & Q \\ \hline \therefore & P\end{array} $$ This is equivalent to

$$ \begin{array}{rl} & P \lor \lnot Q \\ & Q \\ \hline \therefore & P\end{array} $$

Answer 2

I would consider it an application of Disjunctive Syllogism, which is typically stated as:

$P \lor Q$

$\neg P$

$\therefore Q$

Of course, that is not exactly the same pattern, but the basic idea of Disjunctive Syllogism is that you have two options ... but it isn't one of them, and therefore you are left with the other one. Your argument is like that too: it is either P, or $\neg Q$, but given $Q$ it is not $\neg Q$, and so you are left with $P$