Determining the Validity of an Argument by Rules of Inference

Let $P$, $Q$, and $R$ be statements. So far we have discussed the following rules of inference:

• Modus Ponens: $(P \wedge (P \rightarrow Q)) \rightarrow Q$.

• Modus Tollens: $((P \rightarrow Q) \wedge \neg Q) \rightarrow \neg P$.

• The Law of Syllogism: $((P \rightarrow Q) \wedge (Q \rightarrow R)) \rightarrow R$

We will now look at an example of determining if an argument is valid by using these rules of inference. In the examples below, we will list our premises and then list our conclusion below a dotted line.

Example 1

Determine if the following argument is valid:

(1)

\begin{matrix} \mathrm{Premise \: 1} & P\\ \mathrm{Premise \: 2} & P \rightarrow \neg Q\\ \mathrm{Premise \: 3} & \neg Q \rightarrow \neg R\\ & ---\\ \mathrm{Conclusion} & \therefore \neg R \end{matrix}

The argument is valid. Here are the steps:

(2)

\begin{align} \quad 1. & P \rightarrow \neg Q & (\mathrm{Premise \: 2}) \\ \quad 2. & \neg Q \rightarrow \neg R & (\mathrm{Premise \: 3}) \\ \quad 3. & P \rightarrow \neg R & (\mathrm{The \: Law \: of \: Syllogism \: with \: (1) \: and \: (2)}) \\ \quad 4. & P & (\mathrm{Premise \: 1}) \\ \quad 5. & \neg R & (\mathrm{Modus \: Ponens \: with \: (4) \: and \: (3)}) \end{align}