|Definition: Let $P_1$, $P_2$, …, $P_n$ be a collection of statements which we will call Premises and let $Q$ be a statement which we will call a Conclusion. Then an Argument is a formula of the form $(P_1 \wedge P_2 \wedge … \wedge P_n) \rightarrow Q$. The argument is said to be a Valid Argument if under the assumption that $P_1$, $P_2$, …, $P_n$ are true we get that $Q$ is true. If this is a valid argument, we say that the conclusion $Q$ is Deduced or Inferred from the premises $P_1$, $P_2$, …, $P_n$.|
For example, let $P$, $Q$, and $R$ be the following statements:
- $P_1$: If Bob does his homework then he will get time to hang out with his friends.
- $P_2$: If Bob doesn’t play video games then Bob will do his homework.
- $P_3$ Bob does not have time to hang out with his friends.
The three statements above can be broken down. Let $P$, $Q$, and $R$ denote the statements:
- $P$: Bob does his homework.
- $Q$ Bob hangs out with his friends.
- $R$: Bob plays video games.
Then the statements $P_1$, $P_2$, and $P_3$ are given by:
Consider the following argument:
We want to determine if this argument is valid or not. We construct the truth table:
|$P$||$Q$||$R$||$\neg Q$||$\neg R$||$P \rightarrow Q$||$\neg R \rightarrow P$||$\neg Q$||$(P_1 \wedge P_2 \wedge P_3) \rightarrow R$|
We can see that $(P_1 \wedge P_2 \wedge P_3) \rightarrow R$ is a tautology. So this argument is valid.