Valid Arguments

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:

(1)

\begin{align} \quad P_1: P \rightarrow Q \\ \quad P_2: \neg R \rightarrow P \\ \quad P_3: \neg Q \end{align}

Consider the following argument:

(2)

\begin{align} \quad (P_1 \wedge P_2 \wedge P_3) \rightarrow R \end{align}

We want to determine if this argument is valid or not. We construct the truth table:

$P$	$Q$	$R$	$\neg Q$	$\neg R$	$P \rightarrow Q$	$\neg R \rightarrow P$	$\neg Q$	$(P_1 \wedge P_2 \wedge P_3) \rightarrow R$
T	T	T	F	F	T	T	T
T	T	F	F	T	T	T	T
T	F	T	F	F	F	T	T
T	F	F	F	T	F	T	T
F	T	T	T	F	T	T	T
F	T	F	T	T	T	F	T
F	F	T	T	F	T	T	T
F	F	F	T	T	T	F	T

We can see that $(P_1 \wedge P_2 \wedge P_3) \rightarrow R$ is a tautology. So this argument is valid.