Valid Arguments

# 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.