Addition and Simplification Laws

In logic the addition law and simplification law are sometimes misused.

Law of Addition

$p \rightarrow p \vee q$

This law states that if a proposition $p$ is known to be true then the disjunction of  $p$ with any other proposition will also be true.

Law of Simplification

$[(p) \wedge (q)] \rightarrow p$

also,

$[(p) \wedge (q)] \rightarrow q$

This law states that if the conjunction of $p$ and $q$ is true then we can deduce $p$ is true. Similarly, we can deduce $q$ is true. Remember that $[(p) \wedge (q)]$ is true only if both propositions $p$ and $q$ are true. That's why we can deduce $p$  is true and that $q$ is true.

Some students attempt to apply the law of simplification to a disjunction. Without knowing the truth values of $p$ and $q$ some students may deduce from the statement $p \vee q$ that $p$ is true or that $q$ is true. This is not valid. From $p \vee q$ we know that at least one of the propositions is true, but we do not have enough information to know which one of them is true.