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.