We first look at converting conditionals from English to logical statements.
If \(P\) then \(Q\)
The statement "If \(P\) then \(Q\)" means that whenever \(P\) is true, \(Q\) will also be true. The symbol for the conditional is \(\rightarrow\), and is read "implies". Written as a formula: \(P \rightarrow Q\)
For example, let \(P\) be the statement: "You won the lottery." Let \(Q\) be the statement: "You can buy a mansion." In this case, \(P \rightarrow Q\) because "If you win the lottery, then you can buy a mansion".
However, this only states that when \(P\) is true, so is \(Q\). Statement \(Q\) can be true even when \(P\) is false. For example, if you become a famous movie star but you don't win the lottery, you can still buy a mansion. In that case, \(P\) is false and \(Q\) is true, while \(P \rightarrow Q\) is still true.
\(Q\), if \(P\)
Another English statement that means \(P \rightarrow Q\) is one of the form \(Q\), if \(P.\) Sticking with our lottery example, \(Q\), if \(P\) is written "You can buy a mansion, if you win the lottery."
\(P\), only if \(Q\)
A third way to state \(P \rightarrow Q\), which sounds very similar to the second, is \(P\), only if \(Q.\) Using the lottery example, we could say "You have won the lottery, only if you can afford a mansion." Notice that there are different uses for the different sentences in English even though they are logically equivalent.
\(P\) if, and only if, \(Q\)
This statement combines the previous two. It says \(P\), if \(Q\), and \(P\), only if \(Q\). That is, \(Q \rightarrow P\) and \(P \rightarrow Q.\) This means \(P\) and \(Q\) are logically equivalent. They are always either both true, or both false. In symbols, we write \(P \leftrightarrow Q.\)
Here is the truth tables for \(P \rightarrow Q:\)
Here is the truth tables for \(P \leftrightarrow Q:\)
\(P \rightarrow Q=\) T
\(Q \rightarrow P=\) T