Language of Conditionals

We first look at converting conditionals from English to logical statements.

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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.

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\(Q\), 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."

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\(P\), 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.

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\(P\) 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 \(P \leftrightarrow Q.\)

Truth Tables

Here is the truth tables for \(P \rightarrow Q:\)

The third row in this truth table often raises some questions. Think of implies this way:

• A true statement can lead to a true conclusion, so \(T \rightarrow T = T.\)
• A true statement can never lead to a false conclusion, so \(T \rightarrow F = F.\)
• A false statement can lead to anything, so \(F \rightarrow T = T,\) and \(F \rightarrow F = F.\)

The definition here is very important, so we should not gloss over it too quickly.

Often time, we see an event and we want to find the cause. That is, we know \(Q\), and we want to find \(P.\) This sort of analysis involves inductive reasoning, or reasoning from observations, and the scientific method. Whether we start from a true or false hypothesis \(P\), we may be able to conclude that \(P\) implies \(Q.\) This is why nothing can be proven in science. However, we can show that \(P\) does not cause \(Q\) by finding a situation in which \(P\) is true and \(Q\) is false.

In math, we start with statements that we assume are true. So, \(P\) is true by definition. Our goal is to find which statements are true or false when \(P\) is true. By starting with a true statement, we can prove whether other statements are true or false with deductive logic.

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Here is the truth tables for \(P \leftrightarrow Q:\)

The statement \(P \leftrightarrow Q\) is true when \(P\) and \(Q\) are the same, and false when they are different.