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17.2 The Biconditional <->

“Biconditional” just means two conditionals. P↔Q just means P→Q and Q→P.

↔ is the biconditional; if and only if; iff.
<-> (less than-dash-greater than) is how we type the biconditional.

The biconditional is the way we translate the phrase “if and only if”, or “iff” for short.

Pia is guilty if and only if Quinn is guilty

is translated like this:

P<->Q

We write the biconditional with the same method as the conditional: use a dash symbol between the less than and greater than symbols (angle brackets).

Now you try.

“If and only if” is not redundant.

You can learn a lot about conditionals by realizing that “if and only if” is not redundant.

“P if and only if Q”

Says

“P if Q” and “P only if Q”

And those are not the same thing.

Another common expression from logic and mathematics that is translated with the biconditional is “just in case”.

“P just in case Q”

Is translated P<->Q.

P just in case Q: P<->Q

We don’t claim that the biconditional is always what is meant by “just in case” in English. Rather, logicians just got tired of saying “if and only if” all the time, and they now use “just in case” as a synonym for it.

We will follow this common practice, so if you see “just in case”, use the biconditional.