In logic, we see expressions like \(P \implies Q\) a lot.
The symbol \(\implies\) is read as ‘implies’ and is also called the material conditional.
Logically/mathematically, it is a truth-function.
I.e. it merely takes in two yes-or-no bits of information and deterministically spits out a bit of information.
It is expressible as the following table:
| \(P\) | \(Q\) | \(P \implies Q\) |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
A simple characterization is to say that the only way to show that \(P\implies Q\) is false is to show that \(P\) is false and \(Q\) is true.
It’s meaning, in brief, is an assertion that \(Q\) being true can be asserted if \(P\) is true.
There is a gap between these two characterizations, expressed in Lewis Carroll’s parable.
A common example is: If \(x\) is a bachelor, then \(x\) is male.
The \(\implies\) relation is ‘truth functional’ - it only depends on the scenarios in which \(P\) and \(Q\) are true and says nothing about \(P\) and \(Q\) being related to each other in some deeper way. The following examples illustrate this:
If \(1+1=2\), then more than 10 people live on Earth. (this is the first row of the table)
If the moon is made of cheese, then \(1+1=2\). (the third row)
If the moon is made of cheese, then \(1+1=3\). (the fourth row)