Material Conditional

In logic, we see expressions like $P⟹Q$ a lot.

• The symbol $⟹$ 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.
  • A simple characterization is to say that the only way to show that $P⟹Q$ is false is to show that $P$ is false and $Q$ is true.
  • It is expressible as the following table:

$P$	$Q$	$P⟹Q$
T	T	T
T	F	F
F	T	T
F	F	T

• Its 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 $⟹$ 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)