Conditional statement

Let $p$ and $q$ be propositions. The conditional statement $p\to q$, is the proposition that "if $p$ then $q$". This is also called an implication.

$$v(A\to B)=\{\begin{array}{ll}1,& \text{if }v(A)=0\text{ or }v(B)=0\\ 0,& \text{if }v(A)=1\text{ and }v(B)=0\end{array}$$

$p$	$q$	$p\to q$
T	T	T
T	F	F
F	T	T
F	F	T

For example:

• If it is sunny today, we will go to the beach.
• If I am elected, I’ll lower taxes.

You may encounter some weird statements which are still true.
- "If the Moon is made of green cheese then $2\times2=5$." $(F \rightarrow F)$.
- "If the Moon is made of green cheese then $2\times2=4$." $(F \rightarrow T)$.
- "If London is in the UK then $2\times2=4$." $(T \rightarrow T)$.
  This is also known as irrelevant logic.