# What is the simplest way of understanding a conditional

I'm quite poor at mathematics skill and was wondering what the simplest way to understand conditional statements are? i.e. p → q

• Isn't "If p, then q" (that is "If the property p is satisfied, then the property q is automatically also satisfied") simple enough? Or you mean at a more abstract level, why $p\to q=(\textrm{non }p)\textrm{ or }q$? – Taladris Aug 10 '15 at 0:34
• One thing I dont understand is that if p is true & q is false then why does it result with false output? Then why is it acceptable to have q as false and p as false, resulting in true output – Oliver K Aug 10 '15 at 0:35
• @OliverK if $p$ is true and $q$ is false then "$p \rightarrow q$" is false, since "$p \rightarrow q$" is shorthand for "if $p$ is true then $q$ is true". – 727 Aug 10 '15 at 0:37
• Cool, Thanks @LJL :) – Oliver K Aug 10 '15 at 0:39

Here is an intuitive way to think about a conditional $p \rightarrow q$ rather than a strictly mathematical definition:

If it rains, then it will be wet. ($p =$ it rains, $q =$ it will be wet)

If the team loses, then they will not be undefeated. ($p =$ the team loses, $q =$ they will not be undefeated)

If my grade is an F, then I will fail the class. ($p =$ my grade is an F, $q =$ I will fail the class.)

Basically, the $\textit{hypothesis}$ $p$ implies the $\textit{conclusion}$ $q$. i.e. if $p$ then $q$. Hopefully these examples help you understand.

• This is what I was looking for, thanks :) – Oliver K Aug 10 '15 at 1:03
• No problem, glad to help :) – user134593 Aug 10 '15 at 1:59

Typically "$p$" and "$q$" are statements, and "$p \rightarrow q$" is shorthand for "if $p$ is true then $q$ is true", or "$p$ is a sufficient condition for $q$", or "$q$ is a necessary condition for $p$", etc.

Example: let $p$ be the statement "$n$ is a square integer", and let $q$ be the statement "$n$ is a non-negative integer". Substitute those statements in for $p$ and $q$ in the paragraph above and see that all the statements make sense, i.e. "$p \rightarrow q$".

The sentence "if P then Q" asserts that it cannot be possible to be true unless Q also is true. So if Q can ever be false when P is true, then the assertion "P implies Q"is not valid. Untrained people often mistakenly get it backwards, or expect it to say more. But if P is false,the statement "P implies Q" is TRUE, but it gives us no information about Q.

Quite simply:

1. $(\mathrm{TRUE} \rightarrow p)$ always simplifies to $p$.

2. $(\mathrm{FALSE} \rightarrow p)$ always simplies to $\mathrm{TRUE}.$

For example, consider the sentence: "If the sky is green, then I am a Leprechaun." This has the form... "$\mathrm{FALSE} \rightarrow \mathrm{FALSE}$." So by (2), it simplifies to $\mathrm{TRUE}.$

Always look at the expression on the left of $\rightarrow$ to work out what the implication simplifies to.