# What does $\rightarrow$ mean in $p \rightarrow q$

I was looking at an exercise where it asked the following:

$$\begin{array}{ccc} p&q&p\rightarrow q \\ T&T&T \\ &\ldots \end{array}$$

So, for the third column, I just put $T$ which was correct but I didn’t understand what $\rightarrow$ meant. I have seen $\implies$ but I haven’t the arrow. Are they the same thing?

Thanks a bunch!

#### Solutions Collecting From Web of "What does $\rightarrow$ mean in $p \rightarrow q$"

The $\rightarrow$ symbol is a connective. It’s a symbol which connects two propositions in the context of propositional logic (and its extensions, first-order logic, and so on).

The truth table of $\rightarrow$ is defined to be that $p\rightarrow q$ is false if and only if $p$ is true and $q$ is false.

Indeed this is the same meaning of $\implies$, but the difference is that $p\implies q$ is a statement about propositions, whereas $p\rightarrow q$ is a proposition. In some contexts, though, people don’t make this distinction between material implication (the connective) and logical implication (the $\implies$ arrow). But they are not the same thing in every context of propositional logic.

Given $p$, then we have $q$.

or $p$ implies $q$.

The two arrows mean the same thing.

It is a material conditional, or otherwise known as $p$ implies $q$, or if $p$, then $q$

The truth table for that is as follows

p  q  p implies q
T  T  T
T  F  F
F  T  T
F  F  T


$\rightarrow$ can also be written as $\implies$.

In computer science, $p \implies q$ can be rewritten as (not p) or q, or !p||q

Now, although I am only a rising 8th grader taking geometry, I can assure you that there is no difference to the arrows. I have seen a two sided arrow (p<–>q), but that is different. The single arrow just indicates a conditional statement.