Q:

What's the actual value of an if-statement executing a command?

I do not quite understand what the following if-statement actually does, in the sense of what the condition outputs could possibly be:
if linux-command-1 | linux-command-2 | linux-command-3 > /dev/null

I understand the execution as the following:

Linux Command 1 gets executed, it's output is PIPED into Linux-Command-2 as input.
Linux Command 2 gets executed with LC1's input, it's output is PIPED into Linux-Command-3.
Linux Command 3 gets executed with LC2's input, it's output is redirected into /dev/null, basically doesn't appear.

But what about the actual if-statement? What's responsible for it becoming true or false? 
To further elaborate, here's an example with actual commands:
if ps ax | grep -v grep | grep terminator > /dev/null
then
echo "Success"
else
echo "Fail"
fi

I do know that the functionality behaves in the way that if any output occurs (Process is running) in that execution the condition is True, if nothing occurs (Process not running), the condition is False. 
But I don't understand why or how it's coming to that conclusion? Is the shell if statement always expecting a string output as True? 
I've also just discovered pgrep but the question would also remain if the statement were 
if pgreg -f terminator > /dev/null  

A:

In your case you are testing  the exit status of grep itself, which will return false (1) if there was no match and true (0) if there was one 
if ps ax | grep -v grep | grep terminator > /dev/null
then
  echo "Success"
else
  echo "Fail"
fi

you could put a "-q" instead of redirection to /dev/null
if ps ax | grep -v grep | grep -q terminator 
then
  echo "Success"
else
  echo "Fail"
fi

I would execute my commad and test the $?
ps ax | grep -v grep | grep terminator
if [ $? -eq 0 ]; then
    echo 'it is ok'
else
    echo 'is is ko'
fi

If you do:
if linux-command-1 | linux-command-2 | linux-command-3 > /dev/null

only the result of the last command matter
if everything is important put "&&" instead 
if  ps -ae | grep  'bash' | grep  'pty0' && ls . >/dev/null; then     
  echo "bash is in the house" 
fi

that will fail because there is no not_exist
if  ps -ae | grep  'bash' | grep  'pty0' && ls not_exist >/dev/null; then     
  echo "bash is in the house" 
fi

