It's all about operator precedence. For more information about precedence theory checkout perlop.
Regarding your specific problem:
The line  open ($fh,$log) || $down = 'moo'
is evaluated as (open ($fh,$log) || $down) = 'moo';
Obviously, open isn't an lvalue function; you can't assign it a value.

$log = "moooojlfdkafj.txt";
$down="false";
open (FH, $log) || ( $down = "true" );
print $down;
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or

$log = "moooojlfdkafj.txt";
$down="false";
open (FH, $log) or $down = "true";
print $down;
[download]

work.