You'd probably find it easier to store the arrays in an array, e.g.:
$arr[0]=array("cat","dog","pig");
$arr[1]=array("owl","pig","cat");
$arr[2]=array("pig","cat","owl");
Then you can loop over the array, noting that
array_intersect($a1, $a2, $a3, $a4, ...)
is equivalent to
array_intersect($a1,
array_intersect($a2,
array_intersect($a3,
array_intersect($a4, ....))))
which is easily done in a loop a la $ai = array_intersect($ai, $a[$i]);