/execute can't detect player as the target

The /execute command can not detect player as the target.

/execute @e ~ ~ ~ detect ~ ~-1 ~ stone -1 kill @p

This command will kill the player (in survival mode) if any entities stand on a stone block. But in this case the player won't trigger the command, even though the player is an entity as well.
This problem can also be done with @p as the target:

/execute @p ~ ~ ~ detect ~ ~-1 ~ stone -1 kill @p