Bombing Syria for using chemical weapons against its own citizens would violate international law as it currently exists — let's get that straight. But that doesn't answer the question of whether the U.S. should do it anyway.
Some evils are so great that righting them requires violating laws that are inadequate to the situation, such as when the U.S. broke the same international law by bombing Serbia in 1999 to stop what looked a lot like genocide in Kosovo. The real question is: Should we break international law to send the symbolic message that use of chemical weapons violates, well, international law?
The legal analysis is surprisingly simple. If the United Nations Security Council authorizes force, international law allows it. Otherwise, unless acting in self-defense, a country or a group such as the North Atlantic Treaty Organization has no right to attack another.