The new Israeli offensive in the Gaza Strip poses a dilemma for citizens of democracies around the world. It is self-evident that Israel, a thriving if sometimes weed-overrun oasis of democracy in a desert of Middle Eastern despots, has the right to defend itself and to secure its borders. This is a right that other democracies must support, even at great political cost. But it is also growing ever clearer that Israel’s armed action in Gaza is disproportionate and thus illegitimate and, for that reason, ultimately self-defeating.

Like the “shock and awe” doctrine that drove the United States under the second Bush administration to a disastrous and unnecessary war in Iraq, the “all-out war” (in Defense Minister and former Prime Minister Ehud Barak’s description) that Israel has declared on the Hamas organization that dominates Palestinian politics in Gaza will not lead to greater security, but less.

It will deepen Palestinian grievance over Israel’s heavy-handedness, and inflame admittedly combustible public opinion on the so-called Arab street. It may succeed in destroying vital Hamas “infrastructure” in the strip that allows the terrorist organization to launch hundreds of rockets into parts of Israel, but it will complicate the effort to strengthen the moderate forces in Palestinian politics. Not least, it will make future compromise with Israel equivalent, in the eyes of many suffering Palestinians, to a capitulation.

We do not wish to gloss over the great difficulty that Israel confronts. Hamas, which won elections in 2006 at the expense of the traditional and now moderate Fatah, continues to refuse to recognize the existence of Israel or, indeed, its very right to exist. While we acknowledge the growing consensus that Israel’s aggressive settlements policy in the West Bank is a serious complication, and may in fact be the main obstacle to the dream of permanent peace and a two-state solution, Hamas’ primordial refusal to recognize Israel is at the root of the problem too.

That offers Israel with only a few choices, all of them hard. As the current public relations blitz has made clear (Foreign Minister Tzipi Livni, to give only one instance, has given countless interviews since the new Gaza offensive started six days ago), Tel Aviv’s governing coalition has framed the new upsurge of conflict as Israel’s carefully considered plan to finally put an end to the Hamas rocket attacks. As news reports in Israel’s free press have detailed it, Operation Cast Lead has certainly been a military success, from planning (under Barak, Israel’s most decorated soldier) to execution (with the express approval of the three leaders of the coalition, Livni, Barak and outgoing Prime Minister Ehud Olmert).

The conduct of the “all-out war” has been the complete opposite of the way Israel ran its haphazard campaign during the Second Lebanon War in 2006, which helps explain the current high levels of public support inside Israel.

But the brutal efficiency of the Israeli war machine has created a grave humanitarian crisis in the strip, routinely described in press reports as one of the world’s most densely populated areas. Despite official rhetoric about military-only targets, as much as 25 percent of the 400-plus casualties in Gaza thus far has been “collateral damage” — that is to say, civilian casualties. On the other hand, Israeli casualties from rocket attacks number in the low single digits.

It is never easy to use the moral calculus of comparative deaths, but surely no one, not even an Israeli democrat animated by the highest patriotism, can seriously suggest that the number of deaths in southern Israel is equivalent to the number of deaths inflicted in Gaza. Even if we were to limit the number to those of innocent civilians alone, the Israeli action is staggeringly disproportionate, about a hundred to one.

This is the kind of arithmetic that Palestinians, whether they voted for Hamas or Fatah, or whether they did not care enough to vote at all, will remember for a long time; they will memorize it, by rote, like a murderous multiplication table that does not seem to know, or promise, an end.