Ask what the Israeli government is saying about its goals in Gaza and the answer is "nothing". The government, for good reasons, hasn’t stated its goals. When you do such a thing, there's a good chance you’ll later have to explain why the stated goals weren't met. So Israel is more or less mum about its objectives. You can get the usual platitudes: quiet, calm, no rockets, no terror. And of course, peace on earth is what we all want. But the actual achievement Israel is striving for is unclear – it is unclear to the public, and, I suspect, unclear to the leadership as well.
At times one gets the sense that they are just playing it day by day – and that isn't necessarily a bad thing. A cautious government is testing the waters and trying to assess what goals it can realistically hope to achieve. To do that, it will use tactical means – for example, Israel can decide to send forces into Gaza to occupy a less dense area in the north part of the Strip. And it will talk to mediators – Egypt is the mediator that Israel is counting on (the US is the country from which it would like to get quiet backing and as little disruption as possible). And it will measure the temperature within Israel – to make sure it has the backing of the public and the political system.
Working slowly and cautiously is usually a good thing, but in this case it has one caveat. The more the days pass, the more likely it becomes that an incident, in Israel or in Gaza, will force the parties to change their calculations. That is to say: if one rocket passes through and kills many Israelis, the government might be forced to respond in ways that will alter the state of the battle. On the other hand, if the Israeli air force will erroneously target a large group of Palestinian civilians, the pressure on Israel from the outside to curb its actions can become much more significant.
Thinking about the overall possible objectives, there are three groups of possibilities for Israel:
1. To go back to the status quo ante – that is, to have a ceasefire and go back to the understandings following the Pillar of Defense operation a year and a half ago. That is to say: Hamas doesn't get anything from this round, and Israel doesn't get anything from it. The disruption, the loss of life and property, the anxiety, all that for nothing. But Israel can still say that Hamas was the aggressor, that it achieved nothing, and hence that it has possibly learned a lesson. Since Israel didn't initiate the fight, maybe ending it with a result that is essentially a tie is acceptable.
2. To agree to give Hamas something in return for a ceasefire: I don't think Israel is in such a mood, but theoretically speaking, the "something" might not have to be something that Israel will give in return for calm, it can be "something" that Hamas will get from the Egyptians, or from Qatar, or from other international players. This will give the organization an incentive to calm the waters without Israel having to feel that it made concessions. The downside, obviously, is that for Hamas it doesn't really matter which country made the concession to appease it – all that matters is that the round of violence was fruitful. If this round is fruitful it is only a matter of time until Hamas starts another one.
3. To insist on change, and here there are many options. For example, to insist that Hamas loses power. This isn't what Israel wants right now. But there are more modest and possibly more realistic goals. One is to forcibly destroy a large chunk of Hamas' arsenal of rockets. To achieve such a goal Israel would have to launch an expensive and bloody land operation. Another possible goal is to disarm Hamas by diplomatic means – that is, to make it get rid of its arsenal under the threat of force, in a way similar to the one used against Bashar Assad's chemical weapons in Syria. For such a thing to happen, an international front is going to have to back Israel, or do the job.
The likeliest outcome of the current round is the first one. It is more likely because it is the easiest to achieve, and the most convenient for mediators to talk about. It provides for the shortest route from the beginning of talks to the end of hostilities. Of course, that it is likely doesn't necessarily make it desirable to either side. When people go to war they usually hope to get something from it. And they want to win and to feel that the other side, the aggressor, has lost.
In Israel, the notion of Hamas losing has changed its meaning in the course of a couple of years. Not so long ago, it was still common for officials to contend that Israel's goal is to end Hamas rule in Gaza, supposedly replacing it with Fatah rule. Israel, for good reasons and bad reasons, is no longer striving for this goal – not in the short run. (The bad reason is that Israel is less hopeful about the prospect of having a reasonable Palestinian counterpart. The good reason is that Israel has become more realistic about its limited ability to manage the internal politics of its neighbors).
If managing the internal politics of Gaza is no longer a goal, this means that Israel is willing to live with Hamas and is trying to make its presence less of a burden (for example, by forcing it to partially disarm). This is an idea that’s gaining supporters in Israel, because it’s a good one. Whether it is practical for Israel to hope to achieve such a thing is another matter.
The international community, if it were serious about helping to create a more stable situation in Gaza, should have supported such an objective and could have made it more likely to be achieved. It doesn't even need to do much, other than let Israel keep attacking Hamas until it caves. But I don't think the international community is serious about having stability. All it wants is for the scenes of exchanges of fire to disappear as quickly as possible from the TV screens – also for good reasons (stopping the killing) and bad ones (it has little interest in truly understanding the Gaza situation).