Defence secretary Michael Fallon has complained about the “illogicality” of attacking Islamic State in Iraq, but not Syria. He wants to ask this parliament to sanction what its predecessor would not, allowing Britain to join the United States and Arab allies. Fallon was extremely guarded in his speech to the (dispiritingly, half-empty) Commons, and the whole media blitz has the air of a rather perfunctory trial balloon, prompted more by the need to be seen to be doing something, post-Tunisia, than any careful thought.

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But he has a point. Isis has gained territory in Syria since the war began. It has exploited its territorial depth to move men and equipment seamlessly between the two countries, and in May captured the last remaining crossing point. However, Fallon’s impulse to expand the war should be tempered with a sense of strategy.

The first question is what Britain could bring to the Syrian air war that its allies presently lack. Britain is the second most active military power in the coalition, having conducted nearly twice as many air strikes as France; but equally important has been its role in gathering intelligence for targeting. This has freed up American aircraft for Syria. British special forces are certainly active in northern and western Iraq, collecting intelligence to make strikes more accurate and effective. Unless Fallon is willing to deploy many more forces in the region, diverting scarce British assets to Syria could dilute our presence and make it even harder to contain Isis within Iraq. In Iraq, we’ve struggled more with finding targets than finding bombs to put on them. Splitting surveillance aircraft between two theatres will just make this task harder. Britain has the capacity to add more aircraft, but not much.

The Pentagon’s $500m programme to train Syrian rebels is in disarray

This relates to a second, more important question for the defence secretary: who in Syria could take advantage of British bombs? For better or worse, the US is pushing an “Iraq-first” strategy. Ever since the war began, there have been more strikes in Iraq than in Syria every month apart from October last year and this January. In June, the ratio was around 1:2, and it was even more lopsided in May. Why? One reason is that air strikes, particularly in built-up areas, are of limited use without competent ground forces to follow up. In Iraq, the Iraqi Security Forces (ISF), tribal militias, and Kurdish peshmerga were supposed to have played this role. In practice, only the Kurds have been up to the task, along with some Shia militias supported by Iran. After the Iraqi city of Ramadi fell to Isis in May, the US defence secretary Ashton Carter made the remarkable – though slightly unfair – claim that Iraqi forces “showed no will to fight”. US strategy is now in trouble: a centrepiece assault on Mosul, the city that fell to Isis’s blitzkrieg last summer, has been postponed.

The challenge in Syria is even harder. True, Syrian Kurds are proving their mettle. They worked closely with US airpower to push out Isis from Kobani in January and seize Tal Abyad last month, and have carved out their own Kurdish territory in northern Syria. But this brings other problems. It alarms Turkey, because the Kurdish militias are allied to the separatist PKK, designated as terrorists in Europe and America. In any case, Kurds can only do so much. They are focused on defending their lands, not toppling Bashar al-Assad. They have poor relations with the Arab communities that dominate Isis-held areas, and any effort to push further south would meet with resistance. So alliance with the Kurds has its limits.

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Meanwhile, the Pentagon’s $500m programme to train Syrian rebels is in disarray. Fewer than 100 rebels are currently enrolled, making a mockery of the 3,000-5,000 per year target. At this rate, the west’s vaunted rebel army will be ready sometime around Assad’s diamond jubilee. The CIA has a separate covert programme, assisted by allies including Britain, but this is unlikely to be much more successful in producing moderate rebel manpower. Unsurprisingly, Syrian rebels do not respond enthusiastically when told they ought to stick to fighting Isis and ignore Assad. The most powerful factions in Syria today are Isis itself, its al-Qaida-affiliated rival Jabhat al-Nusra, and a competing Islamist coalition called Jaish al-Fateh, which is backed by Saudi Arabia and Turkey. Contrary to popular belief, a Free Syrian Army (FSA) still exists, and western-supplied weapons, including anti-tank missiles, have played a key role in some recent offensives. But western officials, including those from Britain, remain concerned at the close battlefield relationship between moderates and extremists.

The defence secretary therefore faces a simple problem. Britain could start bombing Syria tomorrow, but the coalition’s efforts in that country amount largely to an ad hoc defence of Kurdish territory and pinpricks elsewhere. When Isis advanced into the ancient city of Palmyra in May, American and Arab aircraft stayed away. A serious offensive against Isis’s headquarters in Raqqa is a long way off.

Is Fallon’s aim merely to spread the RAF’s targets across a greater area? If, on the other hand, he has a secret plan to transform coalition strategy into something more ambitious, perhaps he should share it with us. Syria does not want for bombs. What it lacks are suitable ground forces who could drive out and destroy Isis under the coalition’s military umbrella. A more aggressive approach is possible. It would require American leadership, sweeping changes in the programme to train and equip Syrian rebels, and more direct air support in their battles. It would, eventually but inevitably, bring rebels into greater contact with the Assad regime. Such an approach seems unlikely to survive the parliamentary test at home.