PMQS verdict: Harman is right, but it doesn’t save her

She was right about everything. She was right about Greece, she was right about tax credits and the living wage, and she was right about trade union funding for Labour. But it didn't make any difference. Cameron easily won his last clash with Harriet Harman before the summer break.

He did so for two reasons. The first was the easy confidence of victory. Cameron is riding one of those waves of success which make people breezily self-assured in their manner. Harman is temporarily heading a party which is staring electoral oblivion in the face, without any candidate making a convincing case for how to reverse the decline. She spent the early part of the week trying to extinguish the flames of her own rightward shift, not least from the very people vying to lead the party after her. Facing abuse to her front and suspicion to her back, she was unable to assume the same sense of confidence as the prime minister.

The second was the great political art of a one-step argument. More than red lines, this has been the great triumph of Conservative politics in the last few years. Even when it is objectively false, they can explain the argument in one sentence. This makes it sound like common sense.

This is how it goes: Harman observed to Cameron that the Institute of Fiscal Studies (IFS) found there was no arithmetical logic to the living wage announcement. The living wage (clue is in the name) is calculated on the basis of what it takes to live on, so it factors in in-work benefits like tax credits. But George Osborne cut tax credits, then effectively raised the minimum wage and rebranded it a living wage. It's just not the same thing. Osborne knows that. Cameron knows that. Harman knows that. It's like putting a Barack Obama mask on your husband and calling him president of the United States of America.

But, as I've just demonstrated, that takes rather a long time to explain. Cameron didn't bother responding to any of it. He just pointed out that Labour would have set the living wage at £8 while he's going to set it at £9. Argument over. It's literally false. But it doesn't matter, because explaining it makes you sound long-winded, academic and weirdly shifty. Cameron sounds like he's talking common sense.

Harman suffered from the same dynamic on the way in which people go into and out of low paid work. Again, it's true. But her increasingly desperate insistence that he didn't understand the reality of it just sounded tedious compared to the cool, simplistic assurances the prime minister was offering.

By the time she raised the plainly unfair changes to Labour's funding stream from trade union members – changes which will drive Labour towards the arms of big business and make it even less representative of its members than it is now – she was completely lost in a sea of simple answers. Cameron jumped on the 'union paymasters' line and the job was done. No matter that he has to conduct his own funding operations by prostrating himself privately with wealthy donors. He has a one-step argument: Labour is the pocket of its union paymasters. Sounds right. Job done.

So once again we witnessed Harman and Cameron clash with nothing much to come from it, except for a confirmation of the way that simple arguments trump true ones, especially when you've the winds of recent victory at your back.