Peace offerings: A new exhibition reveals how politics often came between the Pablo Picasso and his painting

It sounds like a straight bit of propaganda, with one top artist paying tribute to another. "On the big curtain paint the troublemaking/ Dove of peace of my brother Picasso... " In 1950, the poet and playwright Bertolt Brecht decorated his theatre in East Berlin with Picasso's anti-war bird, the symbol of the international Peace Movement. Who could complain? It's only intended to make trouble for the warmongers, no? But actually this "troublemaking dove" creates all sorts of trouble, artistic and political. It lies at the heart of a new exhibition, Picasso: Peace and Freedom, opening tomorrow at Tate Liverpool. Very good for an argument.

Follow the dove story. In 1944, at the Liberation of Paris, Picasso had joined the French Communist Party, the most Moscow line of the Western communist parties. He stayed a loyal member, through Hungary, Czechoslovakia, for the rest of his long life. He was the most famous artist and the most famous communist in the world. So when the Peace Movement appeared, essentially as a front for Soviet foreign policy in the Cold War, Picasso was naturally approached for an emblem. On the eve of the 1949 Paris Peace Conference, his dove appeared in the communist press. "This poster will be on the walls of the world" its headline ran. So it was, and he made many other versions over the years. But the problem was: Picasso knew something about doves.

So follow the story more close-up. It was the communist poet Louis Aragon who visited Picasso in his studio, looking for something likely. The painter showed little interest. "See if you can find something." "Here we are," Aragon said, picking out a suitable lithograph. Picasso was unimpressed. His father had kept doves. "I never understand how one could make it into a symbol of peace. It is an extremely cruel bird... " But Aragon had other concerns. He needed a Picasso, but the right kind. The communists liked Picasso's fame. They mainly didn't like his art. The Party line was Realism. But Aragon had managed to find a tolerably realistic image. That would do.

And so the first Picasso peace dove was not designed as such. This world-famous image turned out to be a rather odd symbol. It's not actually cruel, but not very positive either. It sits there, semi-solid, slightly helpless, half-bedraggled and half-twee. Later on, Picasso put himself to the job deliberately, producing numerous doves – triumphant, innocent, serene, redeeming, vulnerable, frankly silly. They hover in rainbows, they nest in crushed weaponry, they lead the dance, they carry plenty of olive branches. None of these doves are at all credible; they're playing safe for the cause. And all the while, as an artist, Picasso knew the facts. It is a troublemaking bird. Picasso can perfectly well paint doves when he is painting a creature and not a symbol. He's very good at living animals. It's not difficult to tell the difference, either.

This is a tricky issue, in other words, pulling in many directions. It's much trickier than anything acknowledged in Picasso: Peace and Freedom. This is a show that surveys the last 30 years of the artist's life – he died in 1973 – and looks for connections between his art and various political causes. And basically, that's it. It's one of those exhibitions that should really be a book, except, if it were, it would have to have more critical perspective. What we have is (a) any conceivable cross-reference, however tenuous or implausible, between painting and politics (b) all political causes viewed as good things, simply because they are "politics" (c) the content and form of art treated at the most superficial level. All the same, there is some matter here worth looking at.

Picasso is a passionate believer, that's certainly true. He wants to reflect and address the wider world. But does it help his art? For example, the first big "masterpiece" in this show is The Charnel House, 1945. It's clearly a successor to Guernica, an atrocity scene painted in photo-monochrome. Its declared subject is the killing of a Republican Spanish family, though it could recall many murders committed in the war years. And like Guernica, for political and humane reasons, you might like to believe in it. It doesn't work. The twisted limbed, the interwoven outlines, are overlaid with a pattern of greys. The horror is prettified. The pile of corpses is reduced to a puzzle. The caption should read: how many dismembered body parts can you find hidden in this picture? Picasso believes in the cause. Somehow he can't believe in the image.

There are other outrage paintings that should be here, but which aren't available, like Massacre in Korea and War and Peace. They wouldn't make things any better. Tragedy and terror are not in Picasso's voice. Or perhaps what he can't do is work in a single voice, in a mode of pure sincerity, whatever he's doing. His draughtsmanship is always ironic, with a twist, playful when it should be painful, smart when it should be loving. There's a little drawing here called The Forces of Peace Fighting Against War, and the title gives its oddity away. Peace Fighting? It is more or less a cartoon. Peace raises a sword, shielding a worried dove. War is a war-machine, a head belching fire, running on tanks tracks. Both, especially war, are gleeful. Nobody could mistake this image as a blow for anti-war. It's a funny punch-up.

There is quite a bit of inadvertent hilarity in Picasso: Peace and Freedom too. Look at his most unfortunate and celebratory images devoted to Joseph Stalin. To get the best of them, you have to read the essays in the catalogue, because they give you not the slightest hint that the smallest shadow of unpleasantness might have fallen across the reputation of the tyrant. Picasso greets Stalin's 70th with a glass of wine, dedicated "Staline, À Ta Santé". And when he dies a few years later, the dialectic really hits the fan. Grave offence and upset is given to Party stalwarts. Picasso's memorial portrait has not depicted him as a benevolent Santa Claus, but rather as a romantic young blade. A serious controversy ensues, with fraternal criticism offered. And nobody mentions the millions killed.

Once you start looking for fun at the expense of the curators, it turns up thick and fast. They are such priceless captions, making such hopeful associations. The battle between a Lobster and Cat? "The engagement in conflict and the date of the work suggests that Picasso was making a reference to hostilities between East and West during the Cuban Missile Crisis which began in 1962." (And ended in 1962. The painting is from 1965.) Or a wild variation on Manet's Déjeuner sur l'Herbe? (1960). "It reflected the sexual revolution of the emerging counter culture of the 1960s."

Ah well. But look, one or two very good Picassos, political or not political, have made it through. They're mainly about animals, actually, though I'm not aware that the artist was remotely concerned with Animal Rights. The show would certainly have found this out, if there were any evidence. But when you put the insipid peace doves to one side, these species couldn't be further from a good cause. The Cock of the Liberation is a marvellous construction with its cut-and-thrust shapes. In Still Life with Owl and Three Sea Urchins, the owl-cum-monkey-cum-skull comes at you like a fierce mask. The Studio (the Pigeons) has blobs of avian craziness. His energy of creatures, especially birds, is half violent and half comic. Come to think of it, one of these days, after every other possible theme has been exhausted, they'll have a show simply called "Picasso and Comedy". It'll be a true revelation.

Picasso: Peace and Freedom, Tate Liverpool (0151 702 7400) 21 May to 30 August, £10 (£8 concessions)

For further reading: 'Picasso: Challenging the Past' by E Cowling (Yale University Press)