Ashok Kalyanjee was jailed for life in 2009 and ordered to serve at least 21 years in prison after pleading guilty to murdering Paul Ross, 6, and his two-year-old brother Jay.
Kalyanjee, who was born in India, collected the boys, who lived with their mother in Glasgow, and drove them to lay-by in Crow Road in the Campsie Fells, near Lennoxtown, in East Dunbartonshire, and cut their throats.
Kalyanjee, 52, now claims there was a failure by psychiatrists to identify that he was suffering from a paranoid personality disorder.
But at the Appeal Court in Edinburgh the Crown has argued the disorder was not relevant because it was not "causally linked" to the murders of the children.
One psychiatrist had said he could not see how diminished responsibility could be appropriately pled.