Hørup was born in Torpmagle near Hundested, the son of a North Zealandian schoolteacher, but belonged to the relatively well-to-do middle class.
Already as a student Hørup took interest in politics, early joining the party Venstre after a short conservative intermezzo.
He rejected the military defence as an unrealistic, dangerous and expensive protection of Denmark and he founded a long-standing scepticism against the army that has affected many Danes.
He also connected radical authors to his case, for instance Georg Brandes, whose brother Edvard was Hørups co-editor and party colleague.
He condemned the attempts of compromise with the Right and his relationship to his fighting companion and opponent Christen Berg was switching between co-operation and obstruction.
An attempt of making a compromise himself 1887 failed and after his loss of his parliament seat he had to leave most of the daily leadership of his political fraction to others.
Left liberals and socialists as well as communists have referred to him and quoted his apothegms regarding him as the ideal unselfish and consequent progressive politician.