The Brotherhood of Logicians was an assemblage of intelligent humans who emphasised pure logic. (TV: The Tomb of the Cybermen). It was originally founded by students during the Cyber-War as a secret society who opposed the war on the grounds they believed that humans should become more like Cybermen. (AUDIO: Last of the Cybermen)
The Brotherhood hired Joylove McShane and Yellow Stinker to obtain Cybermen on Grot. Joylove and Stinker were unsuccessful in their job. (COMIC: Junk-Yard Demon II)
Five hundred years after the Cyber-Wars, Eric Klieg and Kaftan (both members of the Brotherhood of Logicians) openly financed an archaeological expedition to the cyber-tombs of Telos, secretly motivated by the belief they could form an alliance with the Telosian Cybermen to help the Brotherhood seize power. Both were both killed by the Cybermen. (TV: The Tomb of the Cybermen)
Behind the scenes
The tongue-in-cheek article "You Know Our Ways" by Jonathan Morris in DWM 504 is formatted as an email sent to the Croydon and Bromley branch of the Brotherhood of Logicians concerning a recently acquired infostamp containing information on the Cybermen.