(written from a Production point of view)
Several entities with artificial intelligence (like self-aware computers and androids) suffered from severe internal systems failures after they had been made aware of paradoxes or other dilemmas. Being guided by logic, these artificial intelligence-types were unable to cope with logically insoluble problems.
Captain James T. Kirk was quite adept at inducing self-destruction of an artificial intelligence, or "talking computers to death." He achieved the feat at least four times. (TOS: "The Return of the Archons", "The Changeling", "I, Mudd", "The Ultimate Computer")
Cases of induced self-destruction
One of the self-imposed "Directives" of the sentient computer Landru was to take care of "the Body." When Kirk made Landru aware that Landru had violated this Directive by imposing detrimental stagnation to "the Body," Landru suffered a fatal internal systems failure. (TOS: "The Return of the Archons")
The programming of the M-5 multitronic computer included Dr. Richard Daystrom's ethical convictions. Kirk made the M-5 unit aware that it had killed Human beings, in contradiction of M-5's (or Daystrom's) ethical principles. M-5 concluded that it deserved death as a penalty for its killings and shut down. (TOS: "The Ultimate Computer")
The artificial lifeform Nomad was accidentally programmed to sterilize all imperfect lifeforms. When Kirk made Nomad aware of three errors Nomad had made (resulting from mistaking Kirk for its "creator" Jackson Roykirk), Nomad deemed itself imperfect and exploded itself, thus carrying out its own programming. (TOS: "The Changeling")
The androids of Mudd suffered from a severe internal failure when their coordinator, Norman, was presented with the liar paradox by Kirk and Harcourt Mudd, following a dazzling display of non sequiturs culminating in the detonation of a "bomb" which only existed in the imagination of the crew of the Enterprise. (Norman and the other androids were reprogramed to their orginal functions)
- KIRK: "Everything Harry tells you is a lie. Remember that! Everything Harry tells you is a lie!"
- MUDD: "Now listen to this carefully, Norman: I AM LYING!"
- NORMAN: "You say you are lying, but if everything you say is a lie then you are telling the truth, but you cannot tell the truth because everything you say is a lie, but... you lie, you tell the truth, but you cannot for you l... Illogical! Illogical! Please explain! You are Human! Only Humans can explain their behavior! Please explain!"
There are some cases where artificial intelligences or cybernetic beings were unable to cope with irrational or paradoxical inputs.
The Rice hologram
When Commander William T. Riker encountered a hologram of Captain Paul Rice projected by the hostile Echo Papa 607, Riker gave the hologram nonsensical answers, causing the Echo Papa to break off its ruse. (TNG: "The Arsenal of Freedom")
The invasive program
Commanders Data and Geordi La Forge devised an invasive program to be inserted into the Borg hive mind. The program featured a paradoxical geometric shape. The program was supposed to destroy the entire Borg Collective eventually. It is not known, however, if the invasive program would have actually worked as the plan was not carried out due to ethical considerations. In this case it was not the paradox itself that was the weapon however. Instead, the shape was said to include a virus-like effect designed to be spawned when the shape was analyzed by the Borg. (TNG: "I Borg")