Induced self-destruction

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 intelligences' routines of thought were unable to cope with logically insoluble problems.

Captain James T. Kirk was quite able at inducing self-destruction in artificial intelligences, or “talking computers into death”. He achieved the feat at least four times.

Landru
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")

M-5
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")

Nomad
The artificial lifeform Nomad was incidentally 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 shut down, thus carrying out its own programming. (TOS: "The Changeling")

Norman
The android Norman suffered from a severe internal failure when he was presented with the liar paradox by Kirk and Harcourt Mudd.


 * ''"Everything Harry tells you is a lie. Remember that! Everything Harry tells you is a lie!"


 * "Now listen to me carefully, Norman laddie: I AM LYING!"


 * "You lie, but if everything you say is a lie then you must be telling the truth, but you cannot be telling the truth because everything you say is a lie... you lie, you tell the truth, you... Illogical! Illogical! Please explain! You are human! Only humans can explain their behavior! Please explain!" (TOS: "I, Mudd")

Rayna Kapec
When the android Rayna Kapec was presented by Kirk with the choice between love for her "father" Flint and for Kirk, Rayna ceased to function, or died. (TOS: "Requiem for Methuselah")

Related cases
Here are some cases where artificial intelligences or cybernetic beings were unable to cope with irrational or paradoxical inputs.

Andrea
After Kirk had kissed the android Andrea she displayed erratic behavior. (TOS: "What Are Little Girls Made Of?")

The invasive program
Commanders Data and Geordi LaForge 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 had actually worked. The plan was never carried out due to ethical considerations. (TNG: "I, Borg")

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")