Data and La Forge are enjoying a Sherlock Holmesholodeck program when the two notice that a character programmed to be left-handed was actually right-handed. They call Lt. Barclay to repair the holodeck, but as he checks the status of the Sherlock Holmes programs, he encounters an area of protected memory. He activates it to find the artificial sentientProfessor James Moriarty (Daniel Davis) character projected into the Holodeck, who appears to have memory since his creation ("Elementary, Dear Data"), including during the period while he was inactive (a feat Picard claims to be impossible). Moriarty wishes to escape the artificial world of the holodeck and was assured by the crew of the Enterprise that they would endeavor to find a way to free him, and is irritated at the lack of results on the part of the crew and their seeming lack of even the tiniest bit of effort. Picard, along with Data and Barclay, attempts to assure Moriarty they are still working on that goal, but Moriarty is dismissive of that.
Moriarty confuses the crew by seemingly willing himself to existence by walking off the holodeck. He explains this to the stunned Picard and Data by saying, "I think, therefore I am.". Moriarty creates a companion for himself, the Countess Regina Bartholomew (Stephanie Beacham), commanding the computer of the Enterprise to place another sentient mind within a female character of the Sherlock Holmes novels. Moriarty then demands that a solution to get Regina off the holodeck be devised. He takes control of the Enterprise through the computer, insisting that a way be found for her to experience life beyond the confines of the holodeck.
While assisting LaForge, Data observes that LaForge's handedness is incorrect, just as they had experienced earlier. Data determines that he himself, Picard, and Barclay never left the holodeck, and everyone and everything that appears to be the Enterprise is part of a holodeck program Moriarty created. At that moment, Picard realizes that he has unwittingly provided Moriarty with the command codes for the Enterprise. With this information, Moriarty takes control of the real Enterprise from within his simulation.
Data finds a way to program the holodeck's simulation of a holodeck to convince Moriarty that he and Regina can be beamed into the real world, though in fact they are only "beamed" in the holodeck's simulation. Moriarty, satisfied with the ruse, releases control of the ship back to Picard. He and the Countess use a shuttlecraft given to them by Commander Riker to leave the Enterprise and explore the galaxy. Picard ends the simulation and returns to the real Enterprise. Barclay extracts the memory cube from the holodeck and sets it in an extended memory device in order to provide Moriarty and the Countess a lifetime of exploration and adventure.
Picard mentions the possibility that the crew's reality may actually be a fabrication generated by "a little device sitting on someone's table." This unnerves Barclay enough for him to test the nature of his own reality one more time: he gives an audible command to "end program" to test whether he is still in a simulation. There is no response.