| This article, Mirror Image, contains spoilers. Be forewarned, plot and/or ending details follow.
R. Daneel Olivaw visits his old friend Elijah Baley on Earth, requesting his assistance in resolving an issue on a Spacer ship. The vessel's passengers include two great Spacer mathematicians headed to a scientific conference on Aurora: the 270-year-old veteran Alfred Barr Humboldt, and the young (under 50) up-and-coming Gennao Sabbat. The two men are accusing each other of plagiarism.
Humboldt claims that prior to boarding the ship, he devised a revolutionary mathematical technique for neural analysis, and discussed the matter with Sabbat while abroad. The younger mathematician was strongly supportive of the idea, and Humboldt wrote a paper on the subject to be presented at the conference, only to discover that Sabbat had done the same in an attempt to falsely claim credit. Sabbat's story is identical, except for the "mirror image" of names: in this version, it was he who created the technique and Humboldt who tried to appropriate it. Each of the men has a personal robot, of identical model and production year, testifying in its master's support.
Neither mathematician will confess to the crime (and there is no guarantee that a confession would result in justice, in any case) or accept a psychic probe, and no documentary evidence is available to resolve the issue. Failure to resolve the dispute before the ship reaches Aurora would result in an undesirable public scandal. Daneel persuaded the captain to take a detour to Earth and obtain the assistance of Baley, avoiding unnecessary publicity.
Neither mathematician will consent to an interview by an Earthman or an inspection of his robot by an Earth robopsychologist, so Baley remotely interviews the two robots in sequence. Each acknowledges that it would lie to protect its master's reputation, to prevent harm under the First Law of Robotics, although whether its master's reputation or that of another person is more important would have to be decided on a case-by-case basis.
To Sabbat's robot, Baley explains that it would be more important to protect Humboldt's reputation than Sabbat's: the latter's offences could be explained away as the errors of a hot-blooded youth and he has a long career ahead of him to recover his reputation, while Humboldt's great historic achievements would be forgotten in the wake of his one crime and he would have little lifespan left to make up for it. To Humboldt's robot, he takes the opposite position: Humboldt's past reputation would shield him from the worst consequences, and he would be excused as an old man no longer sound in judgement; the less secure Sabbat would have a future great career, spanning hundreds of years, closed off by a single mistake of youth.
Sabbat's robot accepts the explanation and confesses to its master's act of plagiarism, while Humboldt's malfunctions and shuts down. Baley interprets their differing reactions as follows: the liar's robot would have been ordered strongly to lie, thereby combining First Law and Second Law, while the truth-teller's robot would receive no such orders (as it could be trusted to support its master on its own) and its actions would be governed only by First Law. Thus, the truthful robot could be persuaded to switch to a lie without too much trouble, while the lying robot would risk breakdown of its positronic circuits in switching to the truth. Baley concludes that Humboldt's robot was the one originally lying.
Presented with the results of the interview, Humboldt breaks down and confesses to the plagiarism. Afterwards, Daneel asks why Baley selected that interpretation, as it could just as easily be argued that it would be easier for a robot to switch from lying to telling the truth than the reverse. Baley explains that he already guessed that Humboldt was the guilty one based on his knowledge of human behavior, and simply interpreted the robots' behavior as evidence against his chosen suspect to intimidate him into a confession. According to Baley, a young academic like Sabbat would be interested in discussing his new idea with a senior peer and would not dare plagiarize such a "revered demigod" in the field, while a senior figure like Humboldt would not think to consult a junior and might well be willing to take advantage of such a newcomer for one last career triumph.