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Given enough assumptions (one of which might be the assumption that there are differences of opinion about whether or not some particular conduct is ethical), it is possible to arrive at the conclusion that ethics is subjective. However, it is important to note that we mentioned ethics. Some assumptions allow us to reach the conclusion: "ethics is subjective." These assumptions are known to be reliable, but they are specialized assumptions that refer specifically to the topic of ethics. If we were to construct analogous assumptions and try to apply them elsewhere, then we might find that they are unreliable and yield false conclusions.
For example, suppose we were to try to apply similar reasoning to chess. What does chess have to do with ethics? Chess is a game with specific rules that are known; those rules specify under what circumstances the game comes to an end. Ethics is completely different. Thus, there are specific assumptions that mention ethics and that are known without any doubt to be reliable, but analogous assumptions that mention chess are of unknown reliability.
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