JoshuaZ comments on Train Philosophers with Pearl and Kahneman, not Plato and Kant - Less Wrong
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Aristotle is a more straightforward example. If you made an effort to understand Aristotle's four types of causes and ten categories of being - if you critically tried out that worldview for a while, tried to understand your own knowledge and experience in those terms, identified where it works and where it doesn't, the logic of the part that works and the problem with the part that doesn't - it would undoubtedly be instructive. Aristotle is such a systematic thinker, you might even fall in love with his system and become a neo-Aristotelian, bringing it up to date and evangelizing its relevance for today's world.
This seems to be more indicative that if one thinks hard enough about any world view it will seem to be useful and make sense. This is essentially as much of an argument to take Aristotle seriously as C. S. Lewis's claim that "I believe in Christianity as I believe that the sun has risen: not only because I see it, but because by it I see everything else." is an argument to take Christianity seriously.
This doesn't answer the question or even the type of question as phrased by Viliam. The claim isn't that you can use a systematic approach to make your own thoughts ordered in some fashion, but how to make the claims pay rent.