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To know what beauty is, shouldn't one observe examples of it? But if one doesn't know what beauty is in the first place, how can one tell if one is observing examples of it?

Nice question! Here's a quick reconstruction of your reasoning: (1) In order to know what X is, you need to observe instances of X. (2) But one could only know one is observing instances of X if one already knew what X is. (3) Hence, one can neither know what X is nor know whether one is observing instances of X. (3) looks like a pretty powerful skeptical conclusion: It would seem like we can't know whether a certain thing is beautiful unless we know what beauty is, but we can't know what beauty is unless we know which things are beautiful. We certainly seem stuck -- and the same reasoning could be used to generate skeptical conclusions about other important philosophical concepts (such as goodness, virtue, and so on). Is there a way out of this conundrum? There are a lot of complex issues here, so let me just mention some possible ways out and leave it to you to assess their plausibility: Testimony Others might be in a credible position to tell us which things are beautiful or what beauty is....