Let's focus on the phrase "something of category X that does not fit category X's definition." One on interpretation, we can't possibly find something of that description: if it doesn't fit category X's definition, then it's not something of category X, as you say. But that interpretation assumes that I've already got a correct definition of category X, a definition that's neither too broad nor too narrow. What if my definition of 'chair' is 'item of furniture with four legs' and you show me a bean-bag chair or an IKEA Poang chair? Haven't you shown me an item of category X that doesn't fit my definition of category X? Haven't you falsified my definition of 'chair', at least as a definition of the word in ordinary use, by showing that it's too narrow? (It's also too broad, as I realize when you show me a four-legged table.)
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Are definitions falsifiable? It seems that if I find something of category X that does not fit category X's definition, then it isn't actually of category X, and thus doesn't prove anything. But on the other hand, if that is the case, it seems no definition cannot be falsified or otherwise demonstrated to be inadequate (unless it is inherently contradictory or so).