Is there a difference between the sentences
"The general case seems to be open."
"The general case appears to be open." ?
Or are they interchangeable?
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Following the principle of not using a certain word if a more precise one is available, I would use seem here, which is the more subjective of the two choices. In your example, the "general case" most likely is not an object that can be seen. If it were, then appear would be the better option, because this verb more precisely indicates the act of perceiving a physical object. But since "case" in this sense sounds like an abstract notion of a question to be decided by some kind of deliberation, the more abstract verb is appropriate. Both verbs can be used to say "gave the general impression of," but only one--seem--mainly means just this, while appear has multiple meanings and therefore could be confusing to a non-native speaker.