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If they're telling the truth then they're not a liar, even though they claim to be.
It's a thought experiment. There's no hidden information.
no they wouldn't and they wouldn't be telling the truth even if they are known compulsive liars saying they are compulsive liars.
The ends do not justify the means. If their intention was to deceive, then they are a liar.
Besides, a liar doesn't get to pretend to be trustworthy because they occasionally say things that are factually correct. If you consider their pattern of behaviour, the truth becomes more clear.
Well, define "liar". If it only means "I often lie", this checks out.