Until you find actual proof that one ever existed, logic dictates that it did not exist. When new evidence appears showing the existence of mermaids, then you can believe they existed. The same goes for God.
...Ok, time to change the ol' Sig ^_^
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If you've read The Iliad, then you'll notice that Homer uses repetitive phrases throughout - my favourite being "rosy fingered dawn", due to me having a childish sense of humour - and that is why there is some truth in his work, as those stock phrases were passed down to him from generations of previous poets. It's why he has some fairly accurate descriptions of armour and possibly names.
The city Homer described was inhabited into the Roman era, it wasn't an unknown place to the ancients, it was just unknown to the modern world, whereas Plato talked about a place nobody at the time had heard of.
I don't think mermaids existed, but than again we have only explored 5% of the ocean. So, who knows what's really out there.
If they existed, they'd have evolved. There would be some record of their history or their transition from terrestrial hominid to aquatic hominid. The 5% of explored ocean as an argument for the possibility of them existed doesn't hold water. That claim is because we haven't cataloged all the small species that exist in abyssal plains and deep ocean trenches. There's not enough nutrient density in those areas to support something that would swim as inefficiently as mer-people.