[@KatherinWinter]
That matches up with what I said. The latter would be true; you'd be afraid of dogs, but not remember why, and any memories of being afraid of dogs (even if you don't remember the cause) would be unaffected by removing the first memory in the chain.

Unless the event that caused the fear of dogs was VERY recent, removing it would have no effect on the overall fear because it has been reinforced over time by other memories in which you experienced the fear of dogs.

It's the not know why that's important in this case. Your personality would remain in tact, you won't remember the original source of personality traits but because the majority of your memory in which those traits were relevant remain in-tact loss of the source memory doesn't impact the trait. It's been reinforced over time by subsequent memories.

There's actually a semi-famous example of that which I can use. There was a man who had the temporal lobes in his brain removed because of severe epilepsy. He lost the ability to learn new things and form new memories, but his personality remained unchanged. In fact, he gave the exact same answer to any question no matter how many times he was asked.

On a more personal (and painful) example. A few years ago, my great grandfather died of Alzheimers disease. Over the course of nearly a decade his mind began to deteriorate and his memories slowly dissapeared. Throughout that time, though, even when he could barely remember his own name, his personality never changed. Not even a little bit. he was still the same stubborn old man that raised me.

EDIT: I keep forgetting to put a mention in. I'll work on that if you decide to let me join.