True.

Logical reasoning: How could they? They do not have umbilical cords.

This is false.

There have been multiple tests done on what the actual attention span is of a goldfish, some of the conclusions depict different times, but defiantly more than three seconds. Take a look at the supporting links.

Supporting Links:
Boston.com – The chemical mind binds us together
theage.com.au – Schoolboy explodes goldfish memory myth