Just being a pedophile or being sexually attracted to children isn't a crime, although it is certainly unhealthy. If, however, that attraction is acted on in certain ways, including possession of CP, than a crime is committed. In Florida, laws governing child pornography can be found in Chapter 847 of the Florida Statutes.
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Common Law is law that grew up over time, starting in England and relied on in the US. Other laws, passed by legislatures, are called statutes. That is why sex with an underage person is commonly called Statutory Rape, although more commononly nowadays, criminal sexual conduct. Very few people convicted of statutory rape are true pedophiles, more often the crime involves a 17 year old and a 15 year old, and similar acts which are still crimes but do not appear to exhibit a sexual perversion. For an actual pedophile, it appears that they are just "wired up wrong" in that they desire (and often only desire) a child as a sexual partner. If the pedophile acts on the urge by molesting a child, penalties are often decades in prison, if not life.
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