haha how is my logic flawed?
Because you don't need to cast "melted ice" (water) in order for the ice to melt and put out the fire. Have you ever tried putting a block of ice on a campfire? Either the fire will be put out, or the fire will subsist, all the ice will melt, and the water will boil. The outcome depends on the relative quantities of said "elements".
we're talking about a fantasy world with unreal physics so the basic rules of fire melts ice should apply...it's a game so it should follow rock/paper/scissors game rules.
Okay, so which is it? Should the world have "unreal physics" (a term which you fail to define, by the way)? Should it have real physics (where imbalances in heat energy tend to balance themselves out, and a fire cut off from its oxygen supply dissipates)? Or should it follow rock/paper/scissors game rules (which have nothing to do with physics)?
Each one beats another just like each skyrim element should beat another.
Imo it should be fire beats ice, ice beats spark and spark beats fire...simple yet effective

Actually, the elemental system in this game doesn't really work that way. Saying that it operates on a rock-paper-scissors system is only correct when you implement different enemy types into the equation. When it comes to two mages battling one another, whichever one is using lightning should most likely win (assuming all other variables are equal), because lightning is the anti-mage element. On the other hand, frost is anti-melee, and fire is general purpose.