Anti-aircraft battleship firepower logic

Could an anti-aircraft battleship be a coherent design?

The idea can make sense because a battleship hull has structure, magazine volume, radar capacity, and enough displacement to carry very heavy anti-aircraft firepower.

The anti-aircraft battleship is a design tradeoff rather than a joke concept. Battleship structure, magazine capacity, radar space, and firepower give the idea a coherent technical basis. Hull volume and survivability matter when navies imagine specialized air-defense ships.