First up, a common misconception in the climate debate is that radiation from a cold body (e.g. the -18C atmosphere) can warm a hot body (e.g. the +15C Earth surface) just because the cold body does indeed send very-low-energy photons to the hot body. Heat transfer (not radiation) from cold to hot is forbidden by the 2nd Law of Thermodynamics on a macro basis, and by the Pauli Exclusion Principle of fundamental quantum theory on an atomic and molecular basis.
Fallacy: misrepresentation