While in many cases an aircraft Limitation (originating from the OEM, but endorsed by the relevant authorities which certified the type) may expressly prohibit takeoffs with contaminated surfaces, an more direct source of control is the relevant operational regulations:
14CFR121 for example, says:
§ 121.629 Operation in icing conditions.
...
(b) No person may take off an aircraft when frost, ice, or snow is adhering to the wings, control surfaces, propellers, engine inlets, or other critical surfaces of the aircraft or when the takeoff would not be in compliance with paragraph (c) of this section. Takeoffs with frost under the wing in the area of the fuel tanks may be authorized by the Administrator.
Most "western" authorities have similar regulations.
Of course, we learned the hard way .. prior to Dryden, the Canadian regulations, for example, allowed a lot more interpretation on the part of the Captain. All of the "in the opinion of" type wording has been long removed, though.