In psychology, the notion of "general intelligence" is associated with the concept of the "g-factor." It is somewhat controversial how robustly this exists as apart from various specialized intelligences.
Mathematical Theory of General Intelligence
Shane Legg and Marcus Hutter have defined intelligence (meaning general intelligence) as "Intelligence measures an agent's ability to achieve goals in a wide variety of environments" and have given a full mathematical formalization of this definition at http://www.vetta.org/documents/ui_benelearn.pdf
A related but less rigorous approach is given in the Appendix to Ben Goertzel's book The Hidden Pattern.
The term general intelligence has been in use for more than 100 years. Note C. Spearman's 1904 essay, General Intelligence: Objectively Determined and Measured.