The main theorem
The А-principle enabled Carnap to prove that (2) is a linear function of n. In other words, Carnap was able to prove the result expressed by (15) below. Yet the А-principle effectively restricts the dependence of (2) to nj and n only when one is considering more than two cells. In the case in which we are considering only two cells, say H and T, like in tossing a coin, the occupation vector is (nH, nT), and the А-principle, being automatically satisfied, does not work. For this reason, Carnap was compelled to state a further condition that he called the principle of linearity (Carnap, 1952, p. 98). This principle states that for two cells, that is,
when d = 2,(2) is a linear function of , j = 1,2. This is tantamount to assuming the validity of the result he was not able to prove. At the end of the 1970s (see Costantini, 1979) one of us proved that both the X-principle and the principle of linearity follow from a single condition, that is, invariance of Qg (n). This proof was very cumbersome. A clue for a new demonstration, simpler and more intuitive (see also Garibaldi and Scalas, 2010), is the following.