We investigated the time evolution of the symmetry of symbols constituting alphabets rooted in the Phoenician script. A diversity of quantitative measures of symmetry of graphemes appearing in Phoenician, Western Greek, Etruscan from Marsiliana, Archaic Etruscan, Neo-Etruscan, Euclidian Greek, Archaic and Classical Latin and Proto-Hebrew scripts, constituting the Phoenician script family, were calculated. The same measures were established for the Hebrew/Ashurit and English scripts. The Shannon-like measures of symmetry were computed. The Shannon diversity index was calculated. Our findings indicate that the Shannon diversity index increased with time in a monotonic way for the studied scripts. The diversity of symmetry groups inherent for addressed alphabets grows with time. We also introduced the symmetry factor of the alphabet. The symmetry factor quantifies the averaged level of symmetrization of the alphabet and the possible parsimony of graphical information necessary for the drawing of the entire set of graphemes constituting the alphabet. We found that the symmetry factor is decreased with time for the alphabets rooted in the Phoenician script. This means that the average level of symmetrization of the studied alphabet increases with time. The parsimony of graphical information necessary for writing graphemes is consequently increased with time. The values of the symmetry factor calculated for the addressed scripts are close to one another, with the pronounced exception of the Hebrew/Ashurit script. Our study supplies the arguments for the point of view, according to which the modern Hebrew/Ashurit script did not emerge from the Phoenician one.