I didn't say the Epson 3800 offers 360/720. Furthermore the 360 or 720 refers to pixels per inch of image resolution. The 1440/2880 is another metric - dots per inch printed.
No, but I referenced it in my post above. The 360/720 you say refers to pixels per inch , but I thought it referred to dithering. The driver clearly lists both 360 and 720 as dpi, as it does for the 1440/2880 metric.
Also, if one selects, for example, 'Enhanced Matte Paper' then there is the additional option of 720dpi. 360 dpi doesn't appear until you get down to 'Photo Quality Inkjet Paper' The four options are described as Normal(360) - Fine(720) - SuperFine(1440) and SuperPhoto(2880).
So if I understand Bart's post the 720ppi is engaged on the Epson printers ONLY by checking the Finest Detail box otherwise it's 360ppi.