Paper is a choice now
In his first answer, my opponent said that he would define a paperless society as "one in which all tasks can be performed without paper in the normal course of business". I still maintain that we are already there, because many of us actually pretty much can perform most all our business tasks without paper. Whether we do or not is a matter of choice at this point, and if we do choose to use paper we use it temporarily and we recycle it. And it's just going to get better from here, as more people see how advantageous and pleasant it is to live with less paper.
As Larry so kindly pointed out, the "marginal cost of a digital document is effectively zero already". Businesses like to cut costs; people really like "free." As free becomes easier and easier to use (we’re already seeing better document reading solutions on mobile devices which make things easier to keep digital than to print out), more individuals and companies will join the paperless pioneers in the less paper-encumbered world that already exists as a reality for early adopters of this worthy ideal.
A generational change
The major factors for whether and how specific companies and specific users dealing with specific tasks shift from the use of paper for those tasks to making them purely electronic is largely determined both by cost and comfort level.
I don't see the cost issues swinging all that wildly against paper in near future, at least to a greater degree than they have in recent years, so people who don't want to give up on paper will be able to keep it without being taken to the cleaners. The cost of converting a paper process may very well not be worth it. On the other hand, if it costs you enough to use paper, you'll put up with an uncomfortable electronic experience.
And as new software systems and business processes - and new people - enter service, they will be much more likely to avoid paper. It's a generational change.