From a computational sense, this logic has fitted well into the more problematic areas that we try to handle in the modern contexts, especially with growing use of computer systems. It does allow a better handling of proofs without falling into the traps of classical logic.
Yet, its entailment is not decidable. Somehow, we need to get that notion better understood. I've used the fact that we don't have 20-20 vision going further (farther, as in, forward movement from a position or state). Of course, we may not have 20-20 in hindsight, either.
Now, why is linear logic so important? It parallels the progress in advanced linear algebra that has been so important to numeric systems and visualization. That is, even non-linear problems are approximated using linear techniques.
And, we, essentially, have overlaid the world with this type of constraining view, despite that we know the limits of monotonicity and have determined the need for non-monotonicity in complex systems. That is, those with numeracy have superimposed upon us what is very well suspect.
That anyone would argue for some caution raises the ad hominem argument of innumeracy.
Now, having before raised the notion of getting technical, we'll do so (actually, we'll plumb to whatever depth is necessary -- the request of the reader? soar as far as we must, to boot). At first, we'll go top down, trying to establish a sound, coherent viewpoint (notice that this is juxtaposed to that approach which has been developed for the past 200 years and which has been deleterious -- disrespect for that which makes us human -- as well as beneficial - attempts at firming up the plumbing which can still leak - outside of our control).
11/21/2010 -- Three years ago, it was said: Computational foci raise miraculous need. Still applies.
11/16/2010 -- It is the fact that there are serious lapses in what we can and do know. Now, our maturity provides the basis for decisioning under the resulting uncertainty. That the computer has become a major player raises a whole bunch of issues. The main one is that the underlying framework is undecidable. Plenty have danced around this subject, but linear logic allows us to look at the problem more realistically.
11/10/2010 -- Jim M. could be a hero if he learned the lesson of undecidability.