See http://cliki.tunes.org/Linearity 101
Also see the papers by "the founding father of Linear Logic", JeanYvesGirard (http://iml.univ-mrs.fr/~girard/Articles.html) and by PhilipWadler (http://www.research.avayalabs.com/user/wadler/topics/linear-logic.html).
LinearLogic is relevant to ComputerScience in the form of LinearTypes. Wadler's tutorial paper "A taste of linear logic" explains how LinearLogic corresponds to type systems for FunctionalProgrammingLanguages via the notion of PropositionsAsTypes.