Symbolic Logic is a branch of philosophy which attempts to create a system that can be used to predict outcomes from known premises to try and prove the validity of an argument.
Essentially sentences are broken down and analysed for validity or invalidity then affixed as variables. Each variable is either true of false, as symbolic logic is a binary system. These variables then can have operators applied to them in an attempt to arrive at a conclusion.
These operators include: NOT, AND, IF, OR, and EQUALS.
From these truth tables can be created to discover all outcomes for all possible premises in an attempt to find a sequence that proves validity. This is where the power of symbolic logic enters. A system called sentential logic has added additional rules that create short cuts to writing out truth tables and finding validity. These are called rules of derevations and rules of replacement.
It is entirely impossible to explain this system effectively on E2, as there are special symbols used in this system not a part of the ASCII chart. If interested in this idea, check your university or college for a course on this, or try searching for a more complete explaination on the web or go here: http://www.earlham.edu/~peters/courses/log/loghome.htm