In the game's menu, please go to EXTRAS and select PERSONAL STATISTICS. Scroll down to find a "Whole Story" checklist there. Make sure that you have everything listed as "complete" there. If not, create a new game and finish the missing quests. Alternatively, you can try the following method:
If you have completed all of the quests, please make a backup of your save files on a memory stick or send it to Sony's cloud system, so as not to lose them.
Having done that, launch the game, go to Play Campaign and delete all your save files (Advanced -> Delete).
Next, start a new campaign, but not from the beginning - choose New Game -> Advanced -> New Game from Quest -> "Pact with Rais".
Now, approach the soldier on the floor near the Tower's exit (the one next to the Quartermaster) and talk to him - he is going to tell you about Alfie. Then, go to Alfie on the same floor and complete his side quest called VOLTAGE.
After completing the quest and receiving the skill points for it, THE WHOLE STORY trophy should appear as achieved.