I think there comes a point in this story when Titus realizes that all of his determination to stick to tradition - to live his life according to the traditions and rules of the Rome that he has served for so many years - has not helped anything turn out the right way in his life. Whether or not he realizes his fault in this - the fact that if he had not sacrificed Tamora's son because of tradition, none of these horrible things would have happened to his family - I really am not sure. But once he sees his daughter, completely destroyed by his enemies, he sets upon a course of revenge that takes him away from the stoic traditionalism that he has lived by for his whole life.
Interestingly, though, once he completes his revenge on Saturninus, Tamora, and her evil sons, he goes back to the tradition of the story of Virginius, which leads him to end Lavinia's suffering by stabbing her to death.
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i think at 2 points: When Tits shoots arrows up into the sky with messages attached to the gods asking for justice. He has no faith in the system anymore and is going over Saturninus's head -- which, if i remember correctly, angers Saturninus. Secondly, at the climactic dinner scene, when Titus reveals it was Chiron&Demetrius that raped Lavinia, Saturninus says "Bring them to me at once," thus Satutninus is ready to dispense justice himself, as emperor. But Titus acts as vigilante by explaining he already killed the 2 sons by himself and then he proceeds to kill Tamora, thus subverting Saturninus's attempt to render justice thru the system.