As a thought experiment, we might imagine changing Critchley's word "certitude" in his summation of thesis 12 to the word "probability" to wit, "the mood [of Greek tragedy] is skeptical, it is about the dissolution of all markers of probability." I know, I know, Critchley wants to tout being skeptical, instead of trying to formulate philosophical probabilities. But isn't it possible that Plato and then Aristotle first grasped the skeptical dimension of Greek tragedies and then tried to the best of their abilities to construct philosophical probabilities as an alternative way to think about the world? In other words, why was Western philosophical reasoning invented?
After all, Critchley himself claims to be developing tragedy's philosophy. But if there were not good reasons for inventing Western philosophy in the first place, can there now possibly be good reasons for developing tragedy's philosophy?
In chapter 21 (pages 98-100), Critchley says, "On the topic of his [Gorgias'] teaching and method, Gorgias did not teach any set of doctrines, but a method, a hodos, which was, in his view, value-free. He gave the highest status to the power of rhetoric" (page 100). But isn't the highest status a value? It doesn't sound value-free.
In any event, the Canadian Jesuit philosopher and theologian Bernard Lonergan (1904-1984) develops a generalized empirical method (hodos) in his philosophical masterpiece Insight: A Study of Human Understanding, 5th ed., edited by Frederick E. Crowe and Robert M. Doran (University of Toronto Press, 1992; 1st ed., 1957).
John Angus Campbell, who is not a Roman Catholic, published the article "Insight and Understanding: The 'Common Sense' Rhetoric of Bernard Lonergan" in the Quarterly Journal of Speech, volume 71 (1985): pages 476-488. Later, he revised and expanded it in the ambitious 1993 anthology Communication and Lonergan: Common Ground for Forging the New Age (pages 3-22), mentioned above. In other words, Campbell considers Lonergan's generalized empirical method for philosophical (and theological) discourse to be a "common sense" rhetoric.
In chapter 26 (pages 119-122), Critchley says, "Tragedy is a symptom of the fifth-century linguistic turn [from what?] that placed enormous value on rationality, argumentation, and persuasion. Reason is essential to the experience of tragedy" (page 119; I've added the question in brackets).
In chapter 4 (pages 17-20), Critchley says, "The most common topics in ancient tragedies are connected with the events preceding or succeeding the Trojan War and the affairs of the House of Atreus and the Palace of Thebes" (pages 17-18).
Because ancient Greek culture represents one example of an honor-shame culture, perhaps we can see Greek epic poetry as exemplifying heroic honor. By contrast, the poetry in Greek tragedies exemplifies shame.
In chapter 3 (pages 12-16), Critchley says, "Greek tragedy provides lessons in shame" (page 16).
When we put these pieces together, "the fifth-century linguistic turn" (page 119) seems to be a turn from poetry centered on heroic honor to poetry centered on tragic shame.
Now, the personal experience of shame is connected with early childhood traumatic experiences, according to John Bradshaw in his book Healing the Shame That Binds You, 2nd ed. (Health Communications, 2005; 1st ed., 1988). Simply stated, if you have ever experienced road rage, such as the road rage we learn about in Sophocles' play Oedipus the King, then you are probably suffering from shame that binds you deep within your psyche.
Now, apart from Critchley's explicitly argued twelve theses, the theme of grief emerges in his book (pages 9, 10, 17-20, 168, 186, and 275-277). But he does not happen to advert explicitly to the death of Socrates and Plato's mourning his death by commemorating his life-spirit in the character known as Socrates in his dialogues. Plato's mourning the death of Socrates probably involved experiencing anger and rage.
In Susan Anderson's book The Journey from Abandonment to Healing, 2nd ed. (Berkley Books/ Penguin Group, 2014; 1st ed., 2000), she describes and explains five stages that people usually go through when they feel abandoned by the loss of an important relationship or the death of a loved one. One stage is rage. Plato most likely experienced rage as part of his mourning the death of Socrates. But Anderson also describes a subsequent stage in the mourning process as lifting.
In the Iliad, King Agamemnon seriously dishonors King Achilles by taking Briseis away from him. Achilles is understandably enraged. He is ready to dispatch Agamemnon on the spot. But the goddess Athena intervenes and stops Achilles forcibly. She urges him, instead, to give Agamemnon a well-deserved tongue-lashing, which Achilles does.
With this famous example in mind, I can imagine Plato writing the Apology in the full force of rage over the death of Socrates. But I suspect that the full force of Plato's rage had lifted (in Anderson's terminology) by the time he wrote the Republic.
Even though mourning is a prominent theme in Critchley's new book, he does not draw on Anderson's book or any comparable books about the mourning process.