The book also includes "Acknowledgments: Why This Book Was Hard to Write and Thanks," pages 283-285; notes, pages 287-300; bibliography, pages 301-305 ; an index, pages 307-322; a note about the author, page 323; and a note about the type, page 324.
Just as Aristotle is part of what Whitehead referred to as footnotes to Plato, so too is Critchley's new book. Indeed, his new book appears to be written by one philosophy professor for "Us [Philosophy Professors]." And perhaps also for graduate students who aspire to become philosophy professors.
But let's consider what Critchley himself says. He says that "when revived" by responsive readers presumably (of the sort that Wadlington tries to develop in his 1987 book Reading Faulknerian Tragedy), "the ancients speak, [and] they do not merely tell us about themselves. They tell us about us. . . . This 'us' is not necessarily existent. It is us, but in some new way, some alien manner. It is us, but not as we have seen ourselves before, [but] turned inside out and upside down" (page 7; I added the bracketed words here).
Let me use C. G. Jung's terminology here. For people in Western culture today, the ancient Greeks, and the ancient Greek tragedies, represent our Western collective unconscious. Now, according to Jung, each adult needs to undertake the task of recognizing and integrating various elements of his or her personal unconscious in healthy and pro-social ways into his or her ego-consciousness. Similarly, each adult also needs to recognize and integrate various elements of our Western collective unconscious in healthy and pro-social ways into his or her ego-consciousness.
The time has now come for me to return to Ong's 1962 book The Barbarian Within: And Other Fugitive Essays and Studies, mentioned above. In it, he reprints his 1954 essay "The Jinnee in the Well-Wrought Urn" (pages 15-25) and his 1958 essay "Voice as Summons for Belief: Literature, Faith, and the Divided Self" (pages 49-67).
For relevant further discussion of Ong's thought, see Thomas D. Zlatic's essay "Faith in Pretext: An Ongian Context for [Melville's Novel] The Confidence-Man" in the anthology Of Ong and Media Ecology, edited by me and Paul A. Soukup (Hampton Press, 2012, pages 241-280).
Later, Critchley says, "Tragedy gives voice to what suffers in us and in others, and how we might become cognizant of that suffering, and work with that suffering, where suffering is that pathos that we undergo, where tragic passion is both something undergone and partially overtaken in action. (I want to emphasize the word 'partially' agency in tragedy is ever partial)" (pages 9-10; his italics).
On the one hand, Critchley, without adverting to Ong's 1954 and 1958 essays as reprinted in his 1962 book, is, in effect, approaching the texts of the Greek tragedies in a way that is consonant with the approach to literary texts that Ong recommends.
On the other hand, Critchley's new 2019 book could not be titled Reading Greek Tragedy as Wadlington's 1987 book is titled Reading Faulknerian Tragedy. But why not? Because Wadlington does not claim to be articulating Faulknerian tragedy's philosophy.
Now, when Critchley started his undergraduate studies in the early 1980s (page 36), the once-fashionable movement of existentialism that Ong refers to in certain essays reprinted in his 1962 book The Barbarian Within: And Other Fugitive Essays and Studies, mentioned about, had waned.
However, in Ong's preface to his book Interfaces of the Word: Studies in the Evolution of Consciousness and Culture (Cornell University Press, 1977, pages 9-13), he says, "At a few points I refer in passing to the work of French and other European structuralists variously psychoanalytic, phenomenological, linguistic, or anthropological in cast such as Jacques Derrida, Michal Foucault, Jacques Lacan, Roland Barthes, and Tzvetan Todorov, not to mention Claude Levi-Strauss and certain cisatlantic critics such as Paul de Man, Geoffrey Hartman, J. Hillis Miller, and Harold Bloom, who are more or less in dialogue with these Europeans. Many readers will doubtless note that the works of these scholars and the present volume share certain themes and perhaps a kind of excitement. In particular, as I am well aware, my treatment of discourse and thought as rooted ineradicably in orality contrasts with Derrida's chirographic and typographic focus in his De la grammatologie  and other works.
"But this book has its own history, traceable through my earlier works and the references embedded in them; it has also, I hope, its own intelligibility. From the time of my studies of Peter Ramus and Ramism, my work has grown into its own kind of phenomenological history of culture and consciousness, so I have often been assured by others, elaborated in terms of noetic operations as these interrelate with primary oral verbalization and later with chirographic and typographic and electronic technologies that reorganize verbalization and thought" (pages 10-11).
But Critchley's project in his new book is to articulate to the best of his abilities Greek tragedy's philosophy and to contextualize that philosophy in the history of Western philosophy starting with Plato and Aristotle. In short, Critchley is not trying simply to get college-educated people who read books to read the Greek tragedies. Rather, he is trying to get college-educated people who read books and who are interested in the history of Western philosophy to understand his articulation of tragedy's philosophy.
In any event, I would draw your attention to chapter 8: "Tragedy as Invention, or the Invention of Tragedy: Twelve Theses" (pages 33-35). Critchley's twelve theses provide an excellent preview/overview of certain key arguments that he develops in the course of his book. However, the sheer multiplicity and diversity of his twelve theses can serve as one reason why I am not going to critique all of them here.
However, I want to offer a nit-picking argument about his wording in his somewhat lengthy summation of thesis 12. The wording involves his characterization of "the dissolution of all the markers of certitude" (page 35). Now, Aristotle is generally credited with inventing the formal study of logic. In syllogistic logic, the conclusion follows from the premises, provided that all of the terms in the premises are operationally defined and used as univocal terms. Then the conclusion expresses a certitude. (In contrast with univocal terms used in Western philosophical discourse, poetry tends to prefer the use of polysemous terms.)
However, apart from syllogistic logic, all terms operational defined and used in philosophical reasoning, such as the reasoning in Plato's various dialogues, do not express certitude, but only probable reasoning claims to the contrary in the dialogues themselves notwithstanding. I know, I know, I am representing Aristotle's footnotes to Plato and Critchley wants to write his own footnotes to Plato.