"All My Sons" falls into the category for Classical Greek Tragedy because the tragic hero, Joe Keller suffers from hubris, or the sin of too much pride, and because of this sin, he makes a tragic mistake which ends up causing his death. Joe's behavior is responsible for his tragic circumstances, it is not by accident that he ends up committing suicide.
There is a relationship between Joe Keller's fate and his free will, he chose to send the faulty parts to the military, thereby putting his fate or destiny in jeopardy by making such a tragic decision. And, in Classical Greek Tragedy, the tragic hero's actual strength becomes the means of his undoing.
There are four specific elements that must be present:
"1. Shame, 2. Suffering, 3. Knowledge,
4. Affirmation of what is right." (See link below for more details)
Or in Joe Keller's case, his ability to support his family, his ability to build his business, work his way up, this very aspect of his life, what he said drove him to make the decision to send the faulty parts, because he was afraid to not meet the contracts deadline, is what leads to his ultimate end.
One important difference between Classical Greek Tragedy which usually involves a tragic hero who is of noble birth and Modern Tragedy is that the latter involves ordinary people in tragic circumstances.
Also, according to Modern Tragedy, once the tragic hero realizes that he has made a terrible mistake, there is no undoing it, you can't take it back, as much as he may want to retrieve the mistake, it is done. Ultimately in the Modern Tragedy, the protagonist takes responsibility for his actions and ends up suffering or dying. And, those around him suffer permanent emotional damage, thereby changing their lives forever.
There are several obvious ways that this play is linked to Classical tragedy. One is the present of the three "unities": time, place, and action. All of the action of the play is about one thing, takes place in one back yard, and occupies approximately a 24 hour period.
If we use Oedipus as an example, the determining influence of fate/the past are key elements in each play. Although Joe seems to have escaped his past, he has not; it has just gone underground, waiting to be discovered.
It is discovered in much the same was as the truth is in Oedipus: the relentless questioning about the past which brings the son (Oedipus viewed as his father's son, not as a ruler) to a horrible knowledge(Oedipus about himself and Chris about his Father). In both cases, through different means, both sons "kill" their fathers, even though some readers may not be as willing to blame Chris as my reading. In both plays, the pursuit of the facts/truth does not lead to the benefits we normally associate with that pursuit.
"All My Sons" also deals with family issues, much as Oedipus and other Classical Greek plays.