Despite their great rivalry, Odysseus recognizes Ajax as a great warrior, wholly deserving of a noble burial. Even one's enemies deserve respect in death, he says. After all, the Greeks allowed Hector's corpse to be buried—eventually—and he was a Trojan.
Odysseus also knows all about the cruel trick that Athena...
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played on Ajax, making him think that the herd of sheep and cattle he slaughtered were the Achaean leaders. Athena is Odysseus's protector among the gods, but Odysseus is under no illusions about her devious, manipulative nature, something she has in common with her fellow immortals. When all's said and done, Odysseus understands that what happened to Ajax could just as easily happen to him. The gods are jealous and capricious and can turn against a mere mortal at the drop of a hat. Odysseus is big enough and smart enough to see the bigger picture. He wouldn't want his own corpse to go unburied and doesn't see why Ajax's should suffer the same fate.
In Sophocles' play Ajax, as in the Homeric epics, Odysseus is an intelligent and devious character, who has complex reasons for everything he does and says. Before Ajax's suicide, Odysseus has been told by Athena that the madness of Ajax was caused by the gods, and she seems to be suggesting that Odysseus should treat him leniently.
The clearest explanation of Odysseus' reasoning occurs in his debate with Agamemnon. He says:
In deference to the gods
don’t be so unyielding you throw Ajax out
without a burial.
The duty of burying corpses, and the notion that an unburied corpse would cause ritual pollution, or "miasma", bringing down the anger of the gods on the entire community, was an important element of Greek religion, as can also be seen in Sophocles' Antigone. This is one major reason why Odysseus wants to bury the corpse.
Next, Odysseus warns Agamemnon that his anger and impulsiveness might lead to bad consequences. An ancient Greek audience would immediately have remembered the opening of the Iliad, where Agamemnon's picking a fight with Achilles nearly caused the Greeks to lose the war, by leading Achilles to withdraw from the fight. Burying Ajax will prevent similar dissension from tearing apart the rather fragile alliance of Greek states in Agamemnon's army.
Also, Ajax was a great warrior who fought on the Greek side and thus deserves the honor of a burial. Finally, Odysseus suggests that burying Ajax will make Agamemnon appear just, benevolent, and statesmanlike.