What kind of deceptions are practiced by Ann and Dmitry's social set in "The Lady with the Pet Dog"?
I assume this question is in reference to the following quote:
"[Dmitri] led a double life--one in public, in the sight of all whom it concerned, full of conventional truth and conventional deception exactly like the lives of his friends and acquaintances, and another which flowed in secret."
Dmitri is commenting on the "lies" of normal life. This is reflective of his situation, and the situations of middle-to-upper class people who have a vested interest in maintaining the lives that they currently possess. Even if they aren't "happy," there's a level of comfort and stability that makes great change (like Dmitri leaving his wife for any of the women he has affairs with) dangerous.
Polite society expects certain outward behavior, even if its practitioners are sinning in secret. Thus, if one wants to maintain or imporove social standing and retain comfort and stability, is imperative to keep up the appearance of following social norms.
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