In this play, Proctor is a tragic hero. The symbolism of him forgetting the adultery commandment is that it is representative of his tragic flaw. The mistake he made was of course having the affair with Abigail. This mistake keeps him from coming forward to “out” Abigail until Elizabeth is condemned. Later, he confesses to lechery, sexual immorality, and forfeits his good name and reputation in an attempt to show the court Abigail is a liar.
Overall, his omission of the adultery commandment contributes to our understanding of Proctor as a tragic hero with a tragic flaw. Nonetheless, he tries to be good again and dies trying.
The reason why John Proctor forgot one of the commandments can be understood if you think of which commandment it was that he forgot. The commandment that John forgot is the seventh commandment. That is the one that says that people must not commit adultery.
Some months before the play is set, John was having an affair with Abigail Williams, who had been working for Mrs. Proctor as a maid. John feels very guilty about this and we have seen that the affair has caused a great deal of tension in their marriage.
So John forgetting this commandment is sort of symbolic of what is causing his problems.