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Semantic communication is a novel communication paradigm that focuses on recognizing and delivering the desired meaning of messages to the destination users. Most existing works in this area focus on delivering explicit semantics, labels or signal features that can be directly identified from the source signals. In this paper, we consider the implicit semantic communication problem in which hidden relations and closely related semantic terms that cannot be recognized from the source signals need to also be delivered to the destination user. We develop a novel adversarial learning-based implicit semantic-aware communication (iSAC) architecture in which the source user, instead of maximizing the total amount of information transmitted to the channel, aims to help the recipient learn an inference rule that can automatically generate implicit semantics based on limited clue information. We prove that by applying iSAC, the destination user can always learn an inference rule that matches the true inference rule of the source messages. Experimental results show that the proposed iSAC can offer up to a 19.69 dB improvement over existing non-inferential communication solutions, in terms of symbol error rate at the destination user.