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What does the crescent mean when it is next to a text message? "Do not disturb" is off.
iPhone 7, iOS 12
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That means that you've turned notifications for that given conversation off. You can re-enable them in the conversation by tapping on the contact name at the top, then tap on 'i' for more info then see the slider for "Hide Alerts". Alternatively, you can swipe to the left on the conversation in the main Messages view and tap "Show Alerts"