* The Sun and Moon don't appear together in the daytime sky: The moon is only visible during the day if it's near new moon, and even then it's very faint and hard to see.
* The Moon's phases are caused by the Sun: The moon doesn't have its own light; it reflects sunlight. The phases we see (new moon, crescent, full moon, etc.) are caused by the angle at which we see the sunlit portion of the moon.
* During the day, the Sun illuminates the Moon: When the moon is in the daytime sky, the sun is directly illuminating it, making it impossible for us to see a "half-lit" moon.
In summary: It's not possible for the moon to be half-lit when the sun is in the daytime sky because the sun's light will fully illuminate the moon, making it hard to see.
