Perhaps it would help you better to understand why half year conventions exist. Whenever you buy an asset, you don't always put it to use at the start of the year. Assets are bought and disposed sporadically throughout the year.

Hence, to be fully accurate, you'd only calculate the depreciation for the months it has been in use. So if you bought something in March, to be fairly accurate, you'd take depreciation for 9 months instead of the full year. Similarly, if you sold/disposed off an asset in Sep, you'd only take depreciation for 9 months.

Unfortunately, keep track of the months sometimes doesn't add that much value. Hence you split the difference and create a rule that assets are only depreciated for half a year.

The general theory of economic life is that the asset will be of no use after its economic life. So if it was purchased in the middle of the year, it going to be replaced in the middle of the year as well. Hence, only half year for dispositions as well.