Why do we use 's when not abbreviating is/was?

Solution 1:

This is called the possessive case or the genitive case of a noun. (Possessive is the more common term, but it implies ownership; genitive is more general and implies a relationship.)

In Old English, there were many ways to put a noun into the genitive case, depending on count and gender. But after the 1500s, a noun could only be put into the genitive case with an -es ending. Eventually the e was dropped, leaving us with 's.

Sometime after, the plural genitive was reintroduced with the following convention: Form a plural genitive by adding just an apostrophe only if the normal plural noun ends in s; in all other cases, add -'s as normal. So beavers' and carpenters', but children's and people's. Sometimes you'll also see the convention applied to singular nouns ending with s (e.g., Jesus' ), but this is often discouraged.

Why my instead of I's? My is a possessive adjective and doesn't follow the normal rules.

Solution 2:

The apostrophe came into English in the sixteenth century as a device for printers to show that a letter had been omitted. It was not until the eighteenth century that its use to indicate possession became widespread. Before then the possessive s could appear at the end of words without the apostrophe with no indication of whether the words were singular or plural. It’s s because when English was a fully inflected language, s was one of the endings, but not the only one, that indicated the genitive case. The present distinction between ’s and s’ is simply a convention to show singular and plural. It is likely to persist for some time, but there signs that the use of the possessive apostrophe in some cases is diminishing.