[AmpsterMan]

I can think of two simple, practical answers to your query:

The expressive power of single value functions is very powerful and the constraint is not necessarily restrictive but may even drive a stronger analysis. (Where does this function have multiple returns? Is it for the whole domain? Etc)

By contrast the expressive power of negation and implication are relatively low given very intuitive and well defined alternatives exist.

Second, there already exist good enough paradigms for dealing with multi-valued functions. Splitting the function up into multiple functions, mapping to an ordered pair, etc.

Defining a function the traditional way is more than just notational convenience. The single value constraint allows for many simplifying assumptions, enough that is worth to pay the cost when dealing with relations that you want to talk about in functional ways