Use '^' for 'but'.
When you say 'but' you add something to your sentence; you add another clause with additional information. It's conjunction.
Using '^' loses the natural language connotation of 'but' as something that somehow qualifies or mitigates the previous clause, but using '^' still does capture the essential truth functional role of 'but'.
In this example:
There is a hurricaine but it is not raining
might be thought to have this sense:
There is a hurricaine, and you'd usually you'd think that with a hurricaine it would be raining, but it is not raining.
But in basic symbolic logic, we usually are willing to lose the "you'd usually think that with a hurricaine it would be raining" part in order to get to the more essential assertions: There is a hurricaine, and it is not raining.