I had a conversation recently about low-priority prefix operators in infix expression grammars, which left me mildly uncertain about what they ought to mean. So here's a quick straw poll.

Suppose I have an expression grammar in which the multiplication operator * and the addition operator + have their usual relative priority (namely, * binds more tightly, i.e. is evaluated first). Then suppose I – perhaps unwisely – introduce a prefix operator, which I'll call PFX for want of a better name, which has intermediate priority between the two, so that

• PFX a + b behaves like (PFX a) + b, i.e. the PFX is evaluated first
• PFX a * b behaves like PFX (a * b), i.e. the PFX is evaluated second.

That's simple enough (if unusual). But things get weirder when PFX appears on the right of another operator. Specifically, what would you imagine happens to this expression:

a * PFX b + c

in which you can't process the operators in priority order (* then PFX then +) because the PFX necessarily has to happen before the *.