Here are some questions I have about teaching formal systems. Suppose I'm teaching an introductory logic course and as part of that course I'm teaching a unit on propositional logic. There is a standard notation for the logical operators, representing premises and conclusions, etc.
1) When (if ever) is it appropriate to change the standard notation? Suppose I think that representing something in a non-standard way is more intuitive, simpler, more easily graspable than when the very same thing is represented using the standard notation. Which notation should I teach? Do I teach the standard notation because students are likely to encounter it in lots of places, or do I teach the non-standard notation because I think it's better?
2) Similarly, when (if ever) is it appropriate to introduce wholly new notational systems in the classroom? Is it ever appropriate to do for pedagogical reasons, or should we make modifications or additions in research and only teach them once they become part of the established body of work that constitutes the field of study (by being published in at least one place)? Or not even then, but only once the non-standard notation starts to become assimilated into the standards of the field (at least once multiple people have published using the previously non-standard notation)?
3) Does it make a difference if the subject is classical logic, or something in a field with less agreement? It seems like the prevalence of classical logic would weigh against using a non-standard notation. And even when there is some persistent differences in notation, those are usually mentioned and then ignored. Many instructors will at least mention that the horseshoe and right arrow are both used interchangeably to represent the material conditional, for instance.
It seems to me that there could be some value in exploring alternative notational systems, especially in places where the notation could use updating.