A conjunction is the logical product of two propositions. What of logical quotients?

An arithmetic product is the result of multiplying two numbers. An arithmetic quotient is derived by dividing the dividend by the divisor. Division is the inverse operation of multiplication.

The logical product is the result of conjoining two propositions. In the proof-theoretic approach, we call this “and introduction” or “conjunction introduction”. We also know what the inverse operation is called: conjunction elimination. However, we don’t speak of logical quotients. Why not?

I can think of three reasons why we don’t extend our understanding of the multiplicative structure of conjunction to discuss logical quotients:

1. It’s not actually a viable concept.
2. Tho viable, it’s not interesting, because the way we tend to deal with propositions doesn’t benefit from any thinking analogous to division.
3. Tho viable, and possibly interesting, it just hasn’t come up….