# Multiplicative Operations

If composition of maps is analogous to multiplication of numbers, what is te analog of division of numbers? (f.08_concep 60)

## Conjunction

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:

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

### TODO Investigate `[0/2]`

`[ ]`

See gabbay04_leibn_frege for a discussion of a notion of logical division discussed by Boole and Peirce.`[ ]`

Study magmas, division in groups, and quasigroups

### TODO Transcribe

./notes-on-logical-quotient.HEIC

# Bibliography

- [f.08_concep] William Lawvere, Conceptual mathematics : a first introduction to categories, Cambridge University Press (2008).
- [gabbay04_leibn_frege] Dov Gabbay & John Woods, The rise of modern logic from Leibniz to Frege, Elsevier (2004).