This is a follow-up to the introductory post on Laws of Form *here*.

There are two kinds of negation: contraries and contradictories, and Laws of Form (LoF) represents both types. Furthermore both types apply to terms and propositions.

Contraries are two complete opposites; the negation of one is the other. The poles of a magnet for example are contraries: not South is North and not North is South. This negation is represented by the Law of Crossing.

Contradictories are partial opposites; the negation of one is different from the other but not necessarily the exact opposite. The negation of a proposition is a proposition that is inconsistent with it. This negation is represented by the Law of Calling.

Calling is a kind of negation but calling again doesn’t return to the original proposition; it reiterates the negation and remains in the same place.

If we negate North as a direction, do we get South? Not necessarily; we could get East or West which are different from North. As directions, North and East are contradictories.

It’s unusual to completely negate a proposition but it can be done. “The place is on North Main Street” is contradicted by “No, it’s on East Main Street.” The contrary proposition is “The place is on South Main Street” in the context of a north-south oriented Main Street.

To model both negations in ordinary arithmetic with 0 and 1, use the two operations: standard multiplication for calling and an alternate multiplication for crossing defined as:

x ◊ y := (x-1) * (y-1).

Then zero represents the marked state and one the unmarked state.

The beauty of LoF is that these two kinds of negation are combined into one symbol — as is the word “not”.