I am confused how one gets the anti-chain preserving negation in Belnap-Dunn's FOUR, which by all accounts underlies their R (which adds implication), when R's valuation model is constructed from a De Morgan monoid... I've read that one eventually drops the monoid identify to get R and not Rt. FOUR's lattice with its De Morgan negation is the one on the left below:
R--the "standard" relevance logic--is not simply that lattice but enriched with an implication table:
→ n f t b n b b b b f n t n b t n f t b b n n n b
As for the def of De Morgan monoid, this can be given in various equivalent ways, but the least confusing formulation, if you want start with the implication (as opposed to negation) as a "basic primitive" is perhaps:
A De Morgan monoid is an algebra A = (A, ∨, ∧, ·, →, 1, 0) such that:
- (A, ∨, ∧) is a distributive lattice,
- (A, ·, 1) is a commutative monoid,
- the equivalence a · b ≤ c iff b ≤ a → c holds for all a, b, c ∈ A,
- the following identities hold in A:
- (4.1) a ≤ a · a
- (4.2) (a → 0) → 0 = a
As usual, we define ∼x = x → 0.
Note that '·' is called "fusion" and is distinct from the lattice join ∧ -- otherwise one gets a Boolean algebra from the above def. The implication connective/operator → is the residuation of '·' fusion (by def, above).
Alternatively, one can take negation as the "built in" primitive, and define → as derived from negation and fusion
An IRL is said to be square-increasing if it satisfies x ≤ x2. [...] A De Morgan monoid is a distributive square-increasing IRL.
In this latter construction/def, which is more easy to picture as a lattice labelled with the monoid elements as they result (via fusion and negation) from some generators (i.e. a kind of Cayley-graph-like labelling for the Hasse diagram, to account for fusion's role), what is the De Morgan monoid that one starts with to get R (with the implication table given above) though? I thought it might be what's sometimes called D4
...and (confusingly perhaps) also called C3 by others.
But I'm not seeing how one gets from that the usual properties of FOUR that neither and both are preserved rather than swapped by negation... because in D4 the anti-chain elements are actually swapped by negation! (It's by definition of a De Morgan monoid that f = ¬e):
we define x → y := ¬(x · ¬y) and f := ¬e,
And in the other paper I found on this (Slanley...) C5, which has 6 elements, seems to have bug/typo in its negation table... so I'm not sure if it's swapping or not its antichain...
So what am I missing, i.e. how one does actually construct R from a De Morgan monoid? Do you start with a 6-element one?