The Relevance of Relevance to Relevance Logic - Logic and Its Applications - page 24

There are at least three treatments of what is standardly called “De Morgan”
negation in relevance logic, and there is also another negation entirely, classical,
or “Boolean” negation which Meyer and Routley showed could be conservatively
added to the system R. Put quickly:

A iff not x ∗ | (“Routley Star”) x | = ∼ = A

(Four-valued) x | = 1 ∼ A iff x | = 0 A

x | = 0 ∼ A iff x | = 1 A

(Perp) x | = ∼ A iff ∀ a ∈ A, a ↥ x .

Depending on subtleties, the first two were both in my dissertation Dunn
(1966). I discussed various representations of De Morgan lattices (the algebraic
counterpart to first-degree entailments) and showed them all equivalent. The
“Routley Star” was in this context anticipated by Bialynicki and Rasiowa in their
representation of De Morgan lattices (they called them quasi-Boolean algebras).
The four-valued semantics was implicit in another representation of De Morgan
lattices using “topics” but was not made explicit until Dunn (1969) and not
published until Dunn (1976). Two important qualifications though - these did
not address the issue of nested implications. We will discuss this some more in
the next section. The Perp treatment of negation arose also in the representation
of De Morgan lattices, but much later. See Dunn (1993). We shall not discuss it
here. For a discussion of these in a more general setting see Dunn (1999).
The Routley Star should really be called the Routleys' Star, since it was intro-
duced in Routley and Routley (1972) as a semantics for first degree entailments.
They required that it satisfy:

a ∗∗ = a

(Period two)

Routley and Meyer go further and combine it with their ternary accessibility
relation to provide a semantics for R, E, etc. and require that * satisfy in addition
to Period Two the following:

If Rabc then Rac ∗ b ∗ .

(Antilogism)

x ∗ can be understood as the sentences not denied by x , which helps us infor-
mally understand the validity of

( A → B ) → ( ∼ B →∼ A ) (Contraposition),

which is formally determined by Antilogism. An informal understanding of Antil-
ogism in terms of contextual relevance goes something like this. Antilogism says
that if b is relevant to c in the context a ,then c ∗ is relevant to b ∗ in the same
context a , that is, the information not denied by c is relevant to the information
not denied by b .Gofigure!