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There are a number of interesting axiom sets for R and E, as well as for
many lesser known relevance logics. The reader who want to learn more “axiom
chopping” is referred to Dunn (1986), as well as of course, Anderson and Belnap
(1975). It is worth pointing out that a kind of necessity can be defined in E
as
A
, and it has roughly the structure of the Lewis modal
system S4. This gives us a way to obtain an axiom set for R from that of E by
simply adding
A
=(
A
→
A
)
→
A
→
A
Demodaliser
It is interesting that one obtains the classical propositional paradox (with
A
→
B
provably equivalent to the material implication
A
Ↄ
B
) if one adds to
either R or E
A
→
(
B
→
A
) Positive Paradox.
We introduced the axioms for E because of E's historical importance, and
because Meyer rails against it, but in the sequel we will focus on the system R.
4 The Semantics of Relevance Logic
Any of us, not just Anderson and Belnap, can lay down a set of axioms, or maybe
even create a natural deduction system with nice rules. But what do they mean?
From the beginnings of relevance logic there has been much controversy about
its semantics. First there was complaint that it had none, and then there was
complaint that it had one, particularly the so-called “Routley-Meyer semantics”
for relevance logic - which used the novelty of a ternary accessibility relation.
We shall speak of the Routley-Meyer semantics even though a number of other
logicians produced similar semantics at about the same time.
Copeland, and van Benthem in his review of Copeland, raised questions about
whether this is a semantics in name only, or merely just a formal device. Similar
issues had of course already been raised in connection with the binary acces-
sibility relation in the so-called “Kripke semantics” for modal logic. But there
have actually been various interpretations made of both the binary and ternary
accessibility relations. The best recent place to read about interpretations of the
ternary accessibility relation is Beall et al. (2012).
We first must explain a little about relevance logic and the Routley-Meyer
semantics in particular. Routley and Meyer published “Semantics of Entailment
I, II, III” in the years 1972 and 1973.
2
Routley and Meyer use a frame (
K,R,∗,
0).
K
is a set, 0
∈ K
,
R ↆ K
3
,and
∗
is a unary operation on
K
. Routley and Meyer
call the members of
K
“set ups,” and put various constraints on a frame, but
we shall not explore these in detail now. We do though note that they defined
2
And “Semantics of Entailment IV” written in 1972 but published as an appendix
in Routley, Meyer, Plumwood, and Brady (1982). As with the “Kripke semantics,”
there were a lot of “competitors” in the early 1970's with essentially the same, or
very similar ideas, including (in alphabetical order) Fine, Gabbay, Maksimova, and
Urquhart.
