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Dunn on relevant o and +
Date: 01 Jan 92 23:14:50 PST (Wed)
To: linear@cs.stanford.edu
I'd hoped to talk to Mike Dunn to confirm my impressions about relevant
disjunction before sending off my message about it, but such does not
always prove compatible with the realtime demands of email conversations.
Anyway Mike's detailed response should not only reliably fill in the
record on the timing of the introduction of o and + in relevance logic
but supply some "relevant" history as well.
-v
------- Forwarded Message
Date: Thu, 2 Jan 92 00:23:05 -0500
From: Mike Dunn <dunn@iuvax.cs.indiana.edu>
To: pratt@cs.stanford.edu
Subject: cotenability
We were away in Chicago when you phoned, and what with the holidays this
is the first chance I have had to get back to you.
The "dot" o (\circ) for cotenability did first appear in my thesis,
"The Algebra of Intensional Logics" (Univ. of Pittsburgh, 1966),
although I called it "consistency." The jargon has shifted over
the years, first from "consistency" to "cotenability" to "fusion"
for various good reasons. I believe that Belnap is responsible for "
the name "cotenability" (used in Entailment, vol 1), and I know that
Bob Meyer is responsible for "fusion."
My choice of "dot" was influenced by the desire for a "multiplicative"
notation, because of Birkhoff, Fuchs, Certaine.
My thesis also seems to be the source of the notation + for the dual.
I was a little surprised at that because Bob Meyer's thesis "Topics in
Modal and Many-valued Logics" (Univ. of Pittsburgh 1966) made some
nice investigations of what Anderson and Belnap had called "intensional
disjunction," but I just looked at Bob's thesis and he used "U" instead
of +. I think that Anderson and Belnap had used no special notation
at all, but just talked of "or". Bob Meyer later called + "fission."
My choice of "+" was undoubtedly motivated by the fact that this notation
had often been used for "or" especially when a multiplicative notation
was used for "and". Also it was on my typewriter (I used lower-case
"o" for "dot").
I saw in the linear logic group that you had posted the thought that
+ has occurred rarely in the relevance logic literature. There is
some kernel of truth there, but you exaggerate. As you can see from
the above it has been there as an explicit formal dual to o
since 1966. And not only did the idea figure in both Bob Meyer's and
my theses, but under the label "intensional disjunction" it was an
important part of the relevance logic motivation in Anderson and Belnap.
The idea was that even though disjunctive syllogism with extensional
disjunction A&(-AvB) -> B is not derivable in relevance logic, that
A&(-A+B) -> B is derivable. This was supposed to explain our intuitive
sense that the argument from A and either not-A or B to B is valid,
the claim being that "or" in English is ambiguous and frequently
expresses intensional disjunction. As I recall you can find this in
the Anderson and Belnap (1962) "Tautological Entailments," Philosophical
Studies, vol. 13, pp. 9-24. It is also in Entailment vol.1, secs.
16.1 and 16.3. It was also in Belnap's 1959 Yale Dissertation "The
Formalization of Entailment."
You will find several references to intensional disjunction in Entailment,
vol. 1, but nonetheless I think you are right that it has not had the
same prominence that o has had. I think the reason is the obvious
connection that o has to implication as its residual (adjoint). The
adjoint ("residual") of + is not logically as natural (unless you
stand on your head and view yourself as trying to build a logic of
refutation rather than assertion).
One last rambling thought. It was Bob Meyer (with Richard Routley) who
freed AoB from having to be thought of as -(A->-B), noticing that
although this equivalence did not hold in the weaker relevance logics,
nonetheless o could be added conservatively relating it to -> by
the residuation law. This was in the (1972) "Algebraic Analysis of
Entailment (I)," Logique et Analyse, vol 15, pp. 407-428. A similar
point can be made about A+B and -A -> B. This is why "fusion" is a better
name for o than cotenatility, and "fission" a better name for + than
intensional disjunction (Anderson and Belnap explicitly construed
intensional disjunction as -A -> B).
Actually, another rambling thought (view this as a p.s.). Back around
1975 or so I noticed the similarity of o to tensor product, and in
fact in 1978 suggested to Fred Szabo that he might apply the ideas
he had in his Algebra of Proofs and combine Cartesian Closed Categories
and Monoidal Closed Categories into a single system with both Cartesian
product and tensor product. This he did (with what success I am not
quite sure) in his (1983) "The Continuous Realizability of Entailment,"
Zeitschrift fuer mathematische Logik und Grundlagen der Mathematik,
vol. 29, pp. 219-233.
Happy new year to you!
- --Mike
P.S. If you think this would be of interest to the linear logic
discussion of notation feel free to repost it, or any excerpts
as you see fit.
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