Fwd: linear and non-linear terms

[Howard Gregory]

Hi Carl,

Many thanks for your message. Just a few comments - I will write again later.

> Good to hear from you, and look forward to reading your papers.
> Did Glyn and Bob argue for no vacuous abstraction in the syntax,
> or in the semantics, or both? Evidently neither of them stuck with it.
> I gather you are advocating relevance logic for the semantics?

They were mainly concerned with coherence / completeness / theta theory, which
I guess is syntax. You're right, I am advocating it primarily for the semantic
meaning language (my original point of departure was Muskens' partialized
Montague semantics, which I was looking at in connection with your and Shalom
Lappin's work on intensionality). But actually I am now looking into using it
in the syntax as well, which would bring it into the territory inhabited by
categorial grammars and glue languages.

> As you're probably aware Curry advocated this too, though I don't know
> whether the term "relevance logic" existed then. My philosopher
> colleague here, Neal Tennant, is another big proponent of it. Has
> anybody tried to base NL syntactic types on relevance logic? It didn't
> occur to me to try this. If your type logic is relevant, what happens
> if you add a Bool type and try to analogize to Church's simple theory
> of types? Where does one go to find about the algebraic and categorical
> models of releance logic?

Glyn and Bob use Curry's terminology rather than relevance logic, as far as I
remember. The original relevance logic literature (Anderson and Belnap 1975
etc) explicitly acknowledges Curry's work as an antecedent, along with similar
work by Moh. I had heard that Neal Tennant was an enthusiast (possibly from
you). I would be interested to know more about his work on it.

I don't know of work on syntactic types using relevance logic, though as I
say, I am trying to do something along those lines. Morrill and Carpenter
mention that Steedman's CCG avoids the use of K, so I guess it is at least
implicit  in quite a lot of work. Unfortunately I am not as well up on
Categorial Grammar literature as I would like to be - it wouldn't surprise me
too much if somebody told me I was trying to re-invent the wheel.

Algebraic models of relevance logic go back to papers by Dunn included in
A&B's original volume - they are a particular type of ordered monoid (now
often known as Dunn algebras), with a closure operation to get modality. I
have written a summary of some of the details, for what it's worth, as part of
an unpublished paper. There is also a very influential Kripke-type frame
semantics due to Routley and Meyer, using a ternary accessibility relation.
This dovetails with the ternary relation frames used for substructural logics
in general, which also have a close relation to Channel Theory (as anticipated
by Barwise 1993). (This is the semantics I actually adopt, largely for that
reason.) I don't know anything very interesting about the category theory I'm
afraid, though I think there are some references in Restall (below).

The main references I use for all this are Dunn's 1986 survey (in Gabbay and
Guenthner's Handbook of Philosophical Logic Vol III), and more
comprehensively, Greg Restall's (2000) Introduction to Substructural Logics.
Plus the two volumes of Anderson and Belnap (1975, 1992).

Best regards,
Howard

___________________________
H. A. O. Gregory
Wissenschaftlicher Mitarbeiter
Georg-August-Universitaet Goettingen
Seminar fuer englische Sprachwissenschaft
Kaete-Hamburger-Weg 3
37073 Goettingen
Deutschland

E-mail: howard.gregory at phil.uni-goettingen.de