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Re: Linear notation

Date: Sat, 4 Jan 92 12:09:31 -0500
To: linear@cs.stanford.edu

This is a footnote to the posting Vaughan Pratt made of my message to
him regarding the notation  o  in relevance logic for what has been
variously called intensional conjunction, consistency, cotenability, 
fusion, tensor product in the relevance logic literature.

I did not mention, because my memory is so poor that I had to go to
the library to check it, but C.I. Lewis and C.H. Langford used this
notation in their Symbolic Logic, 1932, The Century Company, 2nd ed.
1959, Dover Publications, Inc., and they called in "consistency".
It was defined p o q = -(p -> -q), where -> is strict (necessary)
implication (and - is of course negation).  Forgetting the nature
of the -> this is the same definition as can be used for the system
R of relevance logic and also for the standard system of linear logic.

I remember discovering this happy coincidence after I had already
started using  o  for it multiplicative mnemonics and the fact it
was on my typewriter.

One last remark.  Vaughan has provided some helpful history about 
de Morgan, Peirce, etc. and their having two kinds of
"conjunction" in relation algebras (intersection and relative product).
I do not mean to do more than nitpick here, because I think that the
work on relation algebras is an important source of ideas for both
reklevance and linear logics, and is often overlooked.  But to put
things in perspective, it is not clear to me that this should be
counted as anything but a precursor, or even an analog, of the use
of  o  in relevance and linear logics. 

For one thing, o  is usually  made commutative in relevance and 
linear logic, and relative product is not commutative.   Also (this
is related), the definition given above of  o  in terms of 
implication and negation seems not to be modeled in relation 
algebras.  But as Vaughan knows, I like relation algebras too,
having noticed that they are very close to being a model
of the relevance logic R back when I wrote my thesis, but I never
could get them, even under some sensible restrictions, to be a 
characteristic model for R, and I think the same problem holds for 
linear logic. 


Mike Dunn
dunn@iuvax.cs.indiana.edu