# [FOM] About linear logic (was Re:Mismatch)

Alessio Guglielmi Alessio.Guglielmi at inf.tu-dresden.de
Mon Jun 16 08:22:16 EDT 2003

```Hello,

this is a reply to Harvey Friedman's posting on FOM (see
<http://www.cs.nyu.edu/pipermail/fom/2003-June/006833.html>), where
he answers my `mismatch' email and raises some important questions,
especially about linear logic. I'm sending this also to the Proof
Theory list because I believe that there are people there, not on
FOM, who disagree on what I'm saying here, or wish to integrate it.

Let me first make clear my humble opinion on linear logic. Linear
logic, for me, is simply a mathematical object; in particular, it is
an interesting proof theoretical object. I don't believe that linear
logic has a direct relevance for the foundations of mathematics, and,
at least as it stands now, it is not foundational for computer
science. So, for me linear logic is different than intuitionistic
logic, or relevance logic: these logics come with a philosophy, they
aim at being foundational. Linear logic is just an algebraic theory.

On the other hand, I believe that linear logic might have a
beneficial indirect effect on the foundations of mathematics, and
could be further developed to become a foundation for computer
science.

LINEAR LOGIC AND MATHEMATICS

I'd say that proof theory is still seriously underdeveloped, as
witnessed by the fact that we are still not able to answer the
question: `When are two different mathematical proofs essentially the
same?' (it's not even clear how to define being `essentially the
same'); and supposing we know that: `Given two proofs, can we decide
that they are the same?'.

One of the reasons for our inability is that proof theory is
currently suffocated by bureaucracy, meaning that proofs are
cumbersome objects, which leave us very little freedom of
manipulation and analysis.

Linear logic brings into the picture at least two important
novelties: a more refined notion of implication, which leads to a
more refined analysis of proofs, and proof nets, which is a new
syntax with much less bureaucracy than the sequent calculus. The fact
that linear logic deals with `resources' is totally irrelevant in my
opinion.

So, my hope is that linear logic can help proof theory develop in a
way beneficial to classical logic, and so to the foundations of
mathematics. In other words, having to deal with a laboratory-monster
logic is important for proof theory, in order to strengthen the
methods for the normal patient.

This picture is confirmed by recent work in proof theory. What we do
with the Calculus of Structures is an example
<http://www.ki.inf.tu-dresden.de/~guglielm/Research>: properties
first conjectured and proved for linear logic then turned out to be
true (and elegant) also for classical logic. Another example is
research by Fuehrmann and Pym, recently advertised on the PT list
<http://www.cs.bath.ac.uk/~pym/oecm.pdf>, where they attack the
problem of proof equivalence by taking inspiration from linear
logic's research.

In my everyday work, when I have an idea that could be true for
classical logic, I immediately test it also on linear logic, and the
perspective that linear logic offers is almost infallibly very
interesting.

LINEAR LOGIC AND COMPUTER SCIENCE

What could be foundational for computer science? My (very naif!)
philosophy on the subject is the following.

Mathematics has a foundational language: it's the language that
mathematicians speak and write. It's been developed in the millennia
as a specialisation of the human language and benefits from devices
like books and computers for the permanent storage of information. It
relies on the fact that when one says or proves something, this will
never be forgotten, because the information tends to stay in the
brain, and certainly stays in books. In the last century this
language has been studied by logic and we're rather happy. Not
a need for changing the language, because we, human beings, didn't
change.

Human communication is slow, mono-channel, and it relies on a huge
database of shared information. A distributed computer is quite the
opposite: fast multi-channel communication and very little shared
information. After a message is sent, it's also forgotten. So, one
could say, and I would agree, that the logic of the computer must be
different, whatever that is.

Linear logic can be viewed as an attempt at capturing message passing
(or, if you wish, `resources'). Only time will tell whether this will
be successful and useful. I'm a bit skeptical, because I don't see
much hard evidence of this, and also because it looks to me a bit
naif to start from the logic side of the thing.

I'd say that a more promising approach would be first to agree on the
language, or class of languages, one wants to model (I'd bet on
process algebras a la Milner). Then a good idea seems to be trying to
match process algebras and logics *derived and inspired* from linear
logic, but I see too many problems in using linear logic as is. Just
to mention one, there is no clear way of expressing sequential
composition of processes in linear logic. Now, if there is a
fundamental notion in algorithms and recipes, this is: `first beat
the egg, then pour the milk', ...

In conclusion, I'd stress my opinion that in no case linear logic is
to be considered on the same level of classical logic. It's a `logic'
in the sense that it's the study of a language, but this doesn't
mean, for example, that we have to ask for it an intuitive,
philosophically convincing semantics as we do for classical logic. As
for most mathematical *objects*, the models of classical logics are
weird, exotic structures.

* * * * *

Having said all this, I'll briefly answer those of your questions and
remarks I feel confident with:

1   There are complete semantics for linear logic and fragments of
it, including model theoretic semantics. One such semantics is `phase
semantics', which you can find in [1, 2]. The literature is rather
exterminate, I'd leave it to others more knowledgeable than me
pointing you in the right direction. Let me only mention that the
prevailing interest in semantics for linear logic is more in
(denotational) semantics of proofs, not of formulae. In this sense,
model theory is not where most of the research goes.

2   For understanding my posting, it is sufficient to know the syntax
of linear logic in the sequent calculus, which again you can find in
[1], but you might find [2] more concise. This is very elementary:
you just have to read a sequent system. If you want to go deeper, I'd
suggest you try and prove cut elimination for the system. Another
good, quick introduction to some ideas of linear logic is in [3]. In
any case, linear logic in my `mismatch' posting was only supposed to
be an example illustrating a more general point about proof theory.

3   I'd expect FOM subscribers to consider linear logic an important
mathematical idea, not necessarily a foundational one, and not to
expect semantics for it in the same spirit of semantics for classical
logic, because there isn't such a need.

-Alessio

[1] Jean-Yves Girard. Linear logic. Theoretical Computer Science,
50:1-102, 1987.

[2] The Linear Logic Pages: <http://iml.univ-mrs.fr/~lafont/linear/>

[3] Jean-Yves Girard, Yves Lafont and Paul Taylor. Proofs and types.
Cambridge University Press, 1989.
```