1987, with Jon Barwise*The Liar*.

The Liar paradox is an old chestnut: consider "This sentence is
false". The naive analysis runs thus: assume it is true, then it
has to be what it says, that is false, a contradiction; so, assume it
is false, then it actually is what it says, that is true, also a
contradiction! How to solve this? The classical mathematical approach
is to ban such self-referential sentences. But that throws the baby
out with the bathwater: there's nothing *paradoxical* about "This
sentence has five words" or even about "This sentence has
one hundred words".

*The Liar* is about an approach to solving the paradox, rather
than simply banning it. The usual way to analyse such cases is to
build a mathematical model in
set theory, then use it to define and analyse the truth values of
constructs modelled this way. In the case of the Liar paradox, this
involves modelling self-referential propositions. But classical (ZF)
set theory is formulated to avoid self-referential sets: the
Axiom
of Foundation is there to prevent it. Sets have members: these
members can themselves be sets, with members of their own. What the
Axiom of Foundation says is that if you follow this membership
relation downwards, you always eventually get to the "bottom":
atomic members that are not sets (or are the empty set, which has no
members, sets or otherwise). This means there are no infinitely
descending chains of membership (it *isn't* "turtles
all the way down"), and there are no circular membership
relations (a set cannot be a member of itself, which is what makes it
hard to model self-reference).

So, when your mathematics isn't up to the job -- use different
mathematics! In this case, the authors use Aczel's brand of *nonwellfounded
set theory* as a basis for building their models (despite what it
might sound like from its name, it is a perfectly well-defined and
consistent mathematical theory). In chapter 3, the authors summarise
this theory in enough detail to understand how it is being used in
their subsequent modelling of the paradox. The approach has a visual
representation, in modelling sets as graphs (of the membership
relation): wellfounded sets must have acyclic graphs; nonwellfounded
sets can have cycles in their graphs. This give a nice intuition for
what's going on, and the explanations have a good mix of English text
and mathematical rigour. It can sometimes be a bit confusing, however.
For example:

p39.
We said that on Aczel's conception a set
is any collection of objects whose hereditary membership relation can
be pictured by a graph. More precisely, a *graph G* is a set of
nodes and directed edges, as usual.

Here "Aczel's conception of a set" refers to these nonwellfounded sets (or hypersets), now defined in terms of graphs. A graph is "a set of nodes". What kind of set are these nodes? Wellfounded? Nonwellfounded (relying on a circular definition)? Does it make a difference?

(A very minor problem with the exposition is due to the example
atomic members chosen. On p40 we get the equation
*a* = {Max, *a*}. I
had a moment's confusion, trying to work out what was being maximised,
before I remembered that the atoms in the example language include the
authors' children's names Max and Claire. Moral of this tale: if you
are a logician, do not name your children after mathematical
functions!)

Now, these new hypersets look strange. In fact, they initially look so counter-intuitive that they must be "wrong". But that's because we have been brought up on wellfounded set theory, with its assumptions now bedded into our intuition. We have "got used to it". But we can get used to nonwellfounded sets, too:

p58.
The introduction of a new sort of
mathematical object has always met with considerable resistance,
including such now mundane objects as zero, the negative numbers, the
irrationals, the imaginary numbers and infinitesimals. We realize that
some set theorists feel a similar reluctance to admit hypersets as
legitimate mathematical objects. While this reluctance is perhaps
understandable, it is also somewhat ironic. After all, many set
theorists prior to Zermelo were working with a conception which
admitted circularity, as is apparent from the formulation of Russell's
paradox. Furthermore, the axiom of foundation has played almost no
role in mathematics outside of set theory itself. We must admit,
though, that we initially shared this reluctance, having been raised
within the Zermelo tradition. But our own experience has convinced us
that those who take the trouble to master the techniques provided by
AFA will quickly feel at home in the universe of hypersets, and find
important and interesting applications.

So, once they have a mathematical toolkit up to the job, the authors
go ahead with a traditional approach: use this set theory to model the
various self-referential sentences, statements and propositions; give
this a semantics or two; analyse the resulting systems. They analyse
the system in two different ways, which they call "Russellian"
and "Austinian". (They emphasise that these are not actually
the approaches that Russell and
Austin advocated, but that they are
in the spirit of their approaches.) The analyses give different
answers. (What, you wanted *the* answer? But why are you
surprised that the answer depends on how you formulate the question?)

Summarising brutally, and inevitably misleadingly, the analyses run as follows.

The Russellian analysis rests on a subtle distinction between denial
and negation. Negation is a "positive" statement: it states
that there are facts of the world that make proposition *p*
false. Denial is a "negative" statement: it denies that
there are facts that make *p* true. And these are not (in this
formulation) the same thing. (p79.the
fact of it being false is not a fact of the world). The
analysis shows that the naive formulation conflates these two.

The Austinian analysis rests on the approach that propositions are
made in the context of *situations*, and can have different
truth values in different situations. The analysis shows that the
naive formulation confuses different situations. It takes the form of
"diagonalisation" argument: assume you know all the facts of
the world, then construct a new fact that is true, but is not in your
original set.

p171.
Paradoxes in any domain are important:
they force us to make explicit assumptions usually left implicit, and
to test those assumptions in limiting cases. What's more, a common
thread runs through the solution of many of the well-known paradoxes,
namely, the uncovering of some hidden parameter, a parameter whose
value shifts during the reasoning that leads to the paradox.

Whenever one encounters an apparent incoherence in the world, a natural thing to look for is some implicit parameter that is changing values.

Whenever one encounters an apparent incoherence in the world, a natural thing to look for is some implicit parameter that is changing values.

p173.
There is a hidden parameter in the
Russellian account which the Austinian diagnosis makes explicit: the
portion of the world that the proposition is about. The Russellian
assumes that this "portion" encompasses the world in its
entirety. ...
the Liar paradox shows that we cannot in general make statements about
the universe of all facts. If we held on to the Russellian view of
propositions, the Liar would then force us to acknowledge an essential
partiality in the world: there are propositions which aren't true, but
whose falsehood somehow lies outside the universe of facts, outside
the "world."

This is all fascinating and informative. As usual, naive analysis breaks down at extreme or "edge" cases. What we have here is a thorough analysis of these edges that does not shirk the problem by simply banning it, but takes it seriously, and applies a mathematical approach that fits the problem, thereby shedding light, and uncovering assumptions.

Despite it being interesting, I didn't initially set out to read this to discover more about the Liar paradox: I read it to find out more about nonwellfounded set theory. That is because a delegate's throwaway remark during a conference made me wonder if it might be useful for thinking about emergent properties. I hadn't previously come across it (despite the fact that ideas like bisimulation, and some hairier branches of computer science, are apparently based on it, or equivalent to it). So I could have stopped reading after chapter 3, but it was too interesting! Anyway, I was heartened to see the pictorial approach, and even more heartened to see that much of my existing intuition would probably be fine:

p42.
While Aczel's is quite a different
conception from Zermelo's, it turns out that all the usual axioms of
ZFC are true under this conception, except, of course, the axiom of
foundation. This means that we can use all the familiar set-theoretic
operations (intersection, union, power set, ordered pairs, and so
forth) without any change whatsoever. Only when the axiom of
foundation enters (as with inductive definitions ...)
do we need to rethink things.

But that comment about induction is interesting. I'm glad I carried on reading, because in chapter 4 we get:

pp62-3.We
have given a coinductive definition ...,
taking it to be the *largest* collection satisfying the various
clauses. ...
note that every object that *can*
be included in the class *is* so included. Instead of working
from the bottom up, asking which objects are *forced into* the
defined class, we work from the top down, asking which objects are
legitimately *excluded*. This feature guarantees that circular
members ...
are not excluded.

The hairs on the back of my neck rose when I read that. It seems to
imply that the whole basis of *reductionism* is *wellfoundedness*.
(It might be true that the axiom of
foundation has played almost no role in mathematics outside of set
theory itself, but set theory has had an *enormous*
impact on the way scientists model the world.) In this view, we start
at the bottom, with the "atoms", and construct things
inductively. Everything not required (constructible this way) is
forbidden. Because wellfoundedness is so strongly entrenched, this
seems like the natural, maybe the only, way to do it. But
nonwellfoundedness *isn't like this*. There are nonwellfounded
things that *just cannot be built this way*: there are things
with no "bottom" or "beginning" (it *can* be
turtles all the way down, or all the way back in time, in this
model!); there are things that are intrinsically circular,
self-referential, chicken and
egg, strange loops even. Now
everything not forbidden is compulsory; now there is so much more room
to have the kind of things we need. But, what would it mean to have,
even to "construct", material things with such
nonwellfounded structure?

So, I'm off to read more about this, to see if it might be a better way to model emergent and self-organising systems than classical wellfounded set theory.