**Defeasible Reasoning in Topos
Logic**

Michael. Heather, Northumbria University, Newcastle upon Tyne, UK, NE1 8ST, m.heather@unn.ac.uk

B.
Nick Rossiter, Informatics, Northumbria University, nick.rossiter@unn.ac.uk

**Abstract**

The epithet
non-monotonic applied to defeasible reasoning is often understood in the sense
that an addition to a sequence of propositions can negate the consequence.

G ® y G,j ® Øy

It is more like G,j ® y¢ but this meaning does not appear from the
perspective of a 'Boolean World' where
inference A Þ B is defined in set logic by the familiar ØAÚB. It
is as if A ® B and C ´ A Þ ØB because there are parts of C outside of B.

Boolean
implication: defined by A Þ B « ØAÚB

This can lead to
paradoxes arising from the existence of the context C which may be out of the
locality. However the real world does not operate with the logic of an
axiomatised Boolean system but with the constructive logic of intuitionistic
reasoning. Non-monotonic in this sense has the much deeper meaning of
'polytonic'. When a judge reasons in a case the reasoning has to be integrated
with this real-world, that is for example with the laws of physics. It is
perhaps not surprising that legal reasoning has much in common with the new
subject of quantum information processing [2].

Defeasible reasoning
in law requires therefore no more than the standard constructive t‑logic or the logic of a topos [3]. As
this is non-Boolean we need to go beyond Venn diagrams. As can be seen from the
diagram above context C may be significant and we need to use the arrows of
category theory rather than sets. The corresponding intuitionistic inference is
given by the pullback over categories.

Heyting
implication defined by C Ù A £ B

* *«
C £ ( A Þ B)

Where both B and A are
subcategories of an ambient context category C the pullback is:

* *

* *

*
*

Here A Þ B is the largest subcategory of C containing
the limit of A with its context [4]. This pullback generalises the Venn diagram
above in two respects:

·
A is not closed
as in the Boolean version (openness).

·
The interaction
of A with its context C can be anywhere (non-locality) [1].

Taken together these
two mean that terms of logic do not need to be pre-defined and fixed. It is
possible therefore as in natural language to have variable intension as well as
variable extensions. The world as a topos has intuitionistic logic. The
internal logic of the topos is Heyting:

*a *entails ØØ*a * but ØØ*a* may not entail *a*.

De Morgan's law in set
logic gives:

Ø*a *Ù Ø*b *« Ø(*a* Ú *b*)
and Ø*a *Ú Ø*b *« Ø(*a* Ù *b*)

but in topos logic the
same is true for the first entailment and its converse, for the second only Ø*a *Ú Ø*b * ® Ø(*a* Ù *b*)
holds and not its converse [3]. The Heyting term Ø*a *is a pseudo-complement which is very commonly
found in natural language. The word 'non-monotonic' is itself a case in point.

Every Boolean logic is
a Heyting logic but not every Heyting logic is Boolean. This means that Boolean
logic may often work in law but not always as in the case of defeasible
reasoning. This does not resolve the paradox of defeasible logic but avoids
them because they do not exist in full Heyting logic.

[1] Heather, M A,
& Rossiter, B N, Locality, Weak or Strong Anticipation and Quantum
Computing I. Non-locality in Quantum Theory, International Journal Computing
Anticipatory Systems **13** 307-326 (2002).

[2] Heather, M A,
& Rossiter, B N, Locality, Weak or Strong Anticipation and Quantum
Computing. II. Constructivism with Category Theory, International Journal
Computing Anticipatory Systems **13**
327-339 (2002).

[3] Johnstone, P
T, Sketches of an Elephant, A Topos Theory Compendium, Oxford Logic Guides **43**, Clarendon (2002).

[4] Mac Lane, S,
& Moerdijk, I, Sheaves in Geometry and Logic: A First Introduction to Topos
Theory, Springer-Verlag (1992).