Defeasible Reasoning in Topos Logic
Michael. Heather, Northumbria University, Newcastle upon Tyne, UK, NE1 8ST, firstname.lastname@example.org
B. Nick Rossiter, Informatics, Northumbria University, email@example.com
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.
implication: defined by A Þ B « ØAÚB
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 .
Defeasible reasoning in law requires therefore no more than the standard constructive t‑logic or the logic of a topos . 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.
implication defined by C Ù A £ B «
C £ ( A Þ B)
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 . 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) .
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 ØØa but ØØa may not entail a.
De Morgan's law in set logic gives:
Øa Ù Øb « Ø(a Ú b) and Øa Ú Øb « Ø(a Ù b)
but in topos logic the same is true for the first entailment and its converse, for the second only Øa Ú Øb ® Ø(a Ù b) holds and not its converse . The Heyting term Øa 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.
 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).
 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).
 Johnstone, P T, Sketches of an Elephant, A Topos Theory Compendium, Oxford Logic Guides 43, Clarendon (2002).
 Mac Lane, S, & Moerdijk, I, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Springer-Verlag (1992).