## Overview

Defeasible logic, originally created by Donald Nute with a particular concern about efficiency and implementation, is a simple and efficient rule based non-monotonic formalism. Over the year the logic has been developed and extended, and several variants have been proposed.

The main intuition of the logic is to be able to derive “plausible”
conclusions from partial and sometimes conflicting information.
Conclusions are *tentative* conclusions, in the sense that a
conclusion can be withdrawn when we have new pieces of information.

## Language of Defeasible Logic

The knowledge in a *Defeasible Theory* is organised in
*facts* and *rules* and *superiority relation*.
Rules are divided into *strict rules*, *defeasible rules* and
*defeaters*.

- Facts
- Facts are indisputable statements.
- Strict Rules
- Strict rules are rules in the classical sense: whenever the premises are indisputable (e.g., facts) then so is the conclusion.
- Defeasible Rules
- Defeasible rules are rules that can be defeated by contrary evidence.
- Defeaters
- Defeaters are rules that cannot be used to draw any conclusions. Their only use is to prevent some conclusions.
- Superiority Relation
- The superiority relation in a binary relation defined over the set of rules. The superiority relation determines the relative strength of two (conflicting) rules.

## Reasoning

Defeasible Logic is a “skeptical” non-monotonic logic, meaning that it does not support contradictory conclusions. Instead Defeasible Logic seeks to resolve conflicts. In cases where there is some support for concluding A but also support for concluding ¬A, Defeasible Logic does not conclude either of them (thus the name “skeptical”). If the support for A has priority over the support for ¬A then A is concluded.

Conclusions can be classified as *definite* or *defeasible*. A definite conclusion is a conclusion that cannot be withdrawn when new information is available. A defeasible conclusion is a tentative conclusion that might be withdrawn by new pieces of information. In addition the logic is able to tell whether a conclusion is or is not provable. Thus it is possible to have the following 4 types of conclusions:

- Positive definite conclusions: meaning that the conclusion is provable using only facts and strict rules;
- Negative definite conclusions: meaning that it is not possible to prove the conclusion using only facts and strict rules;
- Positive defeasible conclusions: meaning that the conclusions can be defeasible proved;
- Negative defeasible conclusions: meaning that one can show that the conclusion is not even defeasibly provable.

Strict derivations are obtained by forward chaining of strict rules, while a defeasible conclusion A can be derived if there is a rule whose conclusion is A, whose prerequisites (antecedent) have either already been proved or given in the case at hand (i.e., facts), and any stronger rule whose conclusion is ¬A (the negation of A) has prerequisites that fail to be derived. In other words, a conclusion A is (defeasibly) derivable when:

- A is a fact; or
- there is an applicable strict or defeasible rule for A, and either
- all the rules for ¬A are discarded (i.e., not applicable) or
- every applicable rule for ¬A is weaker than an applicable strict or defeasible rule for A.

- Give an argument for the conclusion to be proved
- Consider all possible counter-arguments for the conclusion
- Rebut the counter-arguments
- Show that a counter-argument is not valid (e.g., some of the premises do not hold)
- Defeat a counter-argument by a stronger argument supporting the conclusion

By changing the definitions of “applicable”, “discarded rules”, “arguments” and “counter-arguments” it is possible to capture several different intuitions of non-monotonic reasoning (e.g., ambiguity propagation vs ambiguity blocking).