The logic of defeasible argumentation

Bart Verheij

of the Computer Science Department
at the Universidad Nacional del Sur
in Bahia Blanca, Argentina

April - May, 1999

Questions

1. Give an attack relation with n stage extensions, for any natural number n.

2. I hardly spoke of defeat caused by specificity, which is a central ingredient of Simari and Loui's mathematical treatment of defeasible reasoning and of Simari, Chesñevar and García's Defeasible Logic Programming. Give an example of two arguments, one of which is defeated by the other because it is less specific than the other. Draw its reason/conclusion-structure (you will probably want to use subreasons) and give the corresponding defeater in CumulA. Is it of sentence-type, of step-type or of composite-type? Explain why CumulA's defeaters are not the right tool to distinguish specificity defeat from other types of defeat. Give your (motivated) opinion on whether I should have paid more attention to specificity defeat.

3. Give your (motivated) opinion on the accrual of reasons. Should it be included in a model of defeasible argumentation? Is it harmful to include it? (Don't forget to explain what you think the accrual of reasons is.) Include a discussion of Pollock's argument against the accrual of reasons. (You can find it in his book Cognitive carpentry (1995) and in his 1987 paper 'Defeasible reasoning' in Cognitive Science, Vol. 11, pp. 481-518.)

4. Consider arguments as reason/conclusion-structures, formed by the subordination and the coordination of argument steps (no subreasons). Assume that each pair of sentences (y , j ) is assigned a 'connection value' s with 0 £ s £ 1, that represents the strength of the argument step 'y . Therefore j '. Define the strength of arguments as follows:

a. An argument j (where j is a sentence) has strength 1.
b. Let A be an argument with strength s and conclusion y , let j be a sentence and let the pair (y , j ) be assigned the connection value t. Then the strength of the argument {A} d j is equal to the product of s and t.
c. Let {A1} d j , {A2} d j , …, {An} d j be arguments with strengths s1, s2, …, sn. Then the strength of the argument {A1, A2, …, An} d j is equal to the maximum of s1, s2, …, sn.
Consider the following definition of defeat:
An argument is defeated if and only if its strength is < 1/2.
An argument is undefeated if and only if its strength is ³ 1/2.
Show that this definition obeys the two constraints that, if an argument is undefeated, none of its initials is defeated, and that, if an argument is defeated, none of its narrowings is undefeated. Which set of defeaters corresponds to this defeat definition? Are they of sentence-type, step-type, or of composite-type? What is the attack type (self-defeat, simple defeat, groupwise attack, or collective defeat)? Is there accrual of reasons? Is there defeat by sequential weakening?

5. Consider the following six sentences of Reason-Based Logic:
P, Q, Valid(rule(p, q)), Valid(rule(p, Ø q)), Applies(rule(p, q), p, q), Applies(rule(p, Ø q), p, Ø q) Explain why these sentences in Reason-Based Logic are satisfiable, i.e., there is truth-value assignment obeying Reason-Based Logic's semantic constraints. (An informal explanation suffices.) Explain that in the integrated view on rules and principles the two rules are not typical rules (assuming that these sentences are true).

6. The constraints on the evaluation of the dialectical arguments in ArguMed 2.0 (that constrain which statements are justified, which reasons are justifying, and which exceptions are undercutting) do not lead to a unique evaluation for all finite reason/conclusion/exception-structures. As usual, there can be no or several evaluations. Give examples. (Note that ArguMed 2.0 only allows undercutting exceptions as attacks.)

7. Discuss the distinction between brute and reason-based facts. Give your (motivated) opinion on the existence of brute facts.

Contact information

Bart Verheij
Department of Metajuridica
Universiteit Maastricht
P.O. Box 616
6200 MD Maastricht
The Netherlands

+31 43 3883048
b.verheij at ai dot rug dot nl
http://www.ai.rug.nl/~verheij/