Henry Prakken COMMA 2016 Berlin-Potsdam September 15th, 2016 Rethinking the Rationality Postulates for Argumentation-Based Inference Henry Prakken COMMA 2016 Berlin-Potsdam September 15th, 2016

Rationality postulates (Caminada & Amgoud AIJ 2007) Let E be any acceptable set of arguments and Conc(E) = {| = Conc(A) for some A  E } Then E satisfies: direct consistency iff Conc(E) does not contain two formulas  and ¬ strict closure iff any  deductively implied by Conc(E ) is in Conc(E) indirect consistency iff the strict closure of Conc(E) is directly consistent Assumes a deductive (i.e. strict) consequence notion There is a near universal consensus that systems have to satisfy these three postulates. However, I want to challenge this consensus. 2

The ASPIC+ framework Arguments: DAGs where Nodes are statements in some logical language L containing  Links are applications of inference rules Strict rules  Defeasible rules  Constructed from consistent subsets of a knowledge base K  L Certain premises Kn + uncertain premises Kp Attack: On uncertain premises On defeasible inferences (undercutting) On conclusions of defeasible inferences (rebutting) Defeat: attack + argument ordering Argument evaluation with Dung (1995) An argument is: Fallible if it can be attacked Infallible otherwise My analysis is within ASPIC+ but applies more generally. It applies directly to e.g. CA and ABA, which can be reconstructed as special cases of ASPIC+ The informal arguments apply to structured argumentation in general (e.g. also to DeLp or DL) 3 3

Crucial insight That deduction preserves truth does not imply that deduction preserves justification So that deduction preserves justification should be independently argued But deduction applied to more than one fallible subargument can weaken an argument, Since it can aggregate the amount of fallibility of the subarguments If you want to argue for strict closure you have to give a separate argument why deduction preserves justification 4 4

contrapositive deductive reasoning p & q < (p & q) p q If this is all, then in ASPIC+ (and DeLP, ABA, DL, …) consistency is violated, since arguments cannot be rebutted on a strictly derived conclusion. But if the system allows for contrapositive reasoning in the strict/deductive part, then the argument for –(p & q) can always with all-but-one of the fallible subarguments be combined into an attacker of the remaining fallibel subargument.

Reasonable argument ordering p p & q < (p & q) p q If the system allows for contrapositive reasoning in the strict/deductive part, then the argument for –(p & q) can always with all-but-one of the fallible subarguments be combined into an attacker of the remaining fallible subargument. And if the argument ordering is reasonable, then for at least one fallible subargument the attack succeeds as defeat. However, if deduction can weaken an argument, then the argument ordering may not be reasonable. But then strict closure has to be given up, but then indirect consistency also has to be given up. I will now give an example in which this is OK.

The lottery paradox (Kyburg 1960) Assume: A lottery with 1 million tickets and 1 prize. The probability that some ticket wins is 1 The probability that a given ticket Ti wins is 0.000001. Suppose: a highly probable belief is justified; and what can be deduced from a set of justified beliefs is justified. Then {1,2,3} yields an inconsistent set of justified beliefs The example is probabilistic. However, my argument is entirely general and also applies to non-probabilistic accounts of argument fallibility and argument strength. The choice then is: what to give up? (Any of the three bullets can be given up). All these positions have been defended in the philosophical literature. Giving up two gives up strict closure, giving op three gives up indirect consistency. But giving up one is also problematic. I will illustrate this with Pollock’s modelling of the LP (slightly modified by using CA, to emphasise that defeasible rules are not the cause of the problem).

The lottery paradox in ASPIC+ T1 will win and the other tickets will not win The lottery paradox in ASPIC+ Define:  is justified iff some argument for  is in all S-extensions Kp = {T1,…,T1.000.000} Kn = {X1 xor … xor X1.000.000} (Rs = {S   | S |-PL  and S is finite} Rd = 

Kp = {T1, T2, T3} Kn = {X1 xor X2 xor X3} C1 A2 B A3 A1 Option 1: But then for all i: Ci ≈ Ai So none of {A1,A2,A3} are in all extensions Violates principle that highly probable beliefs are justified C1 T1 T2 X1 xor X2 xor X3 T3 T1 A2 B A3 A1

Excluded by third condition on < Kp = {T1, T2, T3} Kn = {X1 xor X2 xor X3} Excluded by third condition on < Option 2: C1 < A1 But then for all i: Ci < Ai So {A1,A2,A3,B,C1,C2,C3}  E for any extension E Violates direct and indirect consistency C1 T1 Clearly in any sensible account of probabilistic argument strength, A1 must be stronger than C1 (even more clearly if there are a million tickets). T2 X1 xor X2 xor X3 T3 T1 A2 B A3 A1

New rationality postulates Direct consistency should still hold Strict closure and indirect consistency should be restricted to any S  E with at most one fallible argument. 11

Changes in ASPIC+ Allow rebuttal on any strict inference applied to at least two fallible arguments Drop third condition on < Theorem: If strict reasoning contraposes, and for any argument A, Premises(A)  Kn is indirectly consistent and conditions (1) and (2) on < are satisfied Then direct consistency, restricted strict closure and restricted indirect consistency are satisfied Dropping the third condition on < makes sense since strict combinations of at least two fallible arguments aggregate ‘degrees’ of fallibility.

Kp = {T1, T2, T3} Kn = {X1 xor X2 xor X3} C1 A2 B A3 A1 Option 2 again: Ci < Ai Then for all i: Ci < Ai So A1,A2,A3 and B are in extension E, but C1, C2 and C3 are not Violates indirect but not direct consistency Satisfies restricted strict colosure C1 T1 T2 X1 xor X2 xor X3 T3 T1 A2 B A3 A1

Added value of argumentation Deduction is still available in argument construction Applications without attackers are still justified Cannot be undercut applications to a limited number of fallible subarguments can be justifed, depending on the argument ordering Argumentation distinguishes between argument construction and argument evaluation.