Argumentation Logics Lecture 5: Argumentation with structured arguments (1) argument structure

Argumentation LogicsLecture 5:Argumentation with structured arguments (1) argument structure Henry Prakken Chongqing June 2, 2010

Contents • Structured argumentation: • Arguments • Argument schemes • (Attack and defeat)

Merits of Dung (1995) • Framework for nonmonotonic logics • Comparison and properties • Guidance for development • From intuitions to theoretical notions • But should not be used for practical applications

A B E D C

We should lower taxes We should not lower taxes Lower taxes increase productivity Increased productivity is good Lower taxes increase inequality Increased inequality is bad Increased inequality is good Lower taxes do not increase productivity Prof. P says that … Prof. P is not objective People with political ambitions are not objective USA lowered taxes but productivity decreased Increased inequality stimulates competition Prof. P has political ambitions Competition is good

Steps in argumentation • Construct arguments (from a knowledge base) • Determine which arguments attack each other • Determine which attacking arguments defeat each other (with preferences) • Determine the dialectical status of all arguments (justified, defensible or overruled)

ASPIC Framework for rule-based argumentation • Inspired by John Pollock (1987 - 1995) • Developed by • Gerard Vreeswijk (1993,1997) • Leila Amgoud, Martin Caminada, Henry Prakken, ... (2004 - 2009)

Aspic framework: overview Argument structure: • Trees where • Nodes are wff of a logical language L • Links are applications of inference rules • Rs = Strict rules (1, ..., 1  ); or • Rd= Defeasible rules (1, ..., 1  ) • Reasoning starts from a knowledge base K L • Attack: on conclusion, premise or inference • Defeat: attack + preferences • Dialectical status based on Dung (1995)

Argumentation systems An argumentation system is a tuple AS = (L, -,R,) where: L is a logical language - is a contrariness function from L to 2L R = Rs Rd is a set of strict and defeasible inference rules  is a partial preorder on Rd If   -() then: if   -() then  is a contrary of ; if   -() then  and  are contradictories  = _,  = _  Example: classical negation as a contrariness function: -() = {} if does not start with a negation -() = {, }

Knowledge bases A knowledge base in AS = (L, -,R,= ’) is a pair (K, ’) where K L and ’ is a partial preorder on K/Kn. Here: Kn = (necessary) axioms Kp = ordinary premises Ka = assumptions

Structure of arguments • An argumentA on the basis of (K, ’) in (L, -,R, ) is: •  if K with • Conc(A) =  • Sub(A) = {} • DefRules(A) = • A1, ..., An if there is a strict inference rule Conc(A1), ..., Conc(An)   • Conc(A) =  • Sub(A) = Sub(A1)  ...  Sub(An)  {A} • DefRules(A) = DefRules(A1)  ...  DefRules(An) • A1, ..., An if there is a defeasible inference rule Conc(A1), ..., Conc(An)  • Conc(A) =  • Sub(A) = Sub(A1)  ...  Sub(An)  {A} • DefRules(A) = DefRules(A1)  ...  DefRules(An)  {A1, ..., An}

P Q1 Q2 R1 R2 Q1, Q2  P Q1,R1,R2 K R1, R2  Q2

Kp = { (1) Information I concerns health of person P (2) Person P does not agree with publication of information I (3) i is innformation concerning health of person p i is information concerning private life of person p (4) (i is information concerning private of person p & Person p does not agree with publication of information i) It is forbidden to publish information i } Rs = all valid inference rules of propositional and first-order logic Rd = {,     } Forbidden to publish I ,       Rd -elimination Implicit! (i concerns health of p & p does not agree with publication of p ) Forbidden to publish i I concerns private life of P & P does not agree with publication of I 1,2,3,4 K I concerns private life of P P does not agree with publication of I ,   &   Rs I concerns health of P i concerns health of p i concerns private life of p ,      Rs

Example R: • r1: p  q • r2: p,q  r • r3: s  t • r4: t  ¬r1 • r5: u  v • r6: v,q  ¬t • r7: p,v  ¬s • r8: s  ¬p Kn = {p}, Kp = {s,u}

Types of arguments An argument A is: Strict if DefRules(A) =  Defeasible if not strict Firm if Prem(A)  Kn Plausible if not firm S |-  means there is a strict argument A s.t. Conc(A) =  Prem(A)  S

Domain-specific vs. inference general inference rules R1: Bird  Flies R2: Penguin  Bird Penguin K Rd = {,     } Rs = all deductively valid inference rules Bird  Flies K Penguin  Bird K Penguin K Flies Bird Penguin Flies Bird Bird Flies Penguin  Bird Penguin

Argument(ation) schemes: general form Defeasible inference rules! But also critical questions Negative answers are counterarguments Premise 1, … , Premise n Therefore (presumably), conclusion

Expert testimony(Walton 1996) • Critical questions: • Is E biased? • Is P consistent with what other experts say? • Is P consistent with known evidence? E is expert on D E says that P P is within D Therefore (presumably), P is the case

Witness testimony • Critical questions: • Is W sincere? • Does W’s memory function properly? • Did W’s senses function properly? W says P W was in the position to observe P Therefore (presumably), P

Arguments from consequences • Critical questions: • Does A also have bad consequences? • Are there other ways to bring about G? • ... Action A brings about G, G is good Therefore (presumably), A should be done

Temporal persistence(Forward) • Critical questions: • Was P known to be false between T1 and T2? • Is the gap between T1 and T2 too long? P is true at T1 and T2 > T1 Therefore (presumably), P is still true at T2

Temporal persistence(Backward) • Critical questions: • Was P known to be false between T1 and T2? • Is the gap between T1 and T2 too long? P is true at T1 and T2 < T1 Therefore (presumably), P was already true at T2

X murdered Y dmp Y murdered in house at 4:45 X in 4:45 V murdered in L at T & S was in L at T  S murdered V accrual X in 4:45{X in 4:30} X in 4:45{X in 5:00} forw temp pers backw temp pers X in 4:30 X left 5:00 accrual X in 4:30{W1} X in 4:30{W2} testimony testimony testimony W2: “X in 4:30” W1: “X in 4:30” W3: “X left 5:00”

Argumentation Logics Lecture 5: Argumentation with structured arguments (1) argument structure