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Commonsense Reasoning and Argumentation 14/15 HC 9 Structured argumentation (2) Henry Prakken March 4, 2015

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2 Overview Argument schemes Preferences Rationality postulates

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3 Domain-specific vs. inference general inference rules d1: Bird Flies s1: Penguin Bird Penguin K R d = { , } R s includes {S | S |- PL and S is finite} Bird Flies K Penguin Bird K Penguin K Flies Bird Penguin Flies Bird Bird Flies Penguin Penguin Bird

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4 Deriving the strict rules from a monotonic logic For any logic L with (monotonic) consequence notion |- L define S p R s iff S is finite and S |- L p

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5 Argument(ation) schemes: general form But also critical questions Premise 1, …, Premise n Therefore (presumably), conclusion

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6 Argument schemes in ASPIC Argument schemes are defeasible inference rules Critical questions are pointers to counterarguments Some point to undermining attacks Some point to rebutting attacks Some point to undercutting attacks

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Perception Critical questions: Are the observer’s senses OK? Are the circumstances such that reliable observation of P is impossible? … P is observed Therefore (presumably), P

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8 Reasoning with default generalisations But defaults can have exceptions And there can be conflicting defaults P If P then normally/usually/typically Q So (presumably), Q - What experts say is usually true - People with political ambitions are usually not objective about security - People with names typical from country C usually have nationality C - People who flea from a crime scene when the police arrives are normally involved in the crime - Chinese people usually don’t like coffee

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9 How are generalisations justified? Scientific research (induction) Experts Commonsense Individual opinions Prejudice? Very reliable Very unreliable

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10 Inducing generalisations Critical questions: Is the size of the sample large enough? was the sample selection biased? Almost all observed P’s were Q’s Therefore (presumably), If P then usually Q In 16 of 17 tests the ballpoint shot with this bow caused this type of eye injury A ballpoint shot with this type of bow will usually cause this type of eye injury

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11 Expert testimony 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

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Supporting and using generalisations V’s injury was caused by a fall This type of eye injury is usually caused by a fall V has this type of injury E says that his type of injury is usually caused by a fall E is an expert on this type of injury Expert testimony scheme Defeasible modus ponens

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13 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 P is usually of the form “I remember that I observed that...”

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Memory Critical questions: Is the memory contaminated with other information? … P is recalled Therefore (presumably), P

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15 Temporal persistence (Forward) Critical questions: Was P known to be false between T1 and T2? … P is true at T1 and T2 > T1 Therefore (presumably), P is still true at T2

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16 Temporal persistence (Backward) Critical questions: Was P known to be false between T1 and T2? … P is true at T1 and T2 < T1 Therefore (presumably), P was already true at T2

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

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18 Arguments from consequences Critical questions: Does A also have bad (good) consequences? Are there other ways to bring about G?... Action A causes G, G is good (bad) Therefore (presumably), A should (not) be done

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19 Example (arguments pro and con an action) We should lower taxes Lower taxes increase productivity Increased productivity is good We should not lower taxes Lower taxes increase inequality Increased inequality is bad

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20 Example (arguments pro alternative actions) We should lower taxes Lower taxes increase productivity Increased productivity is good We should invest in public infrastructure Investing in public infrastructure increases productivity Increased productivity is good

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21 Refinement: promoting or demoting legal/societal values Critical questions: Are there other ways to cause G? Does A also cause something else that promotes or demotes other values?... Action A causes G, G promotes (demotes) legal/societal value V Therefore (presumably), A should (not) be done

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22 Example (arguments pro and con an action) We should save DNA of all citizens Saving DNA of all citizens leads to solving more crimes Solving more crimes promotes security We should not save DNA of all citizens Saving DNA of all citizens makes more private data publicly accessible Making more private data publicly available demotes privacy

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23 Example (arguments pro alternative actions) We should save DNA of all citizens Saving DNA of all citizens leads to solving more crimes Solving more crimes promotes security We should have more police Having more police leads to solving more crimes Solving more crimes promotes security

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Argument schemes about action (generalised) Action A results in C1 … Action A results in Cn We should achieve C1 … We should achieve Cn Therefore, We should do A Action A results in C1 … Action A results in Cn We should avoid C1 … We should avoid Cn Therefore, We should not do A

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25 Argument preference In general its origin is undefined General constraint: A < a B if B is strict-and- firm and A is defeasible or plausible. Could otherwise be defined in terms of partial preorders (on R d ) and ’ (on K p ) Origins of and ’: domain-specific!

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26 Two example argument orderings (Informal: K p = , no strict-and-firm arguments) Weakest link ordering: Compares the weakest defeasible rule of each argument Last-link ordering: Compares the last defeasible rules of each argument

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27 Example R d : r1: p q r2: p r r3: s t R s : q, r ¬t K: p,s

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28 Comparing ordered sets (elitist ordering, weak version) Ordering s on sets in terms of an ordering (or ’) on their elements: If S1 = then not S1 s S2 If S1 ≠ and S2 = then S1 < s S2 Else S1 s S2 if there exists an s1 S1 such that for all s2 S2: s1 s2

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29 Comparing ordered sets (elitist ordering, strict version) Ordering < s on sets in terms of an ordering (or ’) on their elements: If S1 = then not S1 < s S2 If S1 ≠ and S2 = then S1 < s S2 Else S1 < s S2 if there exists an s1 S1 such that for all s2 S2: s1 < s2

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Weakest-link ordering (formal) A < a B if B is strict-and-firm and A is defeasible or plausible. Otherwise: A a B iff If both A and B are strict, then Prem p (A) s Prem p (A2) If both A and B are firm, then DefRules(A) s DefRules(B); else Prem p (A) s Prem p (A2) and DefRules(A) s DefRules(B) 30

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Last-link ordering (formal) A < a B if B is strict-and-firm and A is defeasible or plausible. Otherwise: A a B iff LDR(A) s LDR(B); or A and B are strict and Prem p (A) s Prem p (B) 31

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32 Last link vs. weakest link (1) r1: Born in Scotland Scottish r2: Scottish Likes Whisky r3: Fitness Lover ¬Likes Whisky K n : Born in Scotland, Fitness Lover r1 < r2, r1 < r3, r2 ≈ r3 Likes Whisky Scottish Born in Scotland Likes Whisky Fitness lover r1 r2 r3

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33 Weakest link r1: Born in Scotland Scottish r2: Scottish Likes Whisky r3: Fitness Lover ¬Likes Whisky K n : Born in Scotland, Fitness Lover r1 < r2, r1 < r3, r2 ≈r3 Likes Whisky Scottish Born in Scotland Likes Whisky Fitness lover r1 r2 r3

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34 Last link r1: Born in Scotland Scottish r2: Scottish Likes Whisky r3: Fitness Lover ¬Likes Whisky K n : Born in Scotland, Fitness Lover r1 < r2, r1 < r3, r2 ≈r3 Likes Whisky Scottish Born in Scotland Likes Whisky Fitness lover r1 r2 r3

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35 Last link vs. weakest link (2) r1: Snores Misbehaves r2: Misbehaves May be removed r3: Professor ¬May be removed K n : Snores, Professor r1 < r2, r1 < r3, r2 ≈r3 May be removed Misbehaves Snores May be removed Professor r1 r2 r3

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36 Consistency in ASPIC+ (with symmetric negation) For any S L S is directly consistent iff S does not contain two formulas and – The strict closure Cl(S) of S is S + everything derivable from S with only R s. S is indirectly consistent iff Cl(S) is directly consistent. Parametrised by choice of strict rules

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Rationality postulates (Caminada & Amgoud 2007) Let E be any Dung-extension and Conc(E) = { | = Conc(A) for some A E } An AT satisfies subargument closure iff B E whenever A E and B Sub(A) direct consistency iff Conc(E) is directly consistent strict closure iff Cl(Conc(E)) = Conc(E) indirect consistency iff Conc(E) is indirectly consistent

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38 Violation of direct and indirect consistency in ASPIC+ s1: r ¬q K n = ; K p = {q,r} r <’ q q qq r s1 > B1 A1 B2 B1 A1 B2

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39 Violation of direct and indirect consistency in ASPIC+ s1: r ¬q s2: q ¬r K n = ; K p = {q,r} r <’ q q qq r s1 rr Constraint on a : If A = B then A ≈ a B > B1 A1 B2 A2 s2

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Trans- and contraposition Transposition: If S p R s then S/{s} U {–p} –s R s Contraposition: If S |- p and s S then S/{s} U {– p} |- –s

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41 Rationality postulates for ASPIC+ (whether consistent premises or not) Closure under subarguments always satisfied Strict closure, direct and indirect consistency: without preferences satisfied if R s closed under transposition or AS closed under contraposition; and K n is indirectly consistent with preferences satisfied if in addition is ‘reasonable’ If A is plausible or defeasible and B is strict-and-firm then A < B If A = B then A ≈ B (Complicated condition) Weakest- and last link ordering are reasonable

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