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Commonsense Reasoning and Argumentation 14/15
HC 9Structured argumentation (2)
Henry PrakkenMarch 4, 2015
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Overview Argument schemes Preferences Rationality postulates
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Domain-specific vs. inference general inference rules
d1: Bird Flies s1: Penguin Bird Penguin K
Rd = {, } Rs 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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Deriving the strict rules from a monotonic logic
For any logic L with (monotonic) consequence notion |-L define
S p Rs iff S is finite and S |-L
p
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Argument(ation) schemes: general form
But also critical questions
Premise 1, … , Premise nTherefore (presumably), conclusion
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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 observedTherefore (presumably), P
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Reasoning with default generalisations
But defaults can have exceptions And there can be conflicting defaults
PIf P then normally/usually/typically QSo (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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How are generalisations justified?
Scientific research (induction) Experts Commonsense Individual opinions Prejudice?
Very reliable
Very unreliable
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Inducing generalisations
Critical questions: Is the size of the sample large enough? was the sample selection biased?
Almost all observed P’s were Q’sTherefore (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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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 DE says that PP 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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Witness testimony
Critical questions: Is W sincere? Does W’s memory function properly? Did W’s senses function properly?
W says PW was in the position to observe PTherefore (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 recalledTherefore (presumably), P
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Temporal persistence(Forward)
Critical questions: Was P known to be false between T1 and T2? …
P is true at T1 and T2 > T1Therefore (presumably), P isstill true at T2
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Temporal persistence(Backward)
Critical questions: Was P known to be false between T1 and T2? …
P is true at T1 and T2 < T1Therefore (presumably), P was already true at T2
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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 testimony
testimony
forwtemp pers
backwtemp pers
d.m.p.
accrual
V murdered in L at T & S was in L at T
S murdered V
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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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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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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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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 VTherefore (presumably), A should (not) be done
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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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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 CnWe should achieve C1…We should achieve CnTherefore, We should do A
Action A results in C1…Action A results in CnWe should avoid C1…We should avoid CnTherefore, We should not do A
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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 Rd) and ’ (on Kp) Origins of and ’: domain-specific!
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Two example argument orderings
(Informal: Kp = , 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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Example Rd: r1: p q r2: p r r3: s t
Rs: q, r ¬t
K: p,s
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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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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 Premp(A) s
Premp(A2) If both A and B are firm, then DefRules(A) s
DefRules(B); else Premp(A) s Premp(A2) and DefRules(A) s
DefRules(B)
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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 Premp(A) s
Premp(B)
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Last link vs. weakest link (1)
r1: Born in Scotland Scottish r2: Scottish Likes Whisky r3: Fitness Lover ¬Likes Whisky
Kn: 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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Weakest link
r1: Born in Scotland Scottish r2: Scottish Likes Whisky r3: Fitness Lover ¬Likes Whisky
Kn: 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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Last link
r1: Born in Scotland Scottish r2: Scottish Likes Whisky r3: Fitness Lover ¬Likes Whisky
Kn: 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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Last link vs. weakest link (2)
r1: Snores Misbehaves r2: Misbehaves May be removed r3: Professor ¬May be removed
Kn: Snores, Professor r1 < r2, r1 < r3, r2 ≈r3
May be removed
Misbehaves
Snores
May be removed
Professor
r1
r2 r3
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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 Rs.
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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Violation of direct and indirect consistency in
ASPIC+
s1: r ¬q Kn = ; Kp = {q,r} r <’ q
q
q
r
s1>
B1A1
B2
B1A1
B2
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Violation of direct and indirect consistency in
ASPIC+
s1: r ¬q s2: q ¬r Kn = ; Kp = {q,r} r <’ q
q
q
r
s1
r
Constraint on a:If A = B then A ≈ a B
>B1A1
B2A2
A2
s2
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Trans- and contraposition Transposition:
If S p Rs then S/{s} U {–p} –s Rs
Contraposition: If S |- p and s S then S/{s} U {– p}
|- –s
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Rationality postulatesfor ASPIC+ (whether consistent
premises or not) Closure under subarguments always satisfied Strict closure, direct and indirect consistency:
without preferences satisfied if Rs closed under transposition or AS closed under
contraposition; and Kn 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