Tackling Defeasible Reasoning in Bochum: the Research Group for - - PowerPoint PPT Presentation

Tackling Defeasible Reasoning in Bochum:

the Research Group for Non-Monotonic Logic and Formal Argumentation

Christian Straßer and Dunja Šešelja April 10, 2017

Outline

The NMLFA Reasoning by Cases Unrestricted Rebut Comparative Studies Sequent-based argumentation (with Ofer Arieli, Tel Aviv) Agent-Based Models

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The NMLFA

The Research Group for Non-Monotonic Logic and Formal Argu- mentation (NMLFA)

• funding: 2015–2019 (Alexander von Humboldt-Foundation)
• aim: study defeasible reasoning with methods of formal

argumentation

• location: Institute for Philosophy II, Ruhr-University

Bochum

• online:
• http://homepage.ruhr-uni-bochum.de/

defeasible-reasoning/index.html

• mailto:defeasible-reasoning@rub.de

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Members

• AnneMarie Borg (PhD candidate)
• Jesse Heyninck (PhD candidate)
• Pere Pardo (PostDoc researcher)
• Christian Straßer (Principal researcher)
• Mathieu Beirlaen (Associated PostDoc researcher)
• Dunja Šešelja (Associated PostDoc researcher)

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Members

• AnneMarie Borg (PhD candidate)
• Jesse Heyninck (PhD candidate)
• Pere Pardo (PostDoc researcher)
• Christian Straßer (Principal researcher)
• Mathieu Beirlaen (Associated PostDoc researcher)
• Dunja Šešelja (Associated PostDoc researcher)

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Members

• AnneMarie Borg (PhD candidate)
• Jesse Heyninck (PhD candidate)
• Pere Pardo (PostDoc researcher)
• Christian Straßer (Principal researcher)
• Mathieu Beirlaen (Associated PostDoc researcher)
• Dunja Šešelja (Associated PostDoc researcher)

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Members

• AnneMarie Borg (PhD candidate)
• Jesse Heyninck (PhD candidate)
• Pere Pardo (PostDoc researcher)
• Christian Straßer (Principal researcher)
• Mathieu Beirlaen (Associated PostDoc researcher)
• Dunja Šešelja (Associated PostDoc researcher)

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Members

• AnneMarie Borg (PhD candidate)
• Jesse Heyninck (PhD candidate)
• Pere Pardo (PostDoc researcher)
• Christian Straßer (Principal researcher)
• Mathieu Beirlaen (Associated PostDoc researcher)
• Dunja Šešelja (Associated PostDoc researcher)

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Members

• AnneMarie Borg (PhD candidate)
• Jesse Heyninck (PhD candidate)
• Pere Pardo (PostDoc researcher)
• Christian Straßer (Principal researcher)
• Mathieu Beirlaen (Associated PostDoc researcher)
• Dunja Šešelja (Associated PostDoc researcher)

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Some research themes

• 1. Extending the expressive power of structured

argumentation

1.1 reasoning by cases and hypothetical reasoning 1.2 expressing doubt – non-greedy argumentative reasoning 1.3 unrestricted rebut

• 2. Comparative studies of different nonmonotonic

formalisms with special attention to argumentation formalisms (ASPIC, ABA, etc.)

• 3. Applications of argumentation theory to deontic logic
• 4. Sequent-based argumentation (with Ofer Arieli, Tel Aviv)
• 5. Agent-based models based on techniques from abstract

argumentation

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Reasoning by Cases

Mathieu Beirlaen, Jesse Heyninck, and Christian Straßer, Reasoning by Cases in Structured Argumentation forthcoming in Proceedings KRR/SAC 2017, ACM Digital Library (2017)

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Reasoning by Cases, Defeasibly

• strict rules (“→”) vs. defeasible rules (“⇒”)
• schematically:

A B A C B C C

• or, more generally:

A B A C B C C

• or, more generally:

A B A C B C C Read A C: “C follows defeasibly from A” or “There is a (defeasible) argument for C based on A.”

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Reasoning by Cases, Defeasibly

• strict rules (“→”) vs. defeasible rules (“⇒”)
• schematically:

A ∨ B A ⇒ C B ⇒ C C

• or, more generally:

A B A C B C C

• or, more generally:

A B A C B C C Read A C: “C follows defeasibly from A” or “There is a (defeasible) argument for C based on A.”

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Reasoning by Cases, Defeasibly

• strict rules (“→”) vs. defeasible rules (“⇒”)
• schematically:

A ∨ B A ⇒ C B ⇒ C C

• or, more generally:

A ∨ B A ⇒ · · · ⇒ C B ⇒ · · · ⇒ C C

• or, more generally:

A B A C B C C Read A C: “C follows defeasibly from A” or “There is a (defeasible) argument for C based on A.”

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Reasoning by Cases, Defeasibly

• strict rules (“→”) vs. defeasible rules (“⇒”)
• schematically:

A ∨ B A ⇒ C B ⇒ C C

• or, more generally:

A ∨ B A ⇒ · · · ⇒ C B ⇒ · · · ⇒ C C

• or, more generally:

A ∨ B A | ∼ C B | ∼ C C Read A | ∼ C: “C follows defeasibly from A” or “There is a (defeasible) argument for C based on A.”

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Are there good formal accounts of defeasible Reasoning by Cases?

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The Meta-Rule Approach: OR

• Rules for rules:

A ⇒ C B ⇒ C A ∨ B ⇒ C [OR]

• Illustration:
• 1. A

C PREM

• 2. B

C PREM

• 3. A

B PREM

• 4. A

B C 1,2; OR

• 5. C

3,4; DefeasibleMP

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The Meta-Rule Approach: OR

• Rules for rules:

A ⇒ C B ⇒ C A ∨ B ⇒ C [OR]

• Illustration:
• 1. A

C PREM

• 2. B

C PREM

• 3. A

B PREM

• 4. A

B C 1,2; OR

• 5. C

3,4; DefeasibleMP

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The Meta-Rule Approach: OR

• Rules for rules:

A ⇒ C B ⇒ C A ∨ B ⇒ C [OR]

• Illustration:
• 1. A ⇒ C

PREM

• 2. B

C PREM

• 3. A

B PREM

• 4. A

B C 1,2; OR

• 5. C

3,4; DefeasibleMP

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The Meta-Rule Approach: OR

• Rules for rules:

A ⇒ C B ⇒ C A ∨ B ⇒ C [OR]

• Illustration:
• 1. A ⇒ C

PREM

• 2. B ⇒ C

PREM

• 3. A

B PREM

• 4. A

B C 1,2; OR

• 5. C

3,4; DefeasibleMP

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The Meta-Rule Approach: OR

• Rules for rules:

A ⇒ C B ⇒ C A ∨ B ⇒ C [OR]

• Illustration:
• 1. A ⇒ C

PREM

• 2. B ⇒ C

PREM

• 3. A ∨ B

PREM

• 4. A

B C 1,2; OR

• 5. C

3,4; DefeasibleMP

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The Meta-Rule Approach: OR

• Rules for rules:

A ⇒ C B ⇒ C A ∨ B ⇒ C [OR]

• Illustration:
• 1. A ⇒ C

PREM

• 2. B ⇒ C

PREM

• 3. A ∨ B

PREM

• 4. A ∨ B ⇒ C

1,2; OR

• 5. C

3,4; DefeasibleMP

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The Meta-Rule Approach: OR

• Rules for rules:

A ⇒ C B ⇒ C A ∨ B ⇒ C [OR]

• Illustration:
• 1. A ⇒ C

PREM

• 2. B ⇒ C

PREM

• 3. A ∨ B

PREM

• 4. A ∨ B ⇒ C

1,2; OR

• 5. C

3,4; DefeasibleMP

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A Problematic Example for OR

Suppose we have Σ = {p ⇒ q ∨ r, q ⇒ s, s ⇒ v, r ⇒ u, u ⇒ v, p}. q s v p q ∨ r v r u v

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A Problematic Example for OR

q s v p q ∨ r v r u v

• by (OR): from s

v and u v

• by (Right-Weakening), from q

s and r u

• by (OR): from q

s u and r s u

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A Problematic Example for OR

q s v p q ∨ r s ∨ u v r u v

• by (OR): from s ⇒ v and u ⇒ v
• by (Right-Weakening), from q

s and r u

• by (OR): from q

s u and r s u

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A Problematic Example for OR

q s v p q ∨ r s ∨ u v r u v

• by (OR): from s ⇒ v and u ⇒ v
• by (Right-Weakening), from q ⇒ s and r ⇒ u
• by (OR): from q

s u and r s u

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A Problematic Example for OR

q s v p q ∨ r s ∨ u v r u v

• by (OR): from s ⇒ v and u ⇒ v
• by (Right-Weakening), from q ⇒ s and r ⇒ u
• by (OR): from q ⇒ s ∨ u and r ⇒ s ∨ u

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A Problematic Example for OR

t ¬s q s v p q ∨ r s ∨ u v r u v !

• Suppose now we also have t and t ⇒ ¬s.
• the possible defeater has no effect on the generalized

path

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A Problematic Example for OR

t ¬s q s v p q ∨ r s ∨ u v t′ ¬r r u v ! !

• Suppose now we also have t′ and t′ ⇒ ¬r.
• the additional possible defeater has no effect on the

generalized path

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Extension-based Approaches: Default Logic (Reiter)

• Input: set of defaults and a set of formulas (“facts”)
• Build extensions by applying Modus Ponens to defaults

while maintaining consistency

• For instance:

Republican Pacifist Nixon Quaker Pacifist

is a is a

• Extensions:

1. Nixon Republican Quaker Pacifist 2. Nixon Republican Quaker Pacifist

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Extension-based Approaches: Default Logic (Reiter)

• Input: set of defaults and a set of formulas (“facts”)
• Build extensions by applying Modus Ponens to defaults

while maintaining consistency

• For instance:

Republican ¬Pacifist Nixon Quaker Pacifist

is a is a

• Extensions:

1. Nixon Republican Quaker Pacifist 2. Nixon Republican Quaker Pacifist

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Extension-based Approaches: Default Logic (Reiter)

• Input: set of defaults and a set of formulas (“facts”)
• Build extensions by applying Modus Ponens to defaults

while maintaining consistency

• For instance:

Republican ¬Pacifist Nixon Quaker Pacifist

is a is a

• Extensions:
• 1. {Nixon, Republican, Quaker, ¬Pacifist}
• 2. {Nixon, Republican, Quaker, Pacifist}

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Extension-based Approaches: Default Logic (Reiter)

• no handling of disjunctive facts “out-of-the-box”
• for instance: Σ = {Republican ∨ Democrat, Republican ⇒

political, Democrat ⇒ political}. Republican Republican ∨ Democrat political Democrat

? ?

• since the default is not triggered by the fact, MP cannot be

applied

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Extension-based Approaches: Default Logic (Reiter)

• idea: split the factual part of the knowledge base

(Gelfond, Lifschitz, Przymusinska, 1991) Republican Republican Republican ∨ Democrat political Democrat Democrat

Base 1 Base 2

• two extensions:
• 1. Republican, political
• 2. Democrat, political

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Extension-based Approaches: Default Logic (Reiter)

• idea: split the factual part of the knowledge base

(Gelfond, Lifschitz, Przymusinska, 1991) Republican Republican Republican ∨ Democrat political Democrat Democrat

Base 1 Base 2

• two extensions:
• 1. Republican, political
• 2. Democrat, political

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Problematic Example for Disjunctive Defaults

Consider the following example:

• 1. Either his left hand or his right hand is broken. lhb ∨ rhb
• 2. If somebody writes legibly then usually the right hand is

not broken. wl rhb

• 3. He writes legibly.

wl With disjunctive default logic we get two extensions:

• 1. wl,

rhb, lhb

• 2. wl, rhb

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Problematic Example for Disjunctive Defaults

Consider the following example:

• 1. Either his left hand or his right hand is broken. lhb ∨ rhb
• 2. If somebody writes legibly then usually the right hand is

not broken. wl ⇒ ¬rhb

• 3. He writes legibly.

wl With disjunctive default logic we get two extensions:

• 1. wl,

rhb, lhb

• 2. wl, rhb

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Problematic Example for Disjunctive Defaults

Consider the following example:

• 1. Either his left hand or his right hand is broken. lhb ∨ rhb
• 2. If somebody writes legibly then usually the right hand is

not broken. wl ⇒ ¬rhb

• 3. He writes legibly.

wl With disjunctive default logic we get two extensions:

• 1. wl,

rhb, lhb

• 2. wl, rhb

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Problematic Example for Disjunctive Defaults

Consider the following example:

• 1. Either his left hand or his right hand is broken. lhb ∨ rhb
• 2. If somebody writes legibly then usually the right hand is

not broken. wl ⇒ ¬rhb

• 3. He writes legibly.

wl With disjunctive default logic we get two extensions:

• 1. wl, ¬rhb, lhb
• 2. wl, rhb

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General Stratagem

So far: Manipulate the database!

• 1. produce new defeasible rules from the given ones
• 2. produce new factual knowledge bases when confronted

with disjunctive information

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Enters: the Argumentative Approach

• Instead of manipulating the knowledge base and

reasoning on top of the manipulated database,

• we will, in what follows, use a more direct approach to the

modeling of Reasoning by Cases in the context of defeasible reasoning, following the inference scheme: A ∨ B A ⇒ · · · ⇒ C B ⇒ · · · ⇒ C C

• r, more generally:

A ∨ B A | ∼ C B | ∼ C C

• This will allow us to have more control over defeating

conditions …

• … and to avoid pitfalls as the ones demonstrated above.

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A New Type of Argument: RbC-Arguments

Basic idea: Given

• an argument a1 ∈ Arg(T) for which Conc(a1) = A ∨ B,
• an argument a2

Arg A with Conc a2 C, and

• an argument a3

Arg B with Conc a3 C, we introduce a new RbC-Argument a1 a2 a3 C .

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A New Type of Argument: RbC-Arguments

Basic idea: Given

• an argument a1 ∈ Arg(T) for which Conc(a1) = A ∨ B,
• an argument a2 ∈ Arg(⟨D, K ∪ {A}⟩) with Conc(a2) = C, and
• an argument a3

Arg B with Conc a3 C, we introduce a new RbC-Argument a1 a2 a3 C .

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A New Type of Argument: RbC-Arguments

Basic idea: Given

• an argument a1 ∈ Arg(T) for which Conc(a1) = A ∨ B,
• an argument a2 ∈ Arg(⟨D, K ∪ {A}⟩) with Conc(a2) = C, and
• an argument a3 ∈ Arg(⟨D, K ∪ {B}⟩) with Conc(a3) = C,

we introduce a new RbC-Argument a1 a2 a3 C .

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A New Type of Argument: RbC-Arguments

Basic idea: Given

• an argument a1 ∈ Arg(T) for which Conc(a1) = A ∨ B,
• an argument a2 ∈ Arg(⟨D, K ∪ {A}⟩) with Conc(a2) = C, and
• an argument a3 ∈ Arg(⟨D, K ∪ {B}⟩) with Conc(a3) = C,

we introduce a new RbC-Argument ⟨a1, [a2], [a3] C⟩ .

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More general: RbC-Argument

Definition Where

• a0 ∈ Arg(T) with Conc(a0) = ∨n

i=1 Ai and

• ai ∈ Arg(⟨D, K ∪ {Ai}⟩) \ Arg(T) (1 ≤ i ≤ n),

⟨a0, [a1], . . . , [an] ∨n

i=1 Conc(Ai)⟩ is an RbC-argument.

• We say that a1

an are hypothetical sub-arguments of a, in signs: a1 an HSub a .

• For each ai, Hyp ai

Ai.

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More general: RbC-Argument

Definition Where

• a0 ∈ Arg(T) with Conc(a0) = ∨n

i=1 Ai and

• ai ∈ Arg(⟨D, K ∪ {Ai}⟩) \ Arg(T) (1 ≤ i ≤ n),

⟨a0, [a1], . . . , [an] ∨n

i=1 Conc(Ai)⟩ is an RbC-argument.

• We say that a1, . . . , an are hypothetical sub-arguments of

a, in signs: a1, . . . , an ∈ HSub(a).

• For each ai, Hyp ai

Ai.

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More general: RbC-Argument

Definition Where

• a0 ∈ Arg(T) with Conc(a0) = ∨n

i=1 Ai and

• ai ∈ Arg(⟨D, K ∪ {Ai}⟩) \ Arg(T) (1 ≤ i ≤ n),

⟨a0, [a1], . . . , [an] ∨n

i=1 Conc(Ai)⟩ is an RbC-argument.

• We say that a1, . . . , an are hypothetical sub-arguments of

a, in signs: a1, . . . , an ∈ HSub(a).

• For each ai, Hyp(ai) = Ai.

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Let T = ⟨D, K⟩ consist of D = {p ⇒ q ∨ r, q ⇒ s, s ⇒ v, r ⇒ u, u ⇒ v, t ⇒ ¬s} and K = {p, t}. We have for instance the arguments:

• a1

p q r Arg T

• a2

q s v Arg q

• a3

r u v Arg r

• a4

a1 a2 a3 v Arg T . a1: a2: a3: q s v p q r v r u v

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Let T = ⟨D, K⟩ consist of D = {p ⇒ q ∨ r, q ⇒ s, s ⇒ v, r ⇒ u, u ⇒ v, t ⇒ ¬s} and K = {p, t}. We have for instance the arguments:

• a1 = ⟨⟨p⟩ ⇒ q ∨ r⟩ ∈ Arg(T)
• a2 = ⟨⟨q⟩ ⇒ s ⇒ v⟩ ∈ Arg(⟨D, K ∪ {q}⟩)
• a3 = ⟨⟨r⟩ ⇒ u ⇒ v⟩ ∈ Arg(⟨D, K ∪ {r}⟩)
• a4 = ⟨a1, [a2], [a3] v⟩ ∈ Arg(T).

a1: a2: a3: q s v p q ∨ r v r u v

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What about attacks?

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• a1 = ⟨⟨p⟩ ⇒ q ∨ r⟩ ∈ Arg(T)
• a2 = ⟨⟨q⟩ ⇒ s ⇒ v⟩ ∈ Arg(⟨S, D, K ∪ {q}⟩)
• a3 = ⟨⟨r⟩ ⇒ u ⇒ v⟩ ∈ Arg(⟨S, D, K ∪ {r}⟩)
• a4 = ⟨a1, [a2], [a3] v⟩ ∈ Arg(T)
• a5

t s Arg T a1: a2: a3: q s v p q ∨ r v r u v

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• a1 = ⟨⟨p⟩ ⇒ q ∨ r⟩ ∈ Arg(T)
• a2 = ⟨⟨q⟩ ⇒ s ⇒ v⟩ ∈ Arg(⟨S, D, K ∪ {q}⟩)
• a3 = ⟨⟨r⟩ ⇒ u ⇒ v⟩ ∈ Arg(⟨S, D, K ∪ {r}⟩)
• a4 = ⟨a1, [a2], [a3] v⟩ ∈ Arg(T)
• a5 = ⟨⟨t⟩ ⇒ ¬s⟩ ∈ Arg(T)

a1: a2: a3: a5: t ¬s q s v p q ∨ r v r u v

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• a1 = ⟨⟨p⟩ ⇒ q ∨ r⟩ ∈ Arg(T)
• a2 = ⟨⟨q⟩ ⇒ s ⇒ v⟩ ∈ Arg(⟨D, K ∪ {q}⟩)
• a3 = ⟨⟨r⟩ ⇒ u ⇒ v⟩ ∈ Arg(⟨D, K ∪ {r}⟩)
• a4 = ⟨a1, [a2], [a3] v⟩ ∈ Arg(T)
• a6

q s Arg q a1: a2: a3: q s v p q ∨ r v r u v

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• a1 = ⟨⟨p⟩ ⇒ q ∨ r⟩ ∈ Arg(T)
• a2 = ⟨⟨q⟩ ⇒ s ⇒ v⟩ ∈ Arg(⟨D, K ∪ {q}⟩)
• a3 = ⟨⟨r⟩ ⇒ u ⇒ v⟩ ∈ Arg(⟨D, K ∪ {r}⟩)
• a4 = ⟨a1, [a2], [a3] v⟩ ∈ Arg(T)
• a6 = ⟨⟨q⟩ ⇒ ¬s⟩ ∈ Arg(⟨D, K ∪ {q}⟩)

a1: a2: a3: a6: q ¬s q s v p q ∨ r v r u v

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Attacks again (non-nested case)

Let HArg(T) be the set of all arguments a for which there is a b ∈ Arg(T) for which a ∈ HSub(b). Argumentation frameworks are now triples Arg T HArg T Attacks T . Altogether attacks are defined as follows: Attacks T Arg T Arg T Arg T HArg T HArg T HArg T where a rebuts b b1 bn B iff Conc a B or B Conc a and

• 1. a

Arg T and b Arg T HArg T or

• 2. a

HArg T and b HArg T and Hyp a Hyp b .

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Attacks again (non-nested case)

Let HArg(T) be the set of all arguments a for which there is a b ∈ Arg(T) for which a ∈ HSub(b). Argumentation frameworks are now triples ⟨Arg(T), HArg(T), Attacks(T)⟩. Altogether attacks are defined as follows: Attacks T Arg T Arg T Arg T HArg T HArg T HArg T where a rebuts b b1 bn B iff Conc a B or B Conc a and

• 1. a

Arg T and b Arg T HArg T or

• 2. a

HArg T and b HArg T and Hyp a Hyp b .

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Attacks again (non-nested case)

Let HArg(T) be the set of all arguments a for which there is a b ∈ Arg(T) for which a ∈ HSub(b). Argumentation frameworks are now triples ⟨Arg(T), HArg(T), Attacks(T)⟩. Altogether attacks are defined as follows: Attacks(T) ⊆ (Arg(T) × Arg(T)) ∪ (Arg(T) × HArg(T)) ∪ (HArg(T) × HArg(T)) where a rebuts b b1 bn B iff Conc a B or B Conc a and

• 1. a

Arg T and b Arg T HArg T or

• 2. a

HArg T and b HArg T and Hyp a Hyp b .

22/43

Attacks again (non-nested case)

Let HArg(T) be the set of all arguments a for which there is a b ∈ Arg(T) for which a ∈ HSub(b). Argumentation frameworks are now triples ⟨Arg(T), HArg(T), Attacks(T)⟩. Altogether attacks are defined as follows: Attacks(T) ⊆ (Arg(T) × Arg(T)) ∪ (Arg(T) × HArg(T)) ∪ (HArg(T) × HArg(T)) where a rebuts b = ⟨b1, . . . , bn ⇒ B⟩ iff Conc(a) = ¬B or B = ¬Conc(a) and

• 1. a ∈ Arg(T) and b ∈ Arg(T) ∪ HArg(T) or
• 2. a

HArg T and b HArg T and Hyp a Hyp b .

22/43

Attacks again (non-nested case)

Let HArg(T) be the set of all arguments a for which there is a b ∈ Arg(T) for which a ∈ HSub(b). Argumentation frameworks are now triples ⟨Arg(T), HArg(T), Attacks(T)⟩. Altogether attacks are defined as follows: Attacks(T) ⊆ (Arg(T) × Arg(T)) ∪ (Arg(T) × HArg(T)) ∪ (HArg(T) × HArg(T)) where a rebuts b = ⟨b1, . . . , bn ⇒ B⟩ iff Conc(a) = ¬B or B = ¬Conc(a) and

• 1. a ∈ Arg(T) and b ∈ Arg(T) ∪ HArg(T) or
• 2. a ∈ HArg(T) and b ∈ HArg(T) and Hyp(a) = Hyp(b).

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Unrestricted Rebut

joint work: Jesse Heyninck and Christian Straßer

22/43

Status quo

• in ASPIC+ only restricted rebut: a rebuts b iff
• 1. the conclusion of a is contrary to the conclusion of b
• 2. and b has a defeasible top rule
• unrestricted rebut: only the first requirement
• pro: natural (Caminada)
• contra: leads to trouble for many semantics such as

preferred, stable, etc

23/43

Status quo

• in ASPIC+ only restricted rebut: a rebuts b iff
• 1. the conclusion of a is contrary to the conclusion of b
• 2. and b has a defeasible top rule
• unrestricted rebut: only the first requirement
• pro: natural (Caminada)
• contra: leads to trouble for many semantics such as

preferred, stable, etc

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Enters: Caminada et al. (COMMA 2014)

• for grounded semantics unrestricted rebut works just fine

(really?)

• Rationality postulates:
• Sub-argument closure: where a

and b Sub a , b .

• Closure under strict rules: where a1

an Arg T and Conc a1 Conc an B, also a1 an B Arg T

• Consistency:

Conc a a is consistent.

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Enters: Caminada et al. (COMMA 2014)

• for grounded semantics unrestricted rebut works just fine

(really?)

• Rationality postulates:
• Sub-argument closure: where a ∈ E and b ∈ Sub(a), b ∈ E.
• Closure under strict rules: where a1

an Arg T and Conc a1 Conc an B, also a1 an B Arg T

• Consistency:

Conc a a is consistent.

24/43

Enters: Caminada et al. (COMMA 2014)

• for grounded semantics unrestricted rebut works just fine

(really?)

• Rationality postulates:
• Sub-argument closure: where a ∈ E and b ∈ Sub(a), b ∈ E.
• Closure under strict rules: where a1, . . . , an ∈ E ∩ Arg(T)

and Conc(a1), . . . , Conc(an) ⊢ B, also ⟨a1, . . . , an → B⟩ ∈ E ∩ Arg(T)

• Consistency:

Conc a a is consistent.

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Enters: Caminada et al. (COMMA 2014)

• for grounded semantics unrestricted rebut works just fine

(really?)

• Rationality postulates:
• Sub-argument closure: where a ∈ E and b ∈ Sub(a), b ∈ E.
• Closure under strict rules: where a1, . . . , an ∈ E ∩ Arg(T)

and Conc(a1), . . . , Conc(an) ⊢ B, also ⟨a1, . . . , an → B⟩ ∈ E ∩ Arg(T)

• Consistency: {Conc(a) | a ∈ E} is consistent.

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Grounded Semantics

c d a b e f h i g

• first select unattacked arguments
• remove the arguments attacked by the selected arguments
• select unattacked arguments
• remove the arguments attacked by the selected arguments
• and so on … until fixed point is reached

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Grounded Semantics

c d a b e f h i g

• first select unattacked arguments
• remove the arguments attacked by the selected arguments
• select unattacked arguments
• remove the arguments attacked by the selected arguments
• and so on … until fixed point is reached

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Grounded Semantics

c d a b e f h i g

• first select unattacked arguments
• remove the arguments attacked by the selected arguments
• select unattacked arguments
• remove the arguments attacked by the selected arguments
• and so on … until fixed point is reached

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Grounded Semantics

c d a b e f h i g

• first select unattacked arguments
• remove the arguments attacked by the selected arguments
• select unattacked arguments
• remove the arguments attacked by the selected arguments
• and so on … until fixed point is reached

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Grounded Semantics

c d a b e f h i g

• first select unattacked arguments
• remove the arguments attacked by the selected arguments
• select unattacked arguments
• remove the arguments attacked by the selected arguments
• and so on … until fixed point is reached

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Grounded Semantics

c d a b e f h i g

• first select unattacked arguments
• remove the arguments attacked by the selected arguments
• select unattacked arguments
• remove the arguments attacked by the selected arguments
• and so on … until fixed point is reached

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Non-Interference

For a set of formulas F let Atoms(F) be the set of all propositional atoms in F.

• Non-interference1 Where T

and T are argumentation theories and A is a formula such that Atoms A Atoms then: T A iff A

1Caminada, Carnielli, Dunne (JLC, 2012). Avron (2016) calls this the basic

relevance criterion.

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Non-Interference

For a set of formulas F let Atoms(F) be the set of all propositional atoms in F.

• Non-interference1 Where T = ⟨D, K⟩ and T′ = ⟨D′, K′⟩ are

argumentation theories and A is a formula such that Atoms(D ∪ K ∪ {A}) ∩ Atoms(D′ ∪ K′) = ∅ then: T | ∼ A iff ⟨D ∪ D′, K ∪ K′⟩ | ∼ A.

1Caminada, Carnielli, Dunne (JLC, 2012). Avron (2016) calls this the basic

relevance criterion.

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The problem with unrestricted rebut in ASPIC−

• Take the knowledge base: {⊤ ⇒ p}.
• Clearly: a

p is in the grounded extension.

• Now, take the knowledge base:

p s s .

• (Let the strict rules be closed under classical logic.)
• Now, a is attacked by

s s p .

• As a consequence, a is not in the grounded extension.
• Thus, Non-Interference doesn’t hold for unrestricted rebut.

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The problem with unrestricted rebut in ASPIC−

• Take the knowledge base: {⊤ ⇒ p}.
• Clearly: a = ⟨⊤ ⇒ p⟩ is in the grounded extension.
• Now, take the knowledge base:

p s s .

• (Let the strict rules be closed under classical logic.)
• Now, a is attacked by

s s p .

• As a consequence, a is not in the grounded extension.
• Thus, Non-Interference doesn’t hold for unrestricted rebut.

27/43

The problem with unrestricted rebut in ASPIC−

• Take the knowledge base: {⊤ ⇒ p}.
• Clearly: a = ⟨⊤ ⇒ p⟩ is in the grounded extension.
• Now, take the knowledge base: {⊤ ⇒ p, ⊤ ⇒ s, ⊤ ⇒ ¬s}.
• (Let the strict rules be closed under classical logic.)
• Now, a is attacked by

s s p .

• As a consequence, a is not in the grounded extension.
• Thus, Non-Interference doesn’t hold for unrestricted rebut.

27/43

The problem with unrestricted rebut in ASPIC−

• Take the knowledge base: {⊤ ⇒ p}.
• Clearly: a = ⟨⊤ ⇒ p⟩ is in the grounded extension.
• Now, take the knowledge base: {⊤ ⇒ p, ⊤ ⇒ s, ⊤ ⇒ ¬s}.
• (Let the strict rules be closed under classical logic.)
• Now, a is attacked by

s s p .

• As a consequence, a is not in the grounded extension.
• Thus, Non-Interference doesn’t hold for unrestricted rebut.

27/43

The problem with unrestricted rebut in ASPIC−

• Take the knowledge base: {⊤ ⇒ p}.
• Clearly: a = ⟨⊤ ⇒ p⟩ is in the grounded extension.
• Now, take the knowledge base: {⊤ ⇒ p, ⊤ ⇒ s, ⊤ ⇒ ¬s}.
• (Let the strict rules be closed under classical logic.)
• Now, a is attacked by ⟨⟨⊤ ⇒ s⟩, ⟨⊤ ⇒ ¬s⟩ → ¬p⟩.
• As a consequence, a is not in the grounded extension.
• Thus, Non-Interference doesn’t hold for unrestricted rebut.

27/43

The problem with unrestricted rebut in ASPIC−

• Take the knowledge base: {⊤ ⇒ p}.
• Clearly: a = ⟨⊤ ⇒ p⟩ is in the grounded extension.
• Now, take the knowledge base: {⊤ ⇒ p, ⊤ ⇒ s, ⊤ ⇒ ¬s}.
• (Let the strict rules be closed under classical logic.)
• Now, a is attacked by ⟨⟨⊤ ⇒ s⟩, ⟨⊤ ⇒ ¬s⟩ → ¬p⟩.
• As a consequence, a is not in the grounded extension.
• Thus, Non-Interference doesn’t hold for unrestricted rebut.

27/43

The problem with unrestricted rebut in ASPIC−

• Take the knowledge base: {⊤ ⇒ p}.
• Clearly: a = ⟨⊤ ⇒ p⟩ is in the grounded extension.
• Now, take the knowledge base: {⊤ ⇒ p, ⊤ ⇒ s, ⊤ ⇒ ¬s}.
• (Let the strict rules be closed under classical logic.)
• Now, a is attacked by ⟨⟨⊤ ⇒ s⟩, ⟨⊤ ⇒ ¬s⟩ → ¬p⟩.
• As a consequence, a is not in the grounded extension.
• Thus, Non-Interference doesn’t hold for unrestricted rebut.

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Prima Facie solution

• sort out inconsistent arguments (Wu, 2012: this works in

ASPIC+)

• however, this doesn’t work with unrestricted rebut

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Prima Facie solution ii

Let {⊤ ⇒1 p, p ⇒1 q, ⊤ ⇒2 ¬(p ∧ q)} be our knowledge base. We have, e.g., the following arguments:

• a = ⟨⊤ ⇒1 p⟩
• b = ⟨a ⇒1 q⟩
• a ⊕ b = ⟨a, b → p ∧ q⟩
• c = ⟨⊤ ⇒2 ¬(p ∧ q)⟩
• a ⊕ c = ⟨a, c → ¬q⟩
• b ⊕ c = ⟨b, c → ¬p⟩

a b c b ⊕ c a ⊕ c a ⊕ b Problem:

• b

c is inconsistent and thus filtered out

• this leaves a and c in but

a c out of the grounded extension.

• Failure of closure!

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Prima Facie solution ii

Let {⊤ ⇒1 p, p ⇒1 q, ⊤ ⇒2 ¬(p ∧ q)} be our knowledge base. We have, e.g., the following arguments:

• a = ⟨⊤ ⇒1 p⟩
• b = ⟨a ⇒1 q⟩
• a ⊕ b = ⟨a, b → p ∧ q⟩
• c = ⟨⊤ ⇒2 ¬(p ∧ q)⟩
• a ⊕ c = ⟨a, c → ¬q⟩
• b ⊕ c = ⟨b, c → ¬p⟩

a b c b ⊕ c a ⊕ c a ⊕ b Problem:

• b ⊕ c is inconsistent and

thus filtered out

• this leaves a and c in but

a ⊕ c out of the grounded extension.

• Failure of closure!

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Enters: ASPIC⊖: generalized unrestricted rebut

• lifting of the contrariness operator to (finite) sets of

formulas, e.g.,

• A1

An

df n i 1 Ai, or

• A1

An

df n i 1 Ai.

• Concs a

df

Conc b b Sub a Definition a gen-rebuts b iff b is defeasible and Conc a for some Concs b . Definition a gen-defeats b iff a gen-rebuts c for some c Sub b and c a.

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Enters: ASPIC⊖: generalized unrestricted rebut

• lifting of the contrariness operator to (finite) sets of

formulas, e.g.,

• {A1, . . . , An} =df

∧n

i=1 Ai, or

• A1

An

df n i 1 Ai.

• Concs a

df

Conc b b Sub a Definition a gen-rebuts b iff b is defeasible and Conc a for some Concs b . Definition a gen-defeats b iff a gen-rebuts c for some c Sub b and c a.

30/43

Enters: ASPIC⊖: generalized unrestricted rebut

• lifting of the contrariness operator to (finite) sets of

formulas, e.g.,

• {A1, . . . , An} =df

∧n

i=1 Ai, or

• {A1, . . . , An} =df

∨n

i=1 Ai.

• Concs a

df

Conc b b Sub a Definition a gen-rebuts b iff b is defeasible and Conc a for some Concs b . Definition a gen-defeats b iff a gen-rebuts c for some c Sub b and c a.

30/43

Enters: ASPIC⊖: generalized unrestricted rebut

• lifting of the contrariness operator to (finite) sets of

formulas, e.g.,

• {A1, . . . , An} =df

∧n

i=1 Ai, or

• {A1, . . . , An} =df

∨n

i=1 Ai.

• Concs(a) =df {Conc(b) | b ∈ Sub(a)}

Definition a gen-rebuts b iff b is defeasible and Conc a for some Concs b . Definition a gen-defeats b iff a gen-rebuts c for some c Sub b and c a.

30/43

Enters: ASPIC⊖: generalized unrestricted rebut

• lifting of the contrariness operator to (finite) sets of

formulas, e.g.,

• {A1, . . . , An} =df

∧n

i=1 Ai, or

• {A1, . . . , An} =df

∨n

i=1 Ai.

• Concs(a) =df {Conc(b) | b ∈ Sub(a)}

Definition a gen-rebuts b iff b is defeasible and Conc a for some Concs b . Definition a gen-defeats b iff a gen-rebuts c for some c Sub b and c a.

30/43

Enters: ASPIC⊖: generalized unrestricted rebut

• lifting of the contrariness operator to (finite) sets of

formulas, e.g.,

• {A1, . . . , An} =df

∧n

i=1 Ai, or

• {A1, . . . , An} =df

∨n

i=1 Ai.

• Concs(a) =df {Conc(b) | b ∈ Sub(a)}

Definition a gen-rebuts b iff b is defeasible and Conc(a) = ∆ for some ∆ ⊆ Concs(b). Definition a gen-defeats b iff a gen-rebuts c for some c ∈ Sub(b) and c ⪯ a.

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Back to the example

a b c b ⊕ c a ⊕ c a ⊕ b

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Rationality

Where the strict rules are obtained from classical logic, for weakest link we get

• sub-argument closure
• closure under strict rules
• consistency
• non-interference

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Comparative Studies

Jesse Heyninck, Christian Straßer: Relations between assumption-based approaches in nonmonotonic logic and formal argumentation (NMR 2016, Cape Town, also available

• n Arxiv)

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The landscape

• ABA: assumption-based argumentation (Dung, Kowalski,

Toni)

• ALs: adaptive logics (Batens)
• DACR: default assumptions (Makinson)
• KLM: preferential semantics (Shoham,

Kraus/Lehman/Magidor)

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On the argumentative side

• ASPIC+:
• defeasible and strict rules
• various attack types: rebuts, undercuts, undermine
• arguments as proof trees
• ABA (Assumption-based argumentation)
• only strict rules
• higher level of abstraction: arguments as sets of defeasible

assumptions

• only “assumption-attacks” (

undermine)

• our translation: without priorities

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On the argumentative side

• ASPIC+:
• defeasible and strict rules
• various attack types: rebuts, undercuts, undermine
• arguments as proof trees
• ABA (Assumption-based argumentation)
• only strict rules
• higher level of abstraction: arguments as sets of defeasible

assumptions

• only “assumption-attacks” (≈ undermine)
• our translation: without priorities

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Currently

• extending (Heyninck, Straßer, NMR 2016) with priorities
• relations between adaptive logics and parametrized logic

programming (Jesse Heyninck, Pere Pardo, Christian Straßer)

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Sequent-based argumentation (with Ofer Arieli, Tel Aviv)

Sequent-based Argumentation

• arguments are
• provable sequents, where
• is a sound and complete sequent-calculus
• of a (Tarskian) core logic L

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Sequent-based Argumentation

• arguments are C-provable sequents, where
• is a sound and complete sequent-calculus
• of a (Tarskian) core logic L

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Sequent-based Argumentation

• arguments are C-provable sequents, where
• C is a sound and complete sequent-calculus
• of a (Tarskian) core logic L

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Sequent-based Argumentation

• arguments are C-provable sequents, where
• C is a sound and complete sequent-calculus
• of a (Tarskian) core logic L

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Sequent-based attacks: elimination rules

Attacker Sequent Conditions Attacked Sequent Eliminated Sequent Examples

• Undercut:

1 1 1 2 2 2 2 2 2 2

• Compact Undercut:

1 2 2 2 2 2

• Rebuttal:

1 1 1 2 2 2 2 2

• Specificity, etc.

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Sequent-based attacks: elimination rules

Attacker Sequent Conditions Attacked Sequent Eliminated Sequent Examples

• Undercut:

Γ1 ⇒ ψ1 ⇒ ψ1 ↔ ¬∧Γ′

2

Γ2, Γ′

2 ⇒ ψ2

Γ2, Γ′

2 ̸⇒ ψ2

• Compact Undercut:

1 2 2 2 2 2

• Rebuttal:

1 1 1 2 2 2 2 2

• Specificity, etc.

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Sequent-based attacks: elimination rules

Attacker Sequent Conditions Attacked Sequent Eliminated Sequent Examples

• Undercut:

Γ1 ⇒ ψ1 ⇒ ψ1 ↔ ¬∧Γ′

2

Γ2, Γ′

2 ⇒ ψ2

Γ2, Γ′

2 ̸⇒ ψ2

• Compact Undercut:

Γ1 ⇒ ¬ ∧ Γ′

2

Γ2, Γ′

2 ⇒ ψ

Γ2, Γ′

2 ̸⇒ ψ

• Rebuttal:

1 1 1 2 2 2 2 2

• Specificity, etc.

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Sequent-based attacks: elimination rules

Attacker Sequent Conditions Attacked Sequent Eliminated Sequent Examples

• Undercut:

Γ1 ⇒ ψ1 ⇒ ψ1 ↔ ¬∧Γ′

2

Γ2, Γ′

2 ⇒ ψ2

Γ2, Γ′

2 ̸⇒ ψ2

• Compact Undercut:

Γ1 ⇒ ¬ ∧ Γ′

2

Γ2, Γ′

2 ⇒ ψ

Γ2, Γ′

2 ̸⇒ ψ

• Rebuttal:

Γ1 ⇒ ψ1 ⇒ ψ1 ↔ ¬ψ2 Γ2 ⇒ ψ2 Γ2 ̸⇒ ψ2

• Specificity, etc.

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Sequent-based attacks: elimination rules

Attacker Sequent Conditions Attacked Sequent Eliminated Sequent Examples

• Undercut:

Γ1 ⇒ ψ1 ⇒ ψ1 ↔ ¬∧Γ′

2

Γ2, Γ′

2 ⇒ ψ2

Γ2, Γ′

2 ̸⇒ ψ2

• Compact Undercut:

Γ1 ⇒ ¬ ∧ Γ′

2

Γ2, Γ′

2 ⇒ ψ

Γ2, Γ′

2 ̸⇒ ψ

• Rebuttal:

Γ1 ⇒ ψ1 ⇒ ψ1 ↔ ¬ψ2 Γ2 ⇒ ψ2 Γ2 ̸⇒ ψ2

• Specificity, etc.

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Dynamic proof theories

1. p ⇒ p Axiom 2. ⇒ p, ¬p [⇒¬], 1 3. ⇒ p ∨ ¬p [⇒∨], 2 4. p ∨ ¬p ⇒ ¬(p ∧ ¬p) . . . 5. ¬(p ∧ ¬p) ⇒ p ∨ ¬p . . . 6. q ⇒ q Axiom 7. ¬p ⇒ ¬p Axiom 8. p ̸⇒ p Ucut, 7, 7, 7, 1 ¬p ⇒ ¬p 9. p ⇒ ¬¬p … 10. ¬¬p ⇒ p … 11. ¬p ̸⇒ ¬p Ucut, 1, 9, 10, 7 p ⇒ p

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Dynamic proof theories (Retraction, Basic Idea)

function Evaluate(D) /* D Attack := ∅; Elim := ∅; Derived := ∅; while (D is not empty) do { if (Top(D) = ⟨i, s, J, ∅⟩) then Derived := Derived ∪ {s}; if (Top(D) = ⟨i, s, J, r⟩) then if (r ̸∈ Elim) then Elim := Elim ∪ {s} and Attack := Attack ∪ {r}; D := Tail(D); } Accept := Derived − Elim; return (Attack, Elim, Accept)

• A derivation must be

coherent: Attack(D) ∩ Elim(D) = ∅

• A sequent A is

finally derived in a dynamic derivation D if A Accept D and D cannot be extended to a dynamic derivation D such that A Elim D .

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Dynamic proof theories (Retraction, Basic Idea)

function Evaluate(D) /* D Attack := ∅; Elim := ∅; Derived := ∅; while (D is not empty) do { if (Top(D) = ⟨i, s, J, ∅⟩) then Derived := Derived ∪ {s}; if (Top(D) = ⟨i, s, J, r⟩) then if (r ̸∈ Elim) then Elim := Elim ∪ {s} and Attack := Attack ∪ {r}; D := Tail(D); } Accept := Derived − Elim; return (Attack, Elim, Accept)

• A derivation must be

coherent: Attack(D) ∩ Elim(D) = ∅

• A sequent A is

finally derived in a dynamic derivation D if A ∈ Accept(D) and D cannot be extended to a dynamic derivation D′ such that A ∈ Elim(D′).

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Sequent-based Argumentation: some publications

• Ofer Arieli, Christian Straßer, Sequent-Based Logical Argumentation, in

Argument and Computation, Vol. 6, Issue 1, pp. 73–99, 2015

• Ofer Arieli, Christian Straßer, Dynamic Derivations for Sequent-Based

Deductive Argumentation, Proceedings of Computational Models of Argument (Editors: S. Parsons, N. Oren, C. Reed, and F. Cerutti) in the series Frontiers in Artificial Intelligence and Applications, Volume 266, IOS Press, pp. 89–100, 2014

• Christian Straßer and Ofer Ariel, Normative Reasoning by

Sequent-Based Argumentation, Journal of Logic and Computation, doi.org/10.1093/logcom/exv050 (2015)

• Ofer Arieli and Christian Straßer, Deductive argumentation by enhanced

sequent calculi and dynamic derivations, Electronic Notes in Theoretical Computer Science, 323, 21–37 (2016).

• Ofer Arieli, Annemarie Borg, and Christian Straßer, Argumentative

Approaches to Reasoning with Consistent Subsets of Premises in proceedings of IEA/AIE’2017 (full paper), Lecture Notes in Artificial Intelligence series, Springer (2017)

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Agent-Based Models

• joint work: AnneMarie Borg, Daniel Frey, Dunja Šešelja and

Christian Straßer

• Borg A., Frey D., Šešelja D. and Straßer C. (accepted) An

Argumentative Agent-Based Model of Scientific Inquiry, forthcoming in the Proceedings of IEA/AIE, Springer-Verlag (extended version at: https://arxiv.org/abs/1612.04432

• Borg A., Frey D., Šešelja D. and Straßer C. (under revision)

Epistemic Effects of Scientific Interaction: approaching the question with an argumentative agent-based model, special issue of Historical Social Research: “Agent Based Modelling across Social Science, Economics, and Philosophy”

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Explanatory Argumentation Frameworks

Šešelja and Straßer, Synthese, 2013, 190:2195–2217

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Abstract argumentation in our ABM

We represent in an abstract way:

• arguments
• discovery relation
• attack relation

Theory 1 Theory 2

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