Assumptive hypersequent-based argumentation Annemarie Borg May 9th 2016 PhDs in Logic VIII Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 1 / 30
Table of contents Preliminaries 1 The idea 2 An example 3 Conclusion 4 Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 2 / 30
Project Research is part of the project An Argumentative Approach to Defeasible Reasoning: Towards a Unifying Base Theory By the Research Group for Non-Monotonic Logic and Formal Argumentation At the Institute of Philosophy II, Ruhr-Universit¨ at Bochum Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 3 / 30
Outline Preliminaries 1 The idea 2 An example 3 Conclusion 4 Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 4 / 30
Aim of this talk Generalization of the sequent approach to argumentation to hypersequents Use this generalization to give an argumentative approach to defeasible reasoning with normality assumptions Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 5 / 30
Defeasible reasoning (DR) Truth of the conclusion not warranted by the truth of the premises Dynamic: inferences can be retracted in view of new information Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 6 / 30
Defeasible reasoning (DR) Truth of the conclusion not warranted by the truth of the premises Dynamic: inferences can be retracted in view of new information Argumentation: a conclusion is drawn unless/until it is attacked Adaptive logics: formulas are derived on explicit and defeasible normality assumptions. In an abnormal situation inferences are retracted Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 6 / 30
Argumentation theory One way of modeling DR is by argumentation frameworks (AFs) Abstract AFs were introduced by Dung These AFs are directed graphs where Nodes are abstract representations of arguments Arcs represent argumentative attacks Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 7 / 30
Argumentation theory One way of modeling DR is by argumentation frameworks (AFs) Abstract AFs were introduced by Dung These AFs are directed graphs where Nodes are abstract representations of arguments Arcs represent argumentative attacks Definition An argumentation framework is a pair AF = �A , →� , where A a set of arguments → ⊆ A × A an attack relation on A Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 7 / 30
Argumentation theory One way of modeling DR is by argumentation frameworks (AFs) Abstract AFs were introduced by Dung These AFs are directed graphs where Nodes are abstract representations of arguments Arcs represent argumentative attacks Definition An argumentation framework is a pair AF = �A , →� , where A a set of arguments → ⊆ A × A an attack relation on A a → b can be read as “ a attacks b ” Acceptance of arguments is calculated by argumentation semantics Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 7 / 30
Argumentation semantics There are many argumentation semantics, here only the notions relevant for this talk: a c d b Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 8 / 30
Argumentation semantics There are many argumentation semantics, here only the notions relevant for this talk: d ∈ A is acceptable w.r.t. C ⊆ A iff every attacker of d is attacked by C a c d b Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 8 / 30
Argumentation semantics There are many argumentation semantics, here only the notions relevant for this talk: d ∈ A is acceptable w.r.t. C ⊆ A iff every attacker of d is attacked by C S ⊆ A is conflict-free iff S does not attack any of its own arguments a c d b Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 8 / 30
Argumentation semantics There are many argumentation semantics, here only the notions relevant for this talk: d ∈ A is acceptable w.r.t. C ⊆ A iff every attacker of d is attacked by C S ⊆ A is conflict-free iff S does not attack any of its own arguments S ⊆ A , conflict free, is admissible iff each argument in S is acceptable w.r.t. S a c d b Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 8 / 30
Argumentation semantics There are many argumentation semantics, here only the notions relevant for this talk: d ∈ A is acceptable w.r.t. C ⊆ A iff every attacker of d is attacked by C S ⊆ A is conflict-free iff S does not attack any of its own arguments S ⊆ A , conflict free, is admissible iff each argument in S is acceptable w.r.t. S S ⊆ A is a preferred extension iff it is a maximal admissible set a c d b Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 8 / 30
Argumentation semantics There are many argumentation semantics, here only the notions relevant for this talk: d ∈ A is acceptable w.r.t. C ⊆ A iff every attacker of d is attacked by C S ⊆ A is conflict-free iff S does not attack any of its own arguments S ⊆ A , conflict free, is admissible iff each argument in S is acceptable w.r.t. S S ⊆ A is a preferred extension iff it is a maximal admissible set a c d b Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 8 / 30
Argumentation semantics There are many argumentation semantics, here only the notions relevant for this talk: d ∈ A is acceptable w.r.t. C ⊆ A iff every attacker of d is attacked by C S ⊆ A is conflict-free iff S does not attack any of its own arguments S ⊆ A , conflict free, is admissible iff each argument in S is acceptable w.r.t. S S ⊆ A is a preferred extension iff it is a maximal admissible set a c d b Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 8 / 30
DR example Consider a bird bird Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 9 / 30
DR example flies Consider a bird Birds fly: this particular bird also flies bird Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 9 / 30
DR example flies Consider a bird Birds fly: this particular bird also flies New information is obtained: The bird is a specific kind of bird: it is a kiwi bird Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 9 / 30
DR example flies Consider a bird Birds fly: this particular bird also flies New information is obtained: The bird is a specific kind of bird: it is a kiwi Kiwis are abnormal birds: they bird do not fly! Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 9 / 30
Logical argumentation Dung’s abstract AFs are sometimes not expressive enough Structured or logical argumentation Arguments are not abstract entities, but contain a logical structure Structure provided by formal languages Several forms, one of which is sequent-based argumentation Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 10 / 30
Sequent-based argumentation I Arguments are sequents provable in the core logic Constructing arguments is done by inference rules: Γ 1 ⇒ ∆ 1 Γ n ⇒ ∆ n . . . Γ ⇒ ∆ Attacks are sequent elimination rules: Γ 1 ⇒ ∆ 1 . . . Γ n ⇒ ∆ n Γ n �⇒ ∆ n Standard logical attacks have their own sequent elimination rule, e.g.: ψ 1 ⇒ ¬ � Γ 2 Γ 1 ⇒ ψ 1 Γ 2 ⇒ ψ 2 Def Γ 2 �⇒ ψ 2 Γ 1 ⇒ ψ 1 ψ 1 ⇒ ¬ ψ 2 ¬ ψ 2 ⇒ ψ 1 Γ 2 ⇒ ψ 2 Reb Γ 2 �⇒ ψ 2 Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 11 / 30
Sequent-based argumentation II Acceptability is based on argumentation semantics applied to the resulting AF Advantages: Different core logics, such as paraconsistent and deontic logics can be used Arguments are automatically constructed and identified Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 12 / 30
Hypersequents Generalization of Gentzen’s sequents: Γ 1 ⇒ ∆ 1 | . . . | Γ n ⇒ ∆ n Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 13 / 30
Hypersequents Generalization of Gentzen’s sequents: Γ 1 ⇒ ∆ 1 | . . . | Γ n ⇒ ∆ n Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 13 / 30
Hypersequents Generalization of Gentzen’s sequents: Γ 1 ⇒ ∆ 1 | . . . | Γ n ⇒ ∆ n Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 13 / 30
Hypersequents Generalization of Gentzen’s sequents: Γ 1 ⇒ ∆ 1 | . . . | Γ n ⇒ ∆ n Independently introduced by Mints (1974), Pottinger (1983) and Avron (1996) Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 13 / 30
Hypersequents Generalization of Gentzen’s sequents: Γ 1 ⇒ ∆ 1 | . . . | Γ n ⇒ ∆ n Independently introduced by Mints (1974), Pottinger (1983) and Avron (1996) Every sequent calculus can be transformed into a hypersequent calculus Annemarie Borg Assumptive hypersequent-based argumentation May 9th 2016 13 / 30
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