X -logics based multivalued reasoning for dialogical agents (ongoing - PowerPoint PPT Presentation

X -logics based multivalued reasoning for dialogical agents (ongoing work) Vincent Risch Aix-Marseille Univ., LSIS – UMR CNRS 7296 Madeira Worshop on Belief Revision and Argumentation, 2015 V. Risch (LSIS) Argumentation, NMatrices, X –logics BRA’15 1 / 29

Motivations A formal attempt around eristic argumentation: human disputes (either explicitely or implicitely) relies on rethoric techniques; V. Risch (LSIS) Argumentation, NMatrices, X –logics BRA’15 2 / 29

Motivations A formal attempt around eristic argumentation: human disputes (either explicitely or implicitely) relies on rethoric techniques; one goal of these techniques, beyond checking the correctness of the reasoning of the speakers, is to help avoiding revision as long as possible ; V. Risch (LSIS) Argumentation, NMatrices, X –logics BRA’15 2 / 29

Motivations A formal attempt around eristic argumentation: human disputes (either explicitely or implicitely) relies on rethoric techniques; one goal of these techniques, beyond checking the correctness of the reasoning of the speakers, is to help avoiding revision as long as possible ; e.g. greek sophists, Socrate, scholastic disputia, Thomas Aquina, Schopenhauer, modern political disputes... V. Risch (LSIS) Argumentation, NMatrices, X –logics BRA’15 2 / 29

Motivations A formal attempt around eristic argumentation: human disputes (either explicitely or implicitely) relies on rethoric techniques; one goal of these techniques, beyond checking the correctness of the reasoning of the speakers, is to help avoiding revision as long as possible ; e.g. greek sophists, Socrate, scholastic disputia, Thomas Aquina, Schopenhauer, modern political disputes... these techniques : V. Risch (LSIS) Argumentation, NMatrices, X –logics BRA’15 2 / 29

Motivations A formal attempt around eristic argumentation: human disputes (either explicitely or implicitely) relies on rethoric techniques; one goal of these techniques, beyond checking the correctness of the reasoning of the speakers, is to help avoiding revision as long as possible ; e.g. greek sophists, Socrate, scholastic disputia, Thomas Aquina, Schopenhauer, modern political disputes... these techniques : assume dialogical agents; V. Risch (LSIS) Argumentation, NMatrices, X –logics BRA’15 2 / 29

Motivations A formal attempt around eristic argumentation: human disputes (either explicitely or implicitely) relies on rethoric techniques; one goal of these techniques, beyond checking the correctness of the reasoning of the speakers, is to help avoiding revision as long as possible ; e.g. greek sophists, Socrate, scholastic disputia, Thomas Aquina, Schopenhauer, modern political disputes... these techniques : assume dialogical agents; aim to represent the mechanisms of a dispute; V. Risch (LSIS) Argumentation, NMatrices, X –logics BRA’15 2 / 29

Motivations A formal attempt around eristic argumentation: human disputes (either explicitely or implicitely) relies on rethoric techniques; one goal of these techniques, beyond checking the correctness of the reasoning of the speakers, is to help avoiding revision as long as possible ; e.g. greek sophists, Socrate, scholastic disputia, Thomas Aquina, Schopenhauer, modern political disputes... these techniques : assume dialogical agents; aim to represent the mechanisms of a dispute; aim to achieve reasoning on arguments; V. Risch (LSIS) Argumentation, NMatrices, X –logics BRA’15 2 / 29

Motivations A formal attempt around eristic argumentation: human disputes (either explicitely or implicitely) relies on rethoric techniques; one goal of these techniques, beyond checking the correctness of the reasoning of the speakers, is to help avoiding revision as long as possible ; e.g. greek sophists, Socrate, scholastic disputia, Thomas Aquina, Schopenhauer, modern political disputes... these techniques : assume dialogical agents; aim to represent the mechanisms of a dispute; aim to achieve reasoning on arguments; One of our far(!) ideal(!!) goal : simulation of strategies for the ’game’ of argumentation V. Risch (LSIS) Argumentation, NMatrices, X –logics BRA’15 2 / 29

Outline X –logics 1 Their use in the context of a dialogical framework 2 Nmatrices, Nsequents 3 Transformation into classical sequents 4 LA , logic of attitudes 5 Links with MSPL (Avron et Al.) 6 Epilog 7 Aim : attempt for defining a dialogical framework in which two ’agents’ can achieve ’some’ reasoning on their arguments. V. Risch (LSIS) Argumentation, NMatrices, X –logics BRA’15 3 / 29

X –logics [Siegel, Forget, 96] Definition Classical Inference: K ⊢ f K ∪ { f } = K iff V. Risch (LSIS) Argumentation, NMatrices, X –logics BRA’15 4 / 29

X –logics [Siegel, Forget, 96] Definition Classical Inference: K ⊢ f K ∪ { f } = K iff Generalisation : K ⊢ X f iff K ∪ { f } ∩ X = K ∩ X V. Risch (LSIS) Argumentation, NMatrices, X –logics BRA’15 4 / 29

X –logics [Siegel, Forget, 96] Definition Classical Inference: K ⊢ f K ∪ { f } = K iff Generalisation : K ⊢ X f iff K ∪ { f } ∩ X = K ∩ X Theorem K ⊢ X f iff K ∪ { f } ∩ X ⊆ K i . e . ( ∀ x ∈ X )( K ∧ f ⊢ x ⇒ K ⊢ x ) V. Risch (LSIS) Argumentation, NMatrices, X –logics BRA’15 4 / 29

X –logics [Siegel, Forget, 96] Definition Classical Inference: K ⊢ f K ∪ { f } = K iff Generalisation : K ⊢ X f iff K ∪ { f } ∩ X = K ∩ X Theorem K ⊢ X f iff K ∪ { f } ∩ X ⊆ K i . e . ( ∀ x ∈ X )( K ∧ f ⊢ x ⇒ K ⊢ x ) Vocabulary f is compatible with K regarding X iff K ⊢ X f , incompatible otherwise. V. Risch (LSIS) Argumentation, NMatrices, X –logics BRA’15 4 / 29

X –logics Properties V. Risch (LSIS) Argumentation, NMatrices, X –logics BRA’15 5 / 29

X –logics Properties ( nonmonotonicity ) K ⊢ X f does not involve K ∪ K ′ ⊢ X f V. Risch (LSIS) Argumentation, NMatrices, X –logics BRA’15 5 / 29

X –logics Properties ( nonmonotonicity ) K ⊢ X f does not involve K ∪ K ′ ⊢ X f ( supraclassicity ) K ⊢ f ⇒ K ⊢ X f V. Risch (LSIS) Argumentation, NMatrices, X –logics BRA’15 5 / 29

X –logics Properties ( nonmonotonicity ) K ⊢ X f does not involve K ∪ K ′ ⊢ X f ( supraclassicity ) K ⊢ f ⇒ K ⊢ X f ( paraconsistancy ) both a formula and its negation can be compatible with K regarding X (resp. incompatibles) V. Risch (LSIS) Argumentation, NMatrices, X –logics BRA’15 5 / 29

X –logics Properties ( nonmonotonicity ) K ⊢ X f does not involve K ∪ K ′ ⊢ X f ( supraclassicity ) K ⊢ f ⇒ K ⊢ X f ( paraconsistancy ) both a formula and its negation can be compatible with K regarding X (resp. incompatibles) ( cumulativity ) If X c = X c then ⊢ x is cumulative V. Risch (LSIS) Argumentation, NMatrices, X –logics BRA’15 5 / 29

X –logics Properties ( nonmonotonicity ) K ⊢ X f does not involve K ∪ K ′ ⊢ X f ( supraclassicity ) K ⊢ f ⇒ K ⊢ X f ( paraconsistancy ) both a formula and its negation can be compatible with K regarding X (resp. incompatibles) ( cumulativity ) If X c = X c then ⊢ x is cumulative ( classical reasoning ) If X = X then ⊢ x is ⊢ V. Risch (LSIS) Argumentation, NMatrices, X –logics BRA’15 5 / 29

X –logics Properties ( nonmonotonicity ) K ⊢ X f does not involve K ∪ K ′ ⊢ X f ( supraclassicity ) K ⊢ f ⇒ K ⊢ X f ( paraconsistancy ) both a formula and its negation can be compatible with K regarding X (resp. incompatibles) ( cumulativity ) If X c = X c then ⊢ x is cumulative ( classical reasoning ) If X = X then ⊢ x is ⊢ Example V. Risch (LSIS) Argumentation, NMatrices, X –logics BRA’15 5 / 29

X –logics Properties ( nonmonotonicity ) K ⊢ X f does not involve K ∪ K ′ ⊢ X f ( supraclassicity ) K ⊢ f ⇒ K ⊢ X f ( paraconsistancy ) both a formula and its negation can be compatible with K regarding X (resp. incompatibles) ( cumulativity ) If X c = X c then ⊢ x is cumulative ( classical reasoning ) If X = X then ⊢ x is ⊢ Example { a } ⊢ {⊥} b and { a , ¬ b } � {⊥} b V. Risch (LSIS) Argumentation, NMatrices, X –logics BRA’15 5 / 29

X –logics Properties ( nonmonotonicity ) K ⊢ X f does not involve K ∪ K ′ ⊢ X f ( supraclassicity ) K ⊢ f ⇒ K ⊢ X f ( paraconsistancy ) both a formula and its negation can be compatible with K regarding X (resp. incompatibles) ( cumulativity ) If X c = X c then ⊢ x is cumulative ( classical reasoning ) If X = X then ⊢ x is ⊢ Example { a } ⊢ {⊥} b and { a , ¬ b } � {⊥} b { b } ⊢ {⊥} a ∧ b and { b } ⊢ {⊥} ¬ ( a ∧ b ) V. Risch (LSIS) Argumentation, NMatrices, X –logics BRA’15 5 / 29

X –logics Properties ( nonmonotonicity ) K ⊢ X f does not involve K ∪ K ′ ⊢ X f ( supraclassicity ) K ⊢ f ⇒ K ⊢ X f ( paraconsistancy ) both a formula and its negation can be compatible with K regarding X (resp. incompatibles) ( cumulativity ) If X c = X c then ⊢ x is cumulative ( classical reasoning ) If X = X then ⊢ x is ⊢ Example { a } ⊢ {⊥} b and { a , ¬ b } � {⊥} b { b } ⊢ {⊥} a ∧ b and { b } ⊢ {⊥} ¬ ( a ∧ b ) { b } � {⊥ , a , ¬ a } a ∧ b { b } � {⊥ , a , ¬ a } ¬ ( a ∧ b ) and V. Risch (LSIS) Argumentation, NMatrices, X –logics BRA’15 5 / 29