Arguments Evaluation in Multi-Agent Systems

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Argument evaluation in multi-agent logics

Alfredo Burrieza and Antonio Yuste-Ginel1 University of M´alaga

Department of Philosophy MBR 2018, Sevilla, October 2018

1

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Outline

Introduction

The logicAE

Syntax Semantics

Assumable properties

Modelling argument evaluation in_AE

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Outline

Introduction

The logicAE

Syntax Semantics

Assumable properties

Modelling argument evaluation in_AE

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

The general question

(i) Lettbe an argument

is t a good argument?

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

The general question

(i) Lettbe an argument

is t a good argument?

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

The general question

(i) Lettbe an argument

is t a good argument?

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

The general question

(i) Lettbe an argument

is t a good argument?

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

The general question

(i) Lettbe an argument

is t a good argument?

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

What does it make a set of premises a good one?

What does it make an inference link a good one?

First Thesis:

Argumentative goodness is a context-dependent notion

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

What does it make a set of premises a good one?

What does it make an inference link a good one?

First Thesis:

Argumentative goodness is a context-dependent notion

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

What does it make a set of premises a good one?

What does it make an inference link a good one?

First Thesis:

Argumentative goodness is a context-dependent notion

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

What does it make a set of premises a good one?

What does it make an inference link a good one?

First Thesis:

Argumentative goodness is a context-dependent notion

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

→A good answer should be sensitive to other questions:

� Good for whom? (agent perspective)

� Good for what? (goal of evaluation)

� How good is it compare to other arguments?

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

→A good answer should be sensitive to other questions:

� Good for whom? (agent perspective)

� Good for what? (goal of evaluation)

� How good is it compare to other arguments?

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

→A good answer should be sensitive to other questions:

� Good for whom? (agent perspective)

� Good for what? (goal of evaluation)

� How good is it compare to other arguments?

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

→A good answer should be sensitive to other questions:

� Good for whom? (agent perspective)

� Good for what? (goal of evaluation)

� How good is it compare to other arguments?

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

→A good answer should be sensitive to other questions:

� Good for whom? (agent perspective)

� Good for what? (goal of evaluation)

� How good is it compare to other arguments?

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

If we combine these three factors we obtain:

(ii) how good ist compare tot�, from a’sperspective if she intends to do Goal?

Some instantiations ofGoal:

� “justifying a given proposition ϕ”

� “deciding if ϕis true”

� “convincing agentb (or group G) thatϕis the case”

� “convincing someone of doing α”

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

If we combine these three factors we obtain:

(ii) how good ist compare tot�, from a’sperspective if she intends to do Goal?

Some instantiations ofGoal:

� “justifying a given proposition ϕ”

� “deciding if ϕis true”

� “convincing agentb (or group G) thatϕis the case”

� “convincing someone of doing α”

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

If we combine these three factors we obtain:

(ii) how good ist compare tot�, from a’sperspective if she intends to do Goal?

Some instantiations ofGoal:

� “justifying a given proposition ϕ”

� “deciding if ϕis true”

� “convincing agentb (or group G) thatϕis the case”

� “convincing someone of doing α”

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

If we combine these three factors we obtain:

(ii) how good ist compare tot�, from a’sperspective if she intends to do Goal?

Some instantiations ofGoal:

� “justifying a given proposition ϕ”

� “deciding if ϕis true”

� “convincing agentb (or group G) thatϕis the case”

� “convincing someone of doing α”

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

If we combine these three factors we obtain:

(ii) how good ist compare tot�, from a’sperspective if she intends to do Goal?

Some instantiations ofGoal:

� “justifying a given proposition ϕ”

� “deciding if ϕis true”

� “convincing agentb (or group G) thatϕis the case”

� “convincing someone of doing α”

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

If we combine these three factors we obtain:

(ii) how good ist compare tot�, from a’sperspective if she intends to do Goal?

Some instantiations ofGoal:

� “justifying a given proposition ϕ”

� “deciding if ϕis true”

� “convincing agentb (or group G) thatϕis the case”

� “convincing someone of doing α”

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

If we combine these three factors we obtain:

(ii) how good ist compare tot�, from a’sperspective if she intends to do Goal?

Some instantiations ofGoal:

� “justifying a given proposition ϕ”

� “deciding if ϕis true”

� “convincing agentb (or group G) thatϕis the case”

� “convincing someone of doing α”

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

If we combine these three factors we obtain:

(ii) how good ist compare tot�, from a’sperspective if she intends to do Goal?

Some instantiations ofGoal:

� “justifying a given proposition ϕ”

� “deciding if ϕis true”

� “convincing agentb (or group G) thatϕis the case”

� “convincing someone of doing α”

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

If we combine these three factors we obtain:

(ii) how good ist compare tot�, from a’sperspective if she intends to do Goal?

Some instantiations ofGoal:

� “justifying a given proposition ϕ”

� “deciding if ϕis true”

� “convincing agentb (or group G) thatϕis the case”

� “convincing someone of doing α”

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

If we combine these three factors we obtain:

(ii) how good ist compare tot�, from a’sperspective if she intends to do Goal?

Some instantiations ofGoal:

� “justifying a given proposition ϕ”

� “deciding if ϕis true”

� “convincing agentb (or group G) thatϕis the case”

� “convincing someone of doing α”

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

If we combine these three factors we obtain:

(ii) how good ist compare tot�, from a’sperspective if she intends to do Goal?

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

If we combine these three factors we obtain:

(ii) how good ist compare tot�, from a’sperspective if she intends to do Goal?

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Outline

Introduction

The logicAE

Syntax Semantics

Assumable properties

Modelling argument evaluation in_AE

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Selecting the tool-kit

� Arguments _→ Justification Logic (Artemov and Nogina, 2005; Artemov, 2008, 2012; Baltag et al., 2012, 2014)

� Doxastic attitudes of evaluative agents_→Doxastic logic(Fagin et al., 2004)

� Expert opinions on different topics_→ Logics for Belief Dependence(Huang, 1990)

� Availability of information → Awareness Logic(Fagin and Halpern, 1987; Baltag et al., 2012)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Selecting the tool-kit

→ Justification Logic (Artemov and Nogina, 2005; Artemov, 2008, 2012; Baltag et al., 2012, 2014)

� Doxastic attitudes of evaluative agents_→Doxastic logic(Fagin et al., 2004)

� Expert opinions on different topics_→ Logics for Belief Dependence(Huang, 1990)

� Availability of information → Awareness Logic(Fagin and Halpern, 1987; Baltag et al., 2012)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Selecting the tool-kit

� Arguments _→ Justification Logic (Artemov and Nogina, 2005; Artemov, 2008, 2012; Baltag et al., 2012, 2014)

� Doxastic attitudes of evaluative agents_→Doxastic logic(Fagin et al., 2004)

� Expert opinions on different topics_→ Logics for Belief Dependence(Huang, 1990)

� Availability of information → Awareness Logic(Fagin and Halpern, 1987; Baltag et al., 2012)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Selecting the tool-kit

� Arguments _→ Justification Logic (Artemov and Nogina, 2005; Artemov, 2008, 2012; Baltag et al., 2012, 2014)

� Doxastic attitudes of evaluative agents

→Doxastic logic(Fagin et al., 2004)

� Expert opinions on different topics_→ Logics for Belief Dependence(Huang, 1990)

� Availability of information → Awareness Logic(Fagin and Halpern, 1987; Baltag et al., 2012)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Selecting the tool-kit

� Arguments _→ Justification Logic (Artemov and Nogina, 2005; Artemov, 2008, 2012; Baltag et al., 2012, 2014)

� Doxastic attitudes of evaluative agents_→Doxastic logic(Fagin et al., 2004)

� Expert opinions on different topics_→ Logics for Belief Dependence(Huang, 1990)

� Availability of information → Awareness Logic(Fagin and Halpern, 1987; Baltag et al., 2012)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Selecting the tool-kit

� Arguments _→ Justification Logic (Artemov and Nogina, 2005; Artemov, 2008, 2012; Baltag et al., 2012, 2014)

� Doxastic attitudes of evaluative agents_→Doxastic logic(Fagin et al., 2004)

� Expert opinions on different topics

→ Logics for Belief Dependence(Huang, 1990)

� Availability of information → Awareness Logic(Fagin and Halpern, 1987; Baltag et al., 2012)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Selecting the tool-kit

� Arguments _→ Justification Logic (Artemov and Nogina, 2005; Artemov, 2008, 2012; Baltag et al., 2012, 2014)

� Doxastic attitudes of evaluative agents_→Doxastic logic(Fagin et al., 2004)

� Expert opinions on different topics_→ Logics for Belief Dependence(Huang, 1990)

� Availability of information → Awareness Logic(Fagin and Halpern, 1987; Baltag et al., 2012)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Selecting the tool-kit

� Arguments _→ Justification Logic (Artemov and Nogina, 2005; Artemov, 2008, 2012; Baltag et al., 2012, 2014)

� Doxastic attitudes of evaluative agents_→Doxastic logic(Fagin et al., 2004)

� Expert opinions on different topics_→ Logics for Belief Dependence(Huang, 1990)

� Availability of information

→ Awareness Logic(Fagin and Halpern, 1987; Baltag et al., 2012)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Selecting the tool-kit

� Arguments _→ Justification Logic (Artemov and Nogina, 2005; Artemov, 2008, 2012; Baltag et al., 2012, 2014)

� Doxastic attitudes of evaluative agents_→Doxastic logic(Fagin et al., 2004)

� Expert opinions on different topics_→ Logics for Belief Dependence(Huang, 1990)

� Availability of information → Awareness Logic(Fagin and Halpern, 1987; Baltag et al., 2012)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Selecting the tool-kit

� Arguments _→ Justification Logic (Artemov and Nogina, 2005; Artemov, 2008, 2012; Baltag et al., 2012, 2014)

� Doxastic attitudes of evaluative agents_→Doxastic logic(Fagin et al., 2004)

� Expert opinions on different topics_→ Logics for Belief Dependence(Huang, 1990)

� Availability of information → Awareness Logic(Fagin and Halpern, 1987; Baltag et al., 2012)

� Complexity of arguments

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Selecting the tool-kit

� Arguments _→ Justification Logic (Artemov and Nogina, 2005; Artemov, 2008, 2012; Baltag et al., 2012, 2014)

� Doxastic attitudes of evaluative agents_→Doxastic logic(Fagin et al., 2004)

� Expert opinions on different topics_→ Logics for Belief Dependence(Huang, 1990)

� Availability of information → Awareness Logic(Fagin and Halpern, 1987; Baltag et al., 2012)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Syntax

Outline

Introduction

The logicAE

Syntax

Semantics

Assumable properties

Modelling argument evaluation in_AE

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Syntax

Language

P (atoms),_A (agents); terms_T and formulas_F are defined:

t::=cϕ |[t+t]|[t·t]

ϕ::=p| ¬ϕ|(ϕ∧ϕ)|t�ϕ|Baϕ|Dabϕ|Eat|Com≤(t, s)

Terms intend to represent arguments

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Syntax

Language

P (atoms),_A (agents);

terms_T and formulas_F are defined:

t::=cϕ |[t+t]|[t·t]

ϕ::=p| ¬ϕ|(ϕ∧ϕ)|t�ϕ|Baϕ|Dabϕ|Eat|Com≤(t, s)

Terms intend to represent arguments

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Syntax

Language

P (atoms),_A (agents); terms_T and formulas_F are defined:

t::=cϕ |[t+t]|[t·t]

ϕ::=p| ¬ϕ|(ϕ∧ϕ)|t�ϕ|Baϕ|Dabϕ|Eat|Com≤(t, s)

Terms intend to represent arguments

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Syntax

Language

P (atoms),_A (agents); terms_T and formulas_F are defined:

t::=cϕ |[t+t]|[t·t]

ϕ::=p| ¬ϕ|(ϕ∧ϕ)|t�ϕ|Baϕ|Dabϕ|Eat|Com≤(t, s)

Terms intend to represent arguments

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Syntax

Language

P (atoms),_A (agents); terms_T and formulas_F are defined:

t::=cϕ |[t+t]|[t·t]

ϕ::=p| ¬ϕ|(ϕ∧ϕ)|t�ϕ|Baϕ|Dabϕ|Eat|Com≤(t, s)

Terms intend to represent arguments

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Syntax

Language

P (atoms),_A (agents); terms_T and formulas_F are defined:

t::=cϕ |[t+t]|[t·t]

ϕ::=p| ¬ϕ|(ϕ∧ϕ)|t�ϕ|Baϕ|Dabϕ|Eat|Com≤(t, s)

Terms intend to represent arguments

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Syntax

Language

P (atoms),_A (agents); terms_T and formulas_F are defined:

t::=cϕ |[t+t]|[t·t]

ϕ::=p| ¬ϕ|(ϕ∧ϕ)|t�ϕ|Baϕ|Dabϕ|Eat|Com≤(t, s)

Terms intend to represent arguments

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Syntax

Language

P (atoms),_A (agents); terms_T and formulas_F are defined:

t::=cϕ |[t+t]|[t·t]

ϕ::=p| ¬ϕ|(ϕ∧ϕ)|t�ϕ|Baϕ|Dabϕ|Eat|Com≤(t, s)

Terms intend to represent arguments

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Syntax

Arguments

t::=cϕ |[t+t]|[t·t]

Taken from (Baltag et al., 2012, 2014).

cϕ:≈“a premise assertingϕ”

t+s:≈“result from combining tand swithout performing any inference”

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Syntax

Arguments

t::=cϕ |[t+t]|[t·t]

Taken from (Baltag et al., 2012, 2014).

cϕ:≈“a premise assertingϕ”

t+s:≈“result from combining tand swithout performing any inference”

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Syntax

Arguments

t::=cϕ |[t+t]|[t·t]

Taken from (Baltag et al., 2012, 2014).

cϕ:≈“a premise assertingϕ”

t+s:≈“result from combining tand swithout performing any inference”

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Syntax

Arguments

t::=cϕ |[t+t]|[t·t]

Taken from (Baltag et al., 2012, 2014).

cϕ:≈“a premise assertingϕ”

t+s:≈“result from combiningt and swithout performing any inference”

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Syntax

Arguments

t::=cϕ |[t+t]|[t·t]

Taken from (Baltag et al., 2012, 2014).

cϕ:≈“a premise assertingϕ”

t+s:≈“result from combiningt and swithout performing any inference”

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Syntax

Arguments

t::=cϕ |[t+t]|[t·t]

Taken from (Baltag et al., 2012, 2014).

cϕ:≈“a premise assertingϕ”

t+s:≈“result from combiningt and swithout performing any inference”

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Syntax

Arguments

t::=cϕ |[t+t]|[t·t]

Note: other operations between terms are definable without any technical problem (as long as they preserve truth)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Syntax

Arguments

t::=cϕ |[t+t]|[t·t]

Note: other operations between terms are definable without any technical problem (as long as they preserve truth)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Syntax

Arguments

t::=cϕ |[t+t]|[t·t]

Note: other operations between terms are definable without any technical problem (as long as they preserve truth)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Syntax

Formulas

ϕ::=p_{| ¬}ϕ_|(ϕ_∧ϕ)_|t_�ϕ_|Baϕ|Dabϕ|Eat|Com≤(t, s)

Baϕ:≈“ agent abelieves thatϕ”

Dabϕ:≈“b is an experton topicϕ according toa’s opinion”

t_�ϕ:_≈“argument thas propositionϕ as itsconclusion”

Com≤(t, s) :≈“argument tisat least as simple as arguments”

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Syntax

Formulas

ϕ::=p_{| ¬}ϕ_|(ϕ_∧ϕ)_|t_�ϕ_|Baϕ|Dabϕ|Eat|Com≤(t, s)

Baϕ:≈“ agent abelieves thatϕ”

Dabϕ:≈“b is an experton topicϕ according toa’s opinion”

t_�ϕ:_≈“argument thas propositionϕ as itsconclusion”

Com≤(t, s) :≈“argument tisat least as simple as arguments”

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Syntax

Formulas

ϕ::=p_{| ¬}ϕ_|(ϕ_∧ϕ)_|t_�ϕ_|Baϕ|Dabϕ|Eat|Com≤(t, s)

Baϕ:≈“ agent abelieves thatϕ”

Dabϕ:≈“b is an experton topicϕ according toa’s opinion”

t_�ϕ:_≈“argument thas propositionϕ as itsconclusion”

Com≤(t, s) :≈“argument tisat least as simple as arguments”

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Syntax

Formulas

ϕ::=p_{| ¬}ϕ_|(ϕ_∧ϕ)_|t_�ϕ_|Baϕ|Dabϕ|Eat|Com≤(t, s)

Baϕ:≈“ agent abelieves thatϕ”

Dabϕ:≈“b is an experton topicϕ according toa’s opinion”

t_�ϕ:_≈“argument thas propositionϕas its conclusion”

Com≤(t, s) :≈“argument tisat least as simple as arguments”

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Syntax

Formulas

ϕ::=p_{| ¬}ϕ_|(ϕ_∧ϕ)_|t_�ϕ_|Baϕ|Dabϕ|Eat|Com≤(t, s)

Baϕ:≈“ agent abelieves thatϕ”

Dabϕ:≈“b is an experton topicϕ according toa’s opinion”

t_�ϕ:_≈“argument thas propositionϕas its conclusion”

Com≤(t, s) :≈“argument tisat least as simple as arguments”

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Syntax

Formulas

ϕ::=p_{| ¬}ϕ_|(ϕ_∧ϕ)_|t_�ϕ_|Baϕ|Dabϕ|Eat|Com≤(t, s)

Baϕ:≈“ agent abelieves thatϕ”

Dabϕ:≈“b is an experton topicϕ according toa’s opinion”

t_�ϕ:_≈“argument thas propositionϕas its conclusion”

Com≤(t, s) :≈“argument tisat least as simple as arguments”

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Semantics

Outline

Introduction

The logicAE

Syntax

Semantics

Assumable properties

Modelling argument evaluation in_AE

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Semantics

Interpreting “syntactic” formulas outside the model

Define_{�⊆ T × T} as the smallest set such that:

� cϕ �ϕ

� If t�ϕ→ψ and s�ϕ, then [t·s]�ψ

� If t_�ϕand s_�ψ, then [t+s]_�ϕ

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Semantics

Interpreting “syntactic” formulas outside the model

Define_{�⊆ T × T} as the smallest set such that:

� cϕ �ϕ

� If t�ϕ→ψ and s�ϕ, then [t·s]�ψ

� If t_�ϕand s_�ψ, then [t+s]_�ϕ

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Semantics

Interpreting “syntactic” formulas outside the model

Define_{�⊆ T × T} as the smallest set such that:

� cϕ �ϕ

� If t�ϕ→ψ and s�ϕ, then [t·s]�ψ

� If t_�ϕand s_�ψ, then [t+s]_�ϕ

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Semantics

Interpreting “syntactic” formulas outside the model

Define_{�⊆ T × T} as the smallest set such that:

� cϕ �ϕ

� If t�ϕ→ψ and s�ϕ, then [t·s]�ψ

� If t_�ϕand s_�ψ, then [t+s]_�ϕ

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Semantics

Interpreting “syntactic” formulas outside the model

Define_{�⊆ T × T} as the smallest set such that:

� cϕ �ϕ

� If t�ϕ→ψ and s�ϕ, then [t·s]�ψ

� If t_�ϕand s_�ψ, then [t+s]_�ϕ

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Semantics

Interpreting “syntactic” formulas outside the model

Define_{�⊆ T × T} as the smallest set such that:

� cϕ �ϕ

� If t�ϕ→ψ and s�ϕ, then [t·s]�ψ

� If t_�ϕand s_�ψ, then [t+s]_�ϕ

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Semantics

Interpreting “syntactic” formulas outside the model

Define_{�⊆ T × T} as the smallest set such that:

� cϕ �ϕ

� If t�ϕ→ψ and s�ϕ, then [t·s]�ψ

� If t_�ϕand s_�ψ, then [t+s]_�ϕ

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Semantics

Interpreting “syntactic” formulas outside the model

Define_{�⊆ T × T} as the smallest set such that:

� cϕ �ϕ

� If t�ϕ→ψ and s�ϕ, then [t·s]�ψ

� If t_�ϕand s_�ψ, then [t+s]_�ϕ

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Semantics

Interpreting “syntactic” formulas outside the model

Define_{�⊆ T × T} as the smallest set such that:

� cϕ �ϕ

� If t�ϕ→ψ and s�ϕ, then [t·s]�ψ

� If t_�ϕand s_�ψ, then [t+s]_�ϕ

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Semantics

Interpreting “syntactic” formulas outside the model

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Semantics

Interpreting “syntactic” formulas outside the model

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Semantics

Models and truth

M:=�W,{Ra},{Da},{Ea}, V�

for eacha∈ A

� W �=∅ (possible worlds)

� Ra⊆W ×W (doxastic relation)

� Da: (A ×W)−→℘(F) (advisor function)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Semantics

Models and truth

M:=�W,{Ra},{Da},{Ea}, V�

for eacha∈ A

� W �=∅ (possible worlds)

� Ra⊆W ×W (doxastic relation)

� Da: (A ×W)−→℘(F) (advisor function)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Semantics

Models and truth

M:=�W,{Ra},{Da},{Ea}, V�

for eacha∈ A

� W �=∅ (possible worlds)

� Ra⊆W ×W (doxastic relation)

� Da: (A ×W)−→℘(F) (advisor function)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Semantics

Models and truth

M:=�W,{Ra},{Da},{Ea}, V�

for eacha∈ A

� W �=∅ (possible worlds)

� Ra⊆W ×W (doxastic relation)

� Da: (A ×W)−→℘(F) (advisor function)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Semantics

Models and truth

M:=�W,{Ra},{Da},{Ea}, V�

for eacha∈ A

� W �=∅ (possible worlds)

� Ra⊆W ×W (doxastic relation)

� Da: (A ×W)−→℘(F) (advisor function)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Semantics

Models and truth

M:=�W, Ra,Da,Ea, V�a∈A

M, w�t�ϕ iff ϕ∈conclusions(t)

M, w�Com≤(t, s) iff length(t) +_|atom(t)_{| ≤}

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Semantics

Models and truth

M:=�W, Ra,Da,Ea, V�a∈A

M, w�t�ϕ iff ϕ∈conclusions(t)

M, w�Com≤(t, s) iff length(t) +_|atom(t)_{| ≤}

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Semantics

Models and truth

M:=�W, Ra,Da,Ea, V�a∈A

M, w�t�ϕ iff ϕ∈conclusions(t)

M, w�Com≤(t, s) iff length(t) +_|atom(t)_{| ≤}

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Semantics

Models and truth

M:=�W, Ra,Da,Ea, V�a∈A

M, w�t�ϕ iff ϕ∈conclusions(t)

M, w�Com≤(t, s) iff length(t) +_|atom(t)_{| ≤}

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Semantics

Models and truth

M:=�W, Ra,Da,Ea, V�a∈A

M, w�Baϕ iff ∀w�(wRaw� ⇒ M, w� �ϕ)

M, w�Dabϕ iff ϕ∈ Da(b, w)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Semantics

Models and truth

M:=�W, Ra,Da,Ea, V�a∈A

M, w�Baϕ iff ∀w�(wRaw� ⇒ M, w� �ϕ)

M, w�Dabϕ iff ϕ∈ Da(b, w)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Semantics

Models and truth

M:=�W, Ra,Da,Ea, V�a∈A

M, w�Baϕ iff ∀w�(wRaw� ⇒ M, w� �ϕ)

M, w�Dabϕ iff ϕ∈ Da(b, w)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Semantics

Models and truth

M:=�W, Ra,Da,Ea, V�a∈A

M, w�Baϕ iff ∀w�(wRaw� ⇒ M, w� �ϕ)

M, w�Dabϕ iff ϕ∈ Da(b, w)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Assumable properties

Outline

Introduction

The logicAE

Syntax Semantics

Assumable properties

Modelling argument evaluation in_AE

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Assumable properties

Minimal system

The minimal logic for argument evaluationLmin

Axioms

All propositional tautologies

�t�ϕwhenevert�ϕ

� ¬(t�ϕ) whenever nott�ϕ (admissibility)

�Ba(ϕ→ψ)→(Baϕ→Baψ) (normality ofBa)

� Com≤(t, s) whenever length(t) +|atom(t)| ≤ length(s) +

|atom(s)|

� ¬Com≤(t, s) wheneverlength(t) +|atom(t)| �length(s) +

|atom(s)| Rules

M P �ϕ,�ϕ→ψ⇒ �ψ

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Assumable properties

Assumable properties

Name Axiom scheme

Consistency of beliefs _{� ¬}Ba⊥

Positive introspection _�Baϕ→BaBaϕ

Negative introspection � ¬Baϕ→Ba¬Baϕ

Neutrality ofDab �Dab¬ϕ→Dabϕ

�Dabϕ→Dab¬ϕwhereϕ�=¬ψ

Safety ofDab �Dabϕ↔BaDabϕ

Uniqueness ofDab �Dabϕ→ ¬Dacϕfor eachb�=c

Transparency of awareness _�Eat↔BaEat

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Assumable properties

Assumable properties

Name Axiom scheme

Consistency of beliefs _{� ¬}Ba⊥

Positive introspection _�Baϕ→BaBaϕ

Negative introspection � ¬Baϕ→Ba¬Baϕ

Neutrality ofDab �Dab¬ϕ→Dabϕ

�Dabϕ→Dab¬ϕwhereϕ�=¬ψ

Safety ofDab �Dabϕ↔BaDabϕ

Uniqueness ofDab �Dabϕ→ ¬Dacϕfor eachb�=c

Transparency of awareness _�Eat↔BaEat

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Assumable properties

Theorem (Soundness, strong completeness and decidability)

Any extensionL• _{of L}min _{with any set of schemes in the table}

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Introduction

The logicAE

Syntax Semantics

Assumable properties

Modelling argument evaluation inAE

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Evaluation and preferences

Note: The result of arguments evaluations can be understood preference relations over arguments, for each agent.

� We can try to establish these relations by the use of other notions expressible in our language.

� Different preference relations should be defined in the object language for different instantiations of:

(ii) how good ist compare tot�, from i’sperspective if she intends to doGoal?

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Evaluation and preferences

Note: The result of arguments evaluations can be understood preference relations over arguments, for each agent.

� We can try to establish these relations by the use of other notions expressible in our language.

� Different preference relations should be defined in the object language for different instantiations of:

(ii) how good ist compare tot�, from i’sperspective if she intends to doGoal?

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Evaluation and preferences

Note: The result of arguments evaluations can be understood preference relations over arguments, for each agent.

� We can try to establish these relations by the use of other notions expressible in our language.

� Different preference relations should be defined in the object language for different instantiations of:

(ii) how good ist compare tot�, from i’sperspective if she intends to doGoal?

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Evaluation and preferences

Note: The result of arguments evaluations can be understood preference relations over arguments, for each agent.

� We can try to establish these relations by the use of other notions expressible in our language.

� Different preference relations should be defined in the object language for different instantiations of:

(ii) how good ist compare tot�, from i’sperspective if she intends to doGoal?

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Evaluation and preferences

Note: The result of arguments evaluations can be understood preference relations over arguments, for each agent.

� We can try to establish these relations by the use of other notions expressible in our language.

� Different preference relations should be defined in the object language for different instantiations of:

(ii) how good ist compare tot�, from i’sperspective if she intends to doGoal?

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Searching the best argument for

ϕ

How to order the different criteria

Tentative order

A.Structural accuracy: have tand sthe proper syntactic shape?

B.Doxastic acceptance: which premises are better regarding

a’s beliefs?

C.Expert opinion: which premises are better according to the agent thataconsiders an expert on topic ϕ?

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Searching the best argument for

ϕ

How to order the different criteria

Tentative order

A.Structural accuracy: have tand sthe proper syntactic shape?

B.Doxastic acceptance: which premises are better regarding

a’s beliefs?

C.Expert opinion: which premises are better according to the agent thataconsiders an expert on topic ϕ?

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Searching the best argument for

ϕ

How to order the different criteria

Tentative order

A.Structural accuracy: have tand sthe proper syntactic shape?

B.Doxastic acceptance: which premises are better regarding

a’s beliefs?

C.Expert opinion: which premises are better according to the agent thataconsiders an expert on topic ϕ?

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Searching the best argument for

ϕ

How to order the different criteria

Tentative order

A.Structural accuracy: have tand sthe proper syntactic shape?

B.Doxastic acceptance: which premises are better regarding

a’s beliefs?

C.Expert opinion: which premises are better according to the agent thataconsiders an expert on topic ϕ?

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Searching the best argument for

ϕ

How to order the different criteria

Tentative order

A.Structural accuracy: have tand sthe proper syntactic shape?

B.Doxastic acceptance: which premises are better regarding

a’s beliefs?

C.Expert opinion: which premises are better according to the agent thataconsiders an expert on topic ϕ?

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Searching the best argument for

ϕ

How to order the different criteria

Tentative order

A.Structural accuracy: have tand sthe proper syntactic shape?

B.Doxastic acceptance: which premises are better regarding

a’s beliefs?

C.Expert opinion: which premises are better according to the agent thataconsiders an expert on topic ϕ?

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Searching the best argument for

ϕ

How to order the different criteria

Tentative order

A.Structural accuracy: have tand sthe proper syntactic shape?

B.Doxastic acceptance: which premises are better regarding

a’s beliefs?

C.Expert opinion: which premises are better according to the agent thataconsiders an expert on topic ϕ?

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Intuitive principles

A.Structural accuracy:

Ifthas the proper structural shape (t_�ϕ) but sdoes not have it (¬(s�ϕ)), thent should be consider strictly better

P_a>(t, s, ϕ)

If bothtand shave defective structures (¬t�ϕ∧ ¬s�ϕ) they should be taken as equally goodP_a≈(t, s, ϕ).

If bothtand shave the proper structure (t_�ϕ_∧s_�ϕ), agentashould go on her evaluation to decide which is better.

ρ1 := (t�ϕ∧s�ϕ) :≈“both arguments satisfy structural

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Intuitive principles

A.Structural accuracy:

Ifthas the proper structural shape (t_�ϕ) but sdoes not have it (¬(s�ϕ)), thent should be consider strictly better

P_a>(t, s, ϕ)

If bothtand shave defective structures (¬t�ϕ∧ ¬s�ϕ) they should be taken as equally goodP_a≈(t, s, ϕ).

If bothtand shave the proper structure (t_�ϕ_∧s_�ϕ), agentashould go on her evaluation to decide which is better.

ρ1 := (t�ϕ∧s�ϕ) :≈“both arguments satisfy structural

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Intuitive principles

A.Structural accuracy:

Ifthas the proper structural shape (t_�ϕ) but sdoes not have it (¬(s�ϕ)), thent should be consider strictly better

P_a>(t, s, ϕ)

If bothtand shave defective structures (¬t�ϕ∧ ¬s�ϕ) they should be taken as equally goodP_a≈(t, s, ϕ).

If bothtand shave the proper structure (t_�ϕ_∧s_�ϕ), agentashould go on her evaluation to decide which is better.

ρ1 := (t�ϕ∧s�ϕ) :≈“both arguments satisfy structural

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Intuitive principles

A.Structural accuracy:

Ifthas the proper structural shape (t_�ϕ) but sdoes not have it (¬(s�ϕ)), thent should be consider strictly better

P_a>(t, s, ϕ)

If bothtand shave defective structures (¬t�ϕ∧ ¬s�ϕ) they should be taken as equally goodP_a≈(t, s, ϕ).

If bothtand shave the proper structure (t_�ϕ_∧s_�ϕ), agentashould go on her evaluation to decide which is better.

ρ1 := (t�ϕ∧s�ϕ) :≈“both arguments satisfy structural

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Intuitive principles

A.Structural accuracy:

Ifthas the proper structural shape (t_�ϕ) but sdoes not have it (¬(s�ϕ)), thent should be consider strictly better

P_a>(t, s, ϕ)

If bothtand shave defective structures (¬t�ϕ∧ ¬s�ϕ) they should be taken as equally goodP_a≈(t, s, ϕ).

If bothtand shave the proper structure (t_�ϕ_∧s_�ϕ), agentashould go on her evaluation to decide which is better.

ρ1 := (t�ϕ∧s�ϕ) :≈“both arguments satisfy structural

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Defining acceptance

B.Doxastic acceptance:

Aat:≈ “aaccepts argumentt, i.e., she believes that all its

premises are true” Baltag et al. (2012)

Rat:≈“ arejects argument t, i.e., she believes that some of its

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Defining acceptance

B.Doxastic acceptance:

Aat:≈“aaccepts argumentt, i.e., she believes that all its

premises are true” Baltag et al. (2012)

Rat:≈“ arejects argument t, i.e., she believes that some of its

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Defining acceptance

B.Doxastic acceptance:

Aat:≈“aaccepts argumentt, i.e., she believes that all its

premises are true” Baltag et al. (2012)

Rat:≈“ arejects argument t, i.e., she believes that some of its

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Defining acceptance

B.Doxastic acceptance:

Aat:≈“aaccepts argumentt, i.e., she believes that all its

premises are true” Baltag et al. (2012)

Rat:≈“ arejects argument t, i.e., she believes that some of its

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Defining acceptance

B.Doxastic acceptance:

Aat:=�_cϕ∈sub(t)Baϕ Rat:�cϕ∈sub(t)Ba¬ϕ

Fact: �Rat→ ¬Aatbut�¬Aat→Rat

A>_a(t, s) :≈“a considerststrictly more acceptable thans” := (Aat∧ ¬Aas)∨(¬Rat∧Ras)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Defining acceptance

B.Doxastic acceptance:

Aat:=�_cϕ∈sub(t)Baϕ

Rat:�cϕ∈sub(t)Ba¬ϕ

Fact: �Rat→ ¬Aatbut�¬Aat→Rat

A>_a(t, s) :≈“a considerststrictly more acceptable thans” := (Aat∧ ¬Aas)∨(¬Rat∧Ras)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Defining acceptance

B.Doxastic acceptance:

Aat:=�_cϕ∈sub(t)Baϕ Rat:�cϕ∈sub(t)Ba¬ϕ

Fact: �Rat→ ¬Aatbut�¬Aat→Rat

A>_a(t, s) :≈“a considerststrictly more acceptable thans” := (Aat∧ ¬Aas)∨(¬Rat∧Ras)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Defining acceptance

B.Doxastic acceptance:

Aat:=�_cϕ∈sub(t)Baϕ Rat:�cϕ∈sub(t)Ba¬ϕ

Fact: �Rat→ ¬Aatbut�¬Aat→Rat

A>_a(t, s) :≈“a considerststrictly more acceptable thans” := (Aat∧ ¬Aas)∨(¬Rat∧Ras)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Defining acceptance

B.Doxastic acceptance:

Aat:=�_cϕ∈sub(t)Baϕ Rat:�cϕ∈sub(t)Ba¬ϕ

Fact: �Rat→ ¬Aatbut�¬Aat→Rat

A>_a(t, s) :≈“a considerststrictly more acceptable thans” := (Aat∧ ¬Aas)∨(¬Rat∧Ras)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Defining acceptance

B.Doxastic acceptance:

Aat:=�_cϕ∈sub(t)Baϕ Rat:�cϕ∈sub(t)Ba¬ϕ

Fact: �Rat→ ¬Aatbut�¬Aat→Rat

A>_a(t, s) :≈“a considerststrictly more acceptable thans” := (Aat∧ ¬Aas)∨(¬Rat∧Ras)

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Intuitive principles

B.Doxastic acceptance:

Ifρ1, and aconsiders the premises of tas strictly more

acceptable (A>_a(t, s)), then she should strictly prefert

(P_a>(t, s, ϕ)).

Ifρ1, and aconsiders both set of premises equally acceptable

(A≈_a(t, s)) she should go on her evaluation.

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Intuitive principles

B.Doxastic acceptance:

Ifρ1, and aconsiders the premises of tas strictly more acceptable (A>_a(t, s)), then she should strictly prefert

(P>

a (t, s, ϕ)).

Ifρ1, and aconsiders both set of premises equally acceptable

(A≈_a(t, s)) she should go on her evaluation.

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Intuitive principles

B.Doxastic acceptance:

Ifρ1, and aconsiders the premises of tas strictly more acceptable (A>_a(t, s)), then she should strictly prefert

(P>

a (t, s, ϕ)).

Ifρ1, and aconsiders both set of premises equally acceptable

(A≈_a(t, s)) she should go on her evaluation.

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Intuitive principles

C.Expert opinion:

Ifρ2, and b(the ϕ-expert of a(Dabϕ)) considers the premises of tstrictly more acceptable (A>_b(t, s)), thena should strictly prefert (P_a>(t, s, ϕ)).

Ifρ2, and b(the ϕ-expert of a(Dabϕ)) considers the premises of tequally acceptable (A≈_b (t, s)), thenashould go on her

evaluation.

Ifρ2, and aknows no expert on ϕ(�b∈A¬Dabϕ), thenashould

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Intuitive principles

C.Expert opinion:

Ifρ2, and b(the ϕ-expert of a(Dabϕ)) considers the premises of tstrictly more acceptable (A>_b(t, s)), thenashould strictly prefert (P>

a (t, s, ϕ)).

Ifρ2, and b(the ϕ-expert of a(Dabϕ)) considers the premises of tequally acceptable (A≈_b (t, s)), thenashould go on her

evaluation.

Ifρ2, and aknows no expert on ϕ(�b∈A¬Dabϕ), thenashould

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Intuitive principles

C.Expert opinion:

Ifρ2, and b(the ϕ-expert of a(Dabϕ)) considers the premises of tstrictly more acceptable (A>_b(t, s)), thenashould strictly prefert (P>

a (t, s, ϕ)).

Ifρ2, and b(the ϕ-expert of a(Dabϕ)) considers the premises of tequally acceptable (A≈_b (t, s)), thenashould go on her

evaluation.

Ifρ2, and aknows no expert on ϕ(�b∈A¬Dabϕ), thenashould

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Intuitive principles

C.Expert opinion:

Ifρ2, and b(the ϕ-expert of a(Dabϕ)) considers the premises of tstrictly more acceptable (A>_b(t, s)), thenashould strictly prefert (P>

a (t, s, ϕ)).

Ifρ2, and b(the ϕ-expert of a(Dabϕ)) considers the premises of tequally acceptable (A≈_b (t, s)), thenashould go on her

evaluation.

Ifρ2, and aknows no expert on ϕ(�b∈A¬Dabϕ), thenashould

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Intuitive principles

C.Expert opinion:

ρ3 :=ρ1∧((Dabϕ∧A≈b (t, s))∨(

�

b∈A

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Intuitive principles

C.Expert opinion:

ρ3 :=ρ1∧((Dabϕ∧A≈b (t, s))∨(

�

b∈A

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Intuitive principles

C.Expert opinion:

ρ3 :=ρ1∧((Dabϕ∧A≈b (t, s))∨(

�

b∈A

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Intuitive principles

D.Simplicity of arguments:

Ifρ3, and t is simpler thans(Com<(t, s)), thenashould prefer ttos(P_a>(t, s, ϕ).

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Intuitive principles

D.Simplicity of arguments:

Ifρ3, and t is simpler thans(Com<(t, s)), thenashould prefer

ttos(P_a>(t, s, ϕ).

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Intuitive principles

D.Simplicity of arguments:

Ifρ3, and t is simpler thans(Com<(t, s)), thenashould prefer

ttos(P_a>(t, s, ϕ).

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Intuitive principles

D.Simplicity of arguments:

Ifρ3, and t is simpler thans(Com<(t, s)), thenashould prefer

ttos(P_a>(t, s, ϕ).

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

P

(

t, s, ϕ

) operator

The following formula allows us to capture all the principles above:

P_a≥(t, s, ϕ) := (¬(t�ϕ)∧ ¬(s�ϕ))∨(t�ϕ∧ ¬(s�ϕ))

∨(ρ1∧A>a(t, s))

∨(ρ2∧Dabϕ∧A>_b (t, s))

∨(ρ3∧Com≤(t, s))

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

P

(

t, s, ϕ

) operator

The following formula allows us to capture all the principles above:

P_a≥(t, s, ϕ) := (¬(t�ϕ)∧ ¬(s�ϕ))∨(t�ϕ∧ ¬(s�ϕ))

∨(ρ1∧A>a(t, s))

∨(ρ2∧Dabϕ∧A>_b (t, s))

∨(ρ3∧Com≤(t, s))

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

P

(

t, s, ϕ

) operator

The following formula allows us to capture all the principles above:

P_a≥(t, s, ϕ) := (¬(t�ϕ)∧ ¬(s�ϕ))∨(t�ϕ∧ ¬(s�ϕ))

∨(ρ1∧A>a(t, s))

∨(ρ2∧Dabϕ∧A>_b (t, s))

∨(ρ3∧Com≤(t, s))

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

P

(

t, s, ϕ

) operator

The following formula allows us to capture all the principles above:

P_a≥(t, s, ϕ) := (¬(t�ϕ)∧ ¬(s�ϕ))∨(t�ϕ∧ ¬(s�ϕ))

∨(ρ1∧A>a(t, s))

∨(ρ2∧Dabϕ∧A>_b (t, s))

∨(ρ3∧Com≤(t, s))

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Justifying the introduction of

E

t

Preferential omniscience

�P_a≥(cϕ, t, ϕ) for any t∈ T

Intuitive reading: mono-premise arguments asserting the conclusion are always better.

Cause: definition of P_a≥ + standard omniscience

Solution: P_ae≥(t, s, ϕ) :=P_a≥(t, s, ϕ)∧Eat∧Eas

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Justifying the introduction of

E

t

Preferential omniscience

�P_a≥(cϕ, t, ϕ) for any t∈ T

Intuitive reading: mono-premise arguments asserting the conclusion are always better.

Cause: definition of P_a≥ + standard omniscience

Solution: P_ae≥(t, s, ϕ) :=P_a≥(t, s, ϕ)∧Eat∧Eas

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Justifying the introduction of

E

t

Preferential omniscience

�P_a≥(cϕ, t, ϕ) for any t∈ T

Intuitive reading: mono-premise arguments asserting the conclusion are always better.

Cause: definition of P_a≥ + standard omniscience

Solution: P_ae≥(t, s, ϕ) :=P_a≥(t, s, ϕ)∧Eat∧Eas

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Justifying the introduction of

E

t

Preferential omniscience

�P_a≥(cϕ, t, ϕ) for any t∈ T

Intuitive reading: mono-premise arguments asserting the conclusion are always better.

Cause: definition of P_a≥ + standard omniscience

Solution: P_ae≥(t, s, ϕ) :=P_a≥(t, s, ϕ)∧Eat∧Eas

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Justifying the introduction of

E

t

Preferential omniscience

�P_a≥(cϕ, t, ϕ) for any t∈ T

Intuitive reading: mono-premise arguments asserting the conclusion are always better.

Cause: definition of P_a≥ + standard omniscience

Solution: P_ae≥(t, s, ϕ) :=P_a≥(t, s, ϕ)∧Eat∧Eas

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Justifying the introduction of

E

t

Preferential omniscience

�P_a≥(cϕ, t, ϕ) for any t∈ T

Intuitive reading: mono-premise arguments asserting the conclusion are always better.

Cause: definition of P_a≥ + standard omniscience

Solution: P_ae≥(t, s, ϕ) :=P_a≥(t, s, ϕ)∧Eat∧Eas

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

Justifying the introduction of

E

t

Preferential omniscience

�P_a≥(cϕ, t, ϕ) for any t∈ T

Intuitive reading: mono-premise arguments asserting the conclusion are always better.

Cause: definition of P_a≥ + standard omniscience

Solution: P_ae≥(t, s, ϕ) :=P_a≥(t, s, ϕ)∧Eat∧Eas

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

An example

� Anne is looking for her best argument to justify r:≈“it has rained”.

� She has no direct arguments cr available, since her office

has no windows (cr∈/Ea(w)).

� She considers twomodus ponens based arguments (_· arguments) with the following premises “If streets are wet, the it has rained (cw→r). And it is the case that streets are

wet (cw)” and “If humidity is high, the it has rained

(ch→r). And it is the case that air humidity is high (ch)”.

Introduction The logicAE Modelling argument evaluation inAE Conclusions and future work References

An example

� Anne is looking for her best argument to justify r:_≈“it has rained”.

� She has no direct arguments cr available, since her office

has no windows (cr∈/Ea(w)).

� She considers twomodus ponens based arguments (_· arguments) with the following premises “If streets are wet, the it has rained (cw→r). And it is the case that streets are

wet (cw)” and “If humidity is high, the it has rained

(ch→r). And it is the case that air humidity is high (ch)”.