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Introduction An analysis of preference between arguments The logical apparatus References

A formal analysis of the notion of preference between deductive arguments

Alfredo Burrieza and Antonio Yuste-Ginel1

University of M´ alaga Department of Philosophy

PhDs in Logic X, Prague, May 2018

1PhD project founded by MECD-FPU 2016 Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Content

1 Introduction 2 An analysis of preference between arguments 3 The logical apparatus 4 References

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

The general question

Let a be an epistemic agent, let ϕ be a proposition and let t and s be a couple of (possibly different) justifications:

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

The general question

Let a be an epistemic agent, let ϕ be a proposition and let t and s be a couple of (possibly different) justifications: (i) Which of the justifications (t or s) should be preferred by a if she pretends to justify ϕ?

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

The general question

Let a be an epistemic agent, let ϕ be a proposition and let t and s be a couple of (possibly different) justifications: (i) Which of the justifications (t or s) should be preferred by a if she pretends to justify ϕ? Note!: we are searching for a normative notion of preference.

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Narrowing down the question

Preference as a context-dependent notion.

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Narrowing down the question

Preference as a context-dependent notion. Possible solution: restrict our analysis to deductive arguments (assuming deductive arguments ⊆ justifications).

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Narrowing down the question

Preference as a context-dependent notion. Possible solution: restrict our analysis to deductive arguments (assuming deductive arguments ⊆ justifications). (ii) Which of the two deductive arguments (t or s) should be preferred by a if she pretends to support ϕ?

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Why to question such a thing

Selection of arguments in an argumentative dialogue

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Why to question such a thing

Selection of arguments in an argumentative dialogue Argument evaluation in formal argumentation theory

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Why to question such a thing

Selection of arguments in an argumentative dialogue Argument evaluation in formal argumentation theory New approach to guaranty the truth of beliefs in order to convert them into knowledge (?).

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Contextualizing the analysis

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Contextualizing the analysis

1 Goal: rational inquiry. Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Contextualizing the analysis

1 Goal: rational inquiry. 2 Logical evaluation ✓. Rhetorical or dialectical

evaluation ✗.

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Contextualizing the analysis

1 Goal: rational inquiry. 2 Logical evaluation ✓. Rhetorical or dialectical

evaluation ✗.

3 No time boundaries (just finiteness). Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Contextualizing the analysis

1 Goal: rational inquiry. 2 Logical evaluation ✓. Rhetorical or dialectical

evaluation ✗.

3 No time boundaries (just finiteness). 4 Claim-related analysis. Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Contextualizing the analysis

1 Goal: rational inquiry. 2 Logical evaluation ✓. Rhetorical or dialectical

evaluation ✗.

3 No time boundaries (just finiteness). 4 Claim-related analysis. 5 Only beliefs are available. Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Contextualizing the analysis

1 Goal: rational inquiry. 2 Logical evaluation ✓. Rhetorical or dialectical

evaluation ✗.

3 No time boundaries (just finiteness). 4 Claim-related analysis. 5 Only beliefs are available. Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Basic thesis

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Basic thesis

Preference between arguments is reducible to other notions:

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Basic thesis

Preference between arguments is reducible to other notions: Syntactic shape or arguments

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Basic thesis

Preference between arguments is reducible to other notions: Syntactic shape or arguments Epistemic attitudes of the agent

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Basic thesis

Preference between arguments is reducible to other notions: Syntactic shape or arguments Epistemic attitudes of the agent Arguments’ source

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Basic thesis

Preference between arguments is reducible to other notions: Syntactic shape or arguments Epistemic attitudes of the agent Arguments’ source Complexity of information

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Basic thesis

Preference between arguments is reducible to other notions: Syntactic shape or arguments Epistemic attitudes of the agent Arguments’ source Complexity of information Quantity of information

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Basic thesis

Preference between arguments is reducible to other notions: Syntactic shape or arguments Epistemic attitudes of the agent Arguments’ source Complexity of information Quantity of information

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Basic thesis

Preference between arguments is reducible to other notions: Syntactic shape or arguments Epistemic attitudes of the agent Arguments’ source Complexity of information Quantity of information

• Informational economy criteria

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Resources used

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Resources used

Basic epistemic logic [6]

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Resources used

Basic epistemic logic [6] Justification logic [1, 3, 2, 4, 5]

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Resources used

Basic epistemic logic [6] Justification logic [1, 3, 2, 4, 5] Logics for belief dependence [9]

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Resources used

Basic epistemic logic [6] Justification logic [1, 3, 2, 4, 5] Logics for belief dependence [9] Preference logic [7, 8]

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Weak and strong preference

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Weak and strong preference

Weak preference: Pa(t, s, ϕ) :≈ “a prefers t at least as much as s in order to support ϕ”.

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Weak and strong preference

Weak preference: Pa(t, s, ϕ) :≈ “a prefers t at least as much as s in order to support ϕ”. Strong preference: P >

a (t, s, ϕ) := Pa(t, s, ϕ) ∧ ¬Pa(s, t, ϕ) :≈ “a

strictly prefers t to s in order to support ϕ”

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Weak and strong preference

Weak preference: Pa(t, s, ϕ) :≈ “a prefers t at least as much as s in order to support ϕ”. Strong preference: P >

a (t, s, ϕ) := Pa(t, s, ϕ) ∧ ¬Pa(s, t, ϕ) :≈ “a

strictly prefers t to s in order to support ϕ” Indifference: P ≈

a (t, s, ϕ) := Pa(t, s, ϕ) ∧ Pa(s, t, ϕ) :≈ “a

considers t and s equally good in order to support ϕ”

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Weak and strong preference

Weak preference: Pa(t, s, ϕ) :≈ “a prefers t at least as much as s in order to support ϕ”. Strong preference: P >

a (t, s, ϕ) := Pa(t, s, ϕ) ∧ ¬Pa(s, t, ϕ) :≈ “a

strictly prefers t to s in order to support ϕ” Indifference: P ≈

a (t, s, ϕ) := Pa(t, s, ϕ) ∧ Pa(s, t, ϕ) :≈ “a

considers t and s equally good in order to support ϕ” Incomparability: P !

a(t, s, ϕ) := ¬Pa(t, s, ϕ) ∧ ¬Pa(s, t, ϕ) :≈ “a

considers t and s incomparable (in terms of preference) in order to support ϕ”

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

The analysis: main idea

When a compares t and s to decide which one is better for justifying ϕ, she puts them to a test consisting of several “filters” or “criteria”. As a result: a will weakly prefer t to s iff t has passed at least as many filters as s.

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

A) First filter: syntactic accuracy

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

A) First filter: syntactic accuracy

Example: trying to justify raising taxes using G¨

• del’s

incompleteness theorem proof.

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

A) First filter: syntactic accuracy

t ≫ ϕ :≈ “t has ϕ as its conclusion”. Note: this a purely syntactic property

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

A) First filter: syntactic accuracy

t ≫ ϕ :≈ “t has ϕ as its conclusion”. Note: this a purely syntactic property ? :≈ “a demands more information related to t and s for deciding which one is better”.

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

A) First filter: syntactic accuracy

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

A) First filter: syntactic accuracy

Relevant cases:

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

A) First filter: syntactic accuracy

Relevant cases: A.1 (¬(t ≫ ϕ) ∧ s ≫ ϕ) →

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

A) First filter: syntactic accuracy

Relevant cases: A.1 (¬(t ≫ ϕ) ∧ s ≫ ϕ) → P >

a (s, t, ϕ)

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

A) First filter: syntactic accuracy

Relevant cases: A.1 (¬(t ≫ ϕ) ∧ s ≫ ϕ) → P >

a (s, t, ϕ)

A.2 (t ≫ ϕ ∧ ¬(s ≫ ϕ)) →

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

A) First filter: syntactic accuracy

Relevant cases: A.1 (¬(t ≫ ϕ) ∧ s ≫ ϕ) → P >

a (s, t, ϕ)

A.2 (t ≫ ϕ ∧ ¬(s ≫ ϕ)) →P >

a (t, s, ϕ)

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

A) First filter: syntactic accuracy

Relevant cases: A.1 (¬(t ≫ ϕ) ∧ s ≫ ϕ) → P >

a (s, t, ϕ)

A.2 (t ≫ ϕ ∧ ¬(s ≫ ϕ)) →P >

a (t, s, ϕ)

A.3 (¬(t ≫ ϕ) ∧ ¬(s ≫ ϕ)) →

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

A) First filter: syntactic accuracy

Relevant cases: A.1 (¬(t ≫ ϕ) ∧ s ≫ ϕ) → P >

a (s, t, ϕ)

A.2 (t ≫ ϕ ∧ ¬(s ≫ ϕ)) →P >

a (t, s, ϕ)

A.3 (¬(t ≫ ϕ) ∧ ¬(s ≫ ϕ)) →P ≈

a (t, s, ϕ)

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

A) First filter: syntactic accuracy

Relevant cases: A.1 (¬(t ≫ ϕ) ∧ s ≫ ϕ) → P >

a (s, t, ϕ)

A.2 (t ≫ ϕ ∧ ¬(s ≫ ϕ)) →P >

a (t, s, ϕ)

A.3 (¬(t ≫ ϕ) ∧ ¬(s ≫ ϕ)) →P ≈

a (t, s, ϕ)

A.4 (t ≫ ϕ ∧ s ≫ ϕ) →

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

A) First filter: syntactic accuracy

Relevant cases: A.1 (¬(t ≫ ϕ) ∧ s ≫ ϕ) → P >

a (s, t, ϕ)

A.2 (t ≫ ϕ ∧ ¬(s ≫ ϕ)) →P >

a (t, s, ϕ)

A.3 (¬(t ≫ ϕ) ∧ ¬(s ≫ ϕ)) →P ≈

a (t, s, ϕ)

A.4 (t ≫ ϕ ∧ s ≫ ϕ) → ?

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

A) First filter: syntactic accuracy

Relevant cases: A.1 (¬(t ≫ ϕ) ∧ s ≫ ϕ) → P >

a (s, t, ϕ)

A.2 (t ≫ ϕ ∧ ¬(s ≫ ϕ)) →P >

a (t, s, ϕ)

A.3 (¬(t ≫ ϕ) ∧ ¬(s ≫ ϕ)) →P ≈

a (t, s, ϕ)

A.4 (t ≫ ϕ ∧ s ≫ ϕ) → ? ρ1 :≈ “first filter has been passed by t and s” := (t ≫ ϕ ∧ s ≫ ϕ)

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

B) Second filter: epistemic attitudes

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

B) Second filter: epistemic attitudes

Aat :≈“ a accepts argument t, i.e., she believes every proposition justified by its components” [4]

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

B) Second filter: epistemic attitudes

Aat :≈“ a accepts argument t, i.e., she believes every proposition justified by its components” [4] Rat :≈“ a rejects argument t, i.e., she believes that some of the proposition justified by its components is false”

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

B) Second filter: epistemic attitudes

Aat :≈“ a accepts argument t, i.e., she believes every proposition justified by its components” [4] Rat :≈“ a rejects argument t, i.e., she believes that some of the proposition justified by its components is false” A>

a (t, s) :≈ “a considers t strictly better than s in doxastic

terms” := (Aat ∧ ¬Aas) ∨ (¬Rat ∧ Ras)

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

B) Second filter: epistemic attitudes

Aat :≈“ a accepts argument t, i.e., she believes every proposition justified by its components” [4] Rat :≈“ a rejects argument t, i.e., she believes that some of the proposition justified by its components is false” A>

a (t, s) :≈ “a considers t strictly better than s in doxastic

terms” := (Aat ∧ ¬Aas) ∨ (¬Rat ∧ Ras) A≈

a := ¬A> a (t, s) ∧ ¬A> a (s, t)

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

B) Second filter: epistemic attitudes

Aat :≈“ a accepts argument t, i.e., she believes every proposition justified by its components” [4] Rat :≈“ a rejects argument t, i.e., she believes that some of the proposition justified by its components is false” A>

a (t, s) :≈ “a considers t strictly better than s in doxastic

terms” := (Aat ∧ ¬Aas) ∨ (¬Rat ∧ Ras) A≈

a := ¬A> a (t, s) ∧ ¬A> a (s, t)

A≥

a (t, s) := A≈ a (t, s) ∨ A> a (t, s)

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

B) Second filter: epistemic attitudes

Aat :≈“ a accepts argument t, i.e., she believes every proposition justified by its components” [4] Rat :≈“ a rejects argument t, i.e., she believes that some of the proposition justified by its components is false” A>

a (t, s) :≈ “a considers t strictly better than s in doxastic

terms” := (Aat ∧ ¬Aas) ∨ (¬Rat ∧ Ras) A≈

a := ¬A> a (t, s) ∧ ¬A> a (s, t)

A≥

a (t, s) := A≈ a (t, s) ∨ A> a (t, s)

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

B) Second filter: epistemic attitudes

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

B) Second filter: epistemic attitudes

Relevant cases:

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

B) Second filter: epistemic attitudes

Relevant cases: B.1 (ρ1 ∧ A>

a (t, s)) →

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

B) Second filter: epistemic attitudes

Relevant cases: B.1 (ρ1 ∧ A>

a (t, s)) →

P >

a (t, s, ϕ)

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

B) Second filter: epistemic attitudes

Relevant cases: B.1 (ρ1 ∧ A>

a (t, s)) →

P >

a (t, s, ϕ)

B.2 (ρ1 ∧ A≈

a (t, s)) →

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

B) Second filter: epistemic attitudes

Relevant cases: B.1 (ρ1 ∧ A>

a (t, s)) →

P >

a (t, s, ϕ)

B.2 (ρ1 ∧ A≈

a (t, s)) →

?

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

B) Second filter: epistemic attitudes

Relevant cases: B.1 (ρ1 ∧ A>

a (t, s)) →

P >

a (t, s, ϕ)

B.2 (ρ1 ∧ A≈

a (t, s)) →

? ρ2 := ρ1 ∧ A≈

a (t, s)

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

C) Third filter: looking for the best source

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

C) Third filter: looking for the best source

Dabϕ :≈ “b is the best advisor or the best source for a regarding topic ϕ” [9]

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

C) Third filter: looking for the best source

Dabϕ :≈ “b is the best advisor or the best source for a regarding topic ϕ” [9] Relevant cases (basic version):

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

C) Third filter: looking for the best source

Dabϕ :≈ “b is the best advisor or the best source for a regarding topic ϕ” [9] Relevant cases (basic version): C.1 (ρ2 ∧ Dabϕ ∧ A>

b (t, s)) →

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

C) Third filter: looking for the best source

Dabϕ :≈ “b is the best advisor or the best source for a regarding topic ϕ” [9] Relevant cases (basic version): C.1 (ρ2 ∧ Dabϕ ∧ A>

b (t, s)) →P > a (t, s, ϕ)

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

C) Third filter: looking for the best source

Dabϕ :≈ “b is the best advisor or the best source for a regarding topic ϕ” [9] Relevant cases (basic version): C.1 (ρ2 ∧ Dabϕ ∧ A>

b (t, s)) →P > a (t, s, ϕ)

C.2 (ρ2 ∧ Dabϕ ∧ A≈

b (t, s)) →

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

C) Third filter: looking for the best source

Dabϕ :≈ “b is the best advisor or the best source for a regarding topic ϕ” [9] Relevant cases (basic version): C.1 (ρ2 ∧ Dabϕ ∧ A>

b (t, s)) →P > a (t, s, ϕ)

C.2 (ρ2 ∧ Dabϕ ∧ A≈

b (t, s)) →

?

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

C) Third filter: looking for the best source

Dabϕ :≈ “b is the best advisor or the best source for a regarding topic ϕ” [9] Relevant cases (basic version): C.1 (ρ2 ∧ Dabϕ ∧ A>

b (t, s)) →P > a (t, s, ϕ)

C.2 (ρ2 ∧ Dabϕ ∧ A≈

b (t, s)) →

? ρ3 := ρ2 ∧ Dabϕ ∧ A≈

b (t, s)

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

D) Fourth filter: complexity of information

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

D) Fourth filter: complexity of information

Len≤(t, s) :≈ “argument t is at least as complex (number of symbols) as s”

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

D) Fourth filter: complexity of information

Len≤(t, s) :≈ “argument t is at least as complex (number of symbols) as s” Len≈(t, s) and Len<(t, s) are defined as expected

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

D) Fourth filter: complexity of information

Len≤(t, s) :≈ “argument t is at least as complex (number of symbols) as s” Len≈(t, s) and Len<(t, s) are defined as expected Relevant cases:

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

D) Fourth filter: complexity of information

Len≤(t, s) :≈ “argument t is at least as complex (number of symbols) as s” Len≈(t, s) and Len<(t, s) are defined as expected Relevant cases: D.1 ρ3 ∧ Len<(t, s) → P >

a (t, s, ϕ)

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

D) Fourth filter: complexity of information

Len≤(t, s) :≈ “argument t is at least as complex (number of symbols) as s” Len≈(t, s) and Len<(t, s) are defined as expected Relevant cases: D.1 ρ3 ∧ Len<(t, s) → P >

a (t, s, ϕ)

D.2 ρ3 ∧ Len≈(t, s) →

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

D) Fourth filter: complexity of information

Len≤(t, s) :≈ “argument t is at least as complex (number of symbols) as s” Len≈(t, s) and Len<(t, s) are defined as expected Relevant cases: D.1 ρ3 ∧ Len<(t, s) → P >

a (t, s, ϕ)

D.2 ρ3 ∧ Len≈(t, s) → ?

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

D) Fourth filter: complexity of information

Len≤(t, s) :≈ “argument t is at least as complex (number of symbols) as s” Len≈(t, s) and Len<(t, s) are defined as expected Relevant cases: D.1 ρ3 ∧ Len<(t, s) → P >

a (t, s, ϕ)

D.2 ρ3 ∧ Len≈(t, s) → ? ρ4 := ρ3 ∧ Len≈(t, s)

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

E) Fifth filter: quantity of atomic information

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

E) Fifth filter: quantity of atomic information

Inf≤(t, s) :≈ “ argument s contains as much atomic information as t”

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

E) Fifth filter: quantity of atomic information

Inf≤(t, s) :≈ “ argument s contains as much atomic information as t” Inf≈(t, s) and Inf<(t, s) are defined as expected

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

E) Fifth filter: quantity of atomic information

Inf≤(t, s) :≈ “ argument s contains as much atomic information as t” Inf≈(t, s) and Inf<(t, s) are defined as expected

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

E) Fifth filter: quantity of atomic information

Inf≤(t, s) :≈ “ argument s contains as much atomic information as t” Inf≈(t, s) and Inf<(t, s) are defined as expected Relevant cases:

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

E) Fifth filter: quantity of atomic information

Inf≤(t, s) :≈ “ argument s contains as much atomic information as t” Inf≈(t, s) and Inf<(t, s) are defined as expected Relevant cases: E.1 (ρ4 ∧ Inf<(t, s)) →

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

E) Fifth filter: quantity of atomic information

Inf≤(t, s) :≈ “ argument s contains as much atomic information as t” Inf≈(t, s) and Inf<(t, s) are defined as expected Relevant cases: E.1 (ρ4 ∧ Inf<(t, s)) → P >

a (t, s, ϕ)

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

E) Fifth filter: quantity of atomic information

Inf≤(t, s) :≈ “ argument s contains as much atomic information as t” Inf≈(t, s) and Inf<(t, s) are defined as expected Relevant cases: E.1 (ρ4 ∧ Inf<(t, s)) → P >

a (t, s, ϕ)

E.2 (ρ4 ∧ Inf≈(t, s)) →

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

E) Fifth filter: quantity of atomic information

Inf≤(t, s) :≈ “ argument s contains as much atomic information as t” Inf≈(t, s) and Inf<(t, s) are defined as expected Relevant cases: E.1 (ρ4 ∧ Inf<(t, s)) → P >

a (t, s, ϕ)

E.2 (ρ4 ∧ Inf≈(t, s)) → P ≈

a (t, s, ϕ)

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

The logic: syntax I

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

The logic: syntax I

DEF (Language) Let Φ (atoms) be numerable, let A = ∅ and finite, define LJBPref = F, T , where F and T are defined by double recursion as follows:

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

The logic: syntax I

DEF (Language) Let Φ (atoms) be numerable, let A = ∅ and finite, define LJBPref = F, T , where F and T are defined by double recursion as follows: ϕ := p |⊥| ϕ → ϕ | t ≫ ϕ | Baϕ | Dabϕ | Inf(t, s) | Len(t, s) where p ∈ Φ1, a ∈ A, t, s ∈ T

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

The logic: syntax I

DEF (Language) Let Φ (atoms) be numerable, let A = ∅ and finite, define LJBPref = F, T , where F and T are defined by double recursion as follows: ϕ := p |⊥| ϕ → ϕ | t ≫ ϕ | Baϕ | Dabϕ | Inf(t, s) | Len(t, s) where p ∈ Φ1, a ∈ A, t, s ∈ T t := cϕ | t + s | t · s where ϕ ∈ F [4]

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

The logic: syntax II

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

The logic: syntax II

DEF (Doxastic acceptance/rejection) Aat :=

cϕ∈sub(t) Baϕ

Rat :=

cϕ∈sub(t) Ba¬ϕ

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

The logic: syntax II

DEF (Doxastic acceptance/rejection) Aat :=

cϕ∈sub(t) Baϕ

Rat :=

cϕ∈sub(t) Ba¬ϕ

Note: it is not generally true that ¬Aat → Rat, i.e., non-acceptance does not imply rejection.

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

The logic: semantics I

DEF (Admissibility [5]) ⊆ T × F. Most concretely, is the smallest relation satisfying the following clauses:

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

The logic: semantics I

DEF (Admissibility [5]) ⊆ T × F. Most concretely, is the smallest relation satisfying the following clauses: (a) cϕ ϕ.

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

The logic: semantics I

DEF (Admissibility [5]) ⊆ T × F. Most concretely, is the smallest relation satisfying the following clauses: (a) cϕ ϕ. (b) If t (ϕ → ψ) and s ϕ, then (t · s) ψ.

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

The logic: semantics I

DEF (Admissibility [5]) ⊆ T × F. Most concretely, is the smallest relation satisfying the following clauses: (a) cϕ ϕ. (b) If t (ϕ → ψ) and s ϕ, then (t · s) ψ. (c) If t ϕ and s ψ, then (t + s) ϕ.

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

The logic: semantics II

DEF (Complexity and atomic info) Define Len : T − → N that maps each term to its number of symbols (excluding parenthesis). Define Atom : T → ℘(Φ) that maps each term to set of different atoms that occur in its certificates. Examples: Atom(cp→(r∨q)) = {p, q, r} Len(Ba((cp∧q→r · cp∧q) ≫ r)) = 14.

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

The logic: semantics III

DEF (Model) A model for L is any tuple M = W, Ra, D, [[·]]a∈A where W = ∅; [[·]] : Φ − → ℘(W); Ra ⊆ W × W and D : A × A × W − → ℘(F) [9] such that it holds:

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

The logic: semantics III

DEF (Model) A model for L is any tuple M = W, Ra, D, [[·]]a∈A where W = ∅; [[·]] : Φ − → ℘(W); Ra ⊆ W × W and D : A × A × W − → ℘(F) [9] such that it holds:

1 ϕ ∈ D(a, b, w) iff ¬ϕ ∈ D(a, b, w) Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

The logic: semantics III

DEF (Model) A model for L is any tuple M = W, Ra, D, [[·]]a∈A where W = ∅; [[·]] : Φ − → ℘(W); Ra ⊆ W × W and D : A × A × W − → ℘(F) [9] such that it holds:

1 ϕ ∈ D(a, b, w) iff ¬ϕ ∈ D(a, b, w) 2 ϕ ∈ D(a, b, w) iff ϕ ∈ D(a, b, w′) for each w′ such that

wRaw′

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

DEF (Truth) Given a model M, define the truth relation as the smallest ⊆ W × F satisfying):

1 w Inf(t, s)

iff |Atom(t)| ≤ |Atom(s)|

2 w |

= Len(t, s) iff Len(t) ≤ Len(s)

3 w t ≫ ϕ

iff t ϕ.

4 w Baϕ

iff ∀w′(wRaw′ ⇒ M, w′ ϕ).

5 w Dabϕ

iff ϕ ∈ D(a, b, w).

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

The logic: proof system

DEF (Proof system) 1 All propositional tautologies 2 t ≫ ϕ whenever t ϕ ¬(t ≫ ϕ) whenever it does not hold t ϕ (admissibility axioms) 3 Ba(ϕ → ψ) → (Baϕ → Baψ) (normality of Ba) 4 Dabϕ ↔ Dab¬ϕ (neutrality of Dab) 5 BaDabϕ ↔ Dabϕ (security about the best advisor)

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

DEF (Proof system) 6 Inf(t, s) whenever |Atom(t)| ≤ |Atom(s)| ¬Inf(t, s) whenever |Atom(t)| |Atom(s)| (atomic information axioms) 7 Len(t, s) whenever Len(t) ≤ Len(s) ¬Len(t, s) whenever Len(s) Len(t) Rules: (MP) ϕ, ϕ → ψ ⊢ ψ (NEC) ϕ ⊢ Baϕ

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

The definition of preference

DEF (Preference between arguments) Pa(t, s, ϕ) := (¬(t ≫ ϕ) ∧ ¬(s ≫ ϕ)) ∨ (t ≫ ϕ ∧ ¬(s ≫ ϕ))∨ ∨(ρ1 ∧ A>

a (t, s))∨

∨(ρ2 ∧ Dabϕ ∧ BaA>

b (t, s))∨

∨(ρ3 ∧ Len<(t, s))∨ ∨(ρ4 ∧ Inf≤(t, s))

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Theorem (Soundness and completeness) The logic JBPref is strongly sound and complete with respect to its semantics. Theorem (Total pre-order on terms) Given a pointed model M, w and an agent a ∈ A we have that P ≥

a induces a total preorder on T .

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Sergei Artemov. The logic of justification. The Review of Symbolic Logic, 1(4):477–513, 2008. Sergei Artemov and Melvin Fitting. Justification logic. In Edward N. Zalta, editor, The Stanford Encyclopedia of

• Philosophy. Metaphysics Research Lab, Stanford University,

winter 2016 edition, 2016. Sergei Artemov and Elena Nogina. Introducing justification into epistemic logic. Journal of Logic and Computation, 15(6):1059–1073, 2005.

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Alexandru Baltag, Bryan Renne, and Sonja Smets. The logic of justified belief change, soft evidence and defeasible knowledge. Logic, Language, Information and Computation, pages 168–190, 2012. Alexandru Baltag, Bryan Renne, and Sonja Smets. The logic of justified belief, explicit knowledge, and conclusive evidence. Annals of Pure and Applied Logic, 165(1):49–81, 2014. Ronald Fagin, Joseph Y Halpern, Yoram Moses, and Moshe Vardi. Reasoning about knowledge. MIT press, 2004.

Burrieza and Yuste-Ginel Preference between deductive arguments

Introduction An analysis of preference between arguments The logical apparatus References

Sven Ove Hansson. Preference logic. In Dov M. Gabbay and Franz Guenthner, editors, Handbook of philosophical logic, pages 319–393. Springer, 2001. Sven Ove Hansson and Till Gr¨ une-Yanoff. Preferences. In Edward N. Zalta, editor, The Stanford Encyclopedia of

• Philosophy. Metaphysics Research Lab, Stanford University,

winter 2017 edition, 2017. Zhisheng Huang. Logics for belief dependence. In International Workshop on Computer Science Logic, pages 274–288. Springer, 1990.

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