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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-Ginel 1 University of M alaga Department

  1. 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

  2. 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

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

  4. Introduction An analysis of preference between arguments The logical apparatus References Weak and strong preference Weak preference : P a ( 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

  5. Introduction An analysis of preference between arguments The logical apparatus References Weak and strong preference Weak preference : P a ( t, s, ϕ ) : ≈ “ a prefers t at least as much as s in order to support ϕ ”. Strong preference : P > a ( t, s, ϕ ) := P a ( t, s, ϕ ) ∧ ¬ P a ( s, t, ϕ ) : ≈ “ a strictly prefers t to s in order to support ϕ ” Burrieza and Yuste-Ginel Preference between deductive arguments

  6. Introduction An analysis of preference between arguments The logical apparatus References Weak and strong preference Weak preference : P a ( t, s, ϕ ) : ≈ “ a prefers t at least as much as s in order to support ϕ ”. Strong preference : P > a ( t, s, ϕ ) := P a ( t, s, ϕ ) ∧ ¬ P a ( s, t, ϕ ) : ≈ “ a strictly prefers t to s in order to support ϕ ” Indifference : P ≈ a ( t, s, ϕ ) := P a ( t, s, ϕ ) ∧ P a ( s, t, ϕ ) : ≈ “ a considers t and s equally good in order to support ϕ ” Burrieza and Yuste-Ginel Preference between deductive arguments

  7. Introduction An analysis of preference between arguments The logical apparatus References Weak and strong preference Weak preference : P a ( t, s, ϕ ) : ≈ “ a prefers t at least as much as s in order to support ϕ ”. Strong preference : P > a ( t, s, ϕ ) := P a ( t, s, ϕ ) ∧ ¬ P a ( s, t, ϕ ) : ≈ “ a strictly prefers t to s in order to support ϕ ” Indifference : P ≈ a ( t, s, ϕ ) := P a ( t, s, ϕ ) ∧ P a ( s, t, ϕ ) : ≈ “ a considers t and s equally good in order to support ϕ ” Incomparability : P ! a ( t, s, ϕ ) := ¬ P a ( t, s, ϕ ) ∧ ¬ P a ( 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

  8. 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

  9. Introduction An analysis of preference between arguments The logical apparatus References A) First filter: syntactic accuracy Burrieza and Yuste-Ginel Preference between deductive arguments

  10. Introduction An analysis of preference between arguments The logical apparatus References A) First filter: syntactic accuracy Example: trying to justify raising taxes using G¨ odel’s incompleteness theorem proof. Burrieza and Yuste-Ginel Preference between deductive arguments

  11. 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

  12. 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

  13. Introduction An analysis of preference between arguments The logical apparatus References A) First filter: syntactic accuracy Burrieza and Yuste-Ginel Preference between deductive arguments

  14. 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

  15. Introduction An analysis of preference between arguments The logical apparatus References A) First filter: syntactic accuracy Relevant cases: ( ¬ ( t ≫ ϕ ) ∧ s ≫ ϕ ) → A.1 Burrieza and Yuste-Ginel Preference between deductive arguments

  16. Introduction An analysis of preference between arguments The logical apparatus References A) First filter: syntactic accuracy Relevant cases: ( ¬ ( t ≫ ϕ ) ∧ s ≫ ϕ ) → P > A.1 a ( s, t, ϕ ) Burrieza and Yuste-Ginel Preference between deductive arguments

  17. Introduction An analysis of preference between arguments The logical apparatus References A) First filter: syntactic accuracy Relevant cases: ( ¬ ( t ≫ ϕ ) ∧ s ≫ ϕ ) → P > A.1 a ( s, t, ϕ ) A.2 ( t ≫ ϕ ∧ ¬ ( s ≫ ϕ )) → Burrieza and Yuste-Ginel Preference between deductive arguments

  18. Introduction An analysis of preference between arguments The logical apparatus References A) First filter: syntactic accuracy Relevant cases: ( ¬ ( t ≫ ϕ ) ∧ s ≫ ϕ ) → P > A.1 a ( s, t, ϕ ) ( t ≫ ϕ ∧ ¬ ( s ≫ ϕ )) → P > A.2 a ( t, s, ϕ ) Burrieza and Yuste-Ginel Preference between deductive arguments

  19. Introduction An analysis of preference between arguments The logical apparatus References A) First filter: syntactic accuracy Relevant cases: ( ¬ ( t ≫ ϕ ) ∧ s ≫ ϕ ) → P > A.1 a ( s, t, ϕ ) ( t ≫ ϕ ∧ ¬ ( s ≫ ϕ )) → P > A.2 a ( t, s, ϕ ) ( ¬ ( t ≫ ϕ ) ∧ ¬ ( s ≫ ϕ )) → A.3 Burrieza and Yuste-Ginel Preference between deductive arguments

  20. Introduction An analysis of preference between arguments The logical apparatus References A) First filter: syntactic accuracy Relevant cases: ( ¬ ( t ≫ ϕ ) ∧ s ≫ ϕ ) → P > A.1 a ( s, t, ϕ ) ( t ≫ ϕ ∧ ¬ ( s ≫ ϕ )) → P > A.2 a ( t, s, ϕ ) ( ¬ ( t ≫ ϕ ) ∧ ¬ ( s ≫ ϕ )) → P ≈ A.3 a ( t, s, ϕ ) Burrieza and Yuste-Ginel Preference between deductive arguments

  21. Introduction An analysis of preference between arguments The logical apparatus References A) First filter: syntactic accuracy Relevant cases: ( ¬ ( t ≫ ϕ ) ∧ s ≫ ϕ ) → P > A.1 a ( s, t, ϕ ) ( t ≫ ϕ ∧ ¬ ( s ≫ ϕ )) → P > A.2 a ( t, s, ϕ ) ( ¬ ( t ≫ ϕ ) ∧ ¬ ( s ≫ ϕ )) → P ≈ A.3 a ( t, s, ϕ ) A.4 ( t ≫ ϕ ∧ s ≫ ϕ ) → Burrieza and Yuste-Ginel Preference between deductive arguments

  22. Introduction An analysis of preference between arguments The logical apparatus References A) First filter: syntactic accuracy Relevant cases: ( ¬ ( t ≫ ϕ ) ∧ s ≫ ϕ ) → P > A.1 a ( s, t, ϕ ) ( t ≫ ϕ ∧ ¬ ( s ≫ ϕ )) → P > A.2 a ( t, s, ϕ ) ( ¬ ( t ≫ ϕ ) ∧ ¬ ( s ≫ ϕ )) → P ≈ A.3 a ( t, s, ϕ ) A.4 ( t ≫ ϕ ∧ s ≫ ϕ ) → ? Burrieza and Yuste-Ginel Preference between deductive arguments

  23. Introduction An analysis of preference between arguments The logical apparatus References A) First filter: syntactic accuracy Relevant cases: ( ¬ ( t ≫ ϕ ) ∧ s ≫ ϕ ) → P > A.1 a ( s, t, ϕ ) ( t ≫ ϕ ∧ ¬ ( s ≫ ϕ )) → P > A.2 a ( t, s, ϕ ) ( ¬ ( t ≫ ϕ ) ∧ ¬ ( s ≫ ϕ )) → P ≈ A.3 a ( t, s, ϕ ) A.4 ( t ≫ ϕ ∧ s ≫ ϕ ) → ? ρ 1 : ≈ “first filter has been passed by t and s ” := ( t ≫ ϕ ∧ s ≫ ϕ ) Burrieza and Yuste-Ginel Preference between deductive arguments

  24. Introduction An analysis of preference between arguments The logical apparatus References B) Second filter: epistemic attitudes Burrieza and Yuste-Ginel Preference between deductive arguments

  25. Introduction An analysis of preference between arguments The logical apparatus References B) Second filter: epistemic attitudes A a t : ≈ “ a accepts argument t , i.e., she believes every proposition justified by its components” [4] Burrieza and Yuste-Ginel Preference between deductive arguments

  26. Introduction An analysis of preference between arguments The logical apparatus References B) Second filter: epistemic attitudes A a t : ≈ “ a accepts argument t , i.e., she believes every proposition justified by its components” [4] R a t : ≈ “ 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

  27. Introduction An analysis of preference between arguments The logical apparatus References B) Second filter: epistemic attitudes A a t : ≈ “ a accepts argument t , i.e., she believes every proposition justified by its components” [4] R a t : ≈ “ 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” := ( A a t ∧ ¬ A a s ) ∨ ( ¬ R a t ∧ R a s ) Burrieza and Yuste-Ginel Preference between deductive arguments

  28. Introduction An analysis of preference between arguments The logical apparatus References B) Second filter: epistemic attitudes A a t : ≈ “ a accepts argument t , i.e., she believes every proposition justified by its components” [4] R a t : ≈ “ 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” := ( A a t ∧ ¬ A a s ) ∨ ( ¬ R a t ∧ R a s ) A ≈ a := ¬ A > a ( t, s ) ∧ ¬ A > a ( s, t ) Burrieza and Yuste-Ginel Preference between deductive arguments

  29. Introduction An analysis of preference between arguments The logical apparatus References B) Second filter: epistemic attitudes A a t : ≈ “ a accepts argument t , i.e., she believes every proposition justified by its components” [4] R a t : ≈ “ 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” := ( A a t ∧ ¬ A a s ) ∨ ( ¬ R a t ∧ R a s ) 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

  30. Introduction An analysis of preference between arguments The logical apparatus References B) Second filter: epistemic attitudes A a t : ≈ “ a accepts argument t , i.e., she believes every proposition justified by its components” [4] R a t : ≈ “ 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” := ( A a t ∧ ¬ A a s ) ∨ ( ¬ R a t ∧ R a s ) 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

  31. Introduction An analysis of preference between arguments The logical apparatus References B) Second filter: epistemic attitudes Burrieza and Yuste-Ginel Preference between deductive arguments

  32. 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

  33. Introduction An analysis of preference between arguments The logical apparatus References B) Second filter: epistemic attitudes Relevant cases: ( ρ 1 ∧ A > B.1 a ( t, s )) → Burrieza and Yuste-Ginel Preference between deductive arguments

  34. Introduction An analysis of preference between arguments The logical apparatus References B) Second filter: epistemic attitudes Relevant cases: ( ρ 1 ∧ A > P > B.1 a ( t, s )) → a ( t, s, ϕ ) Burrieza and Yuste-Ginel Preference between deductive arguments

  35. Introduction An analysis of preference between arguments The logical apparatus References B) Second filter: epistemic attitudes Relevant cases: ( ρ 1 ∧ A > P > B.1 a ( t, s )) → a ( t, s, ϕ ) ( ρ 1 ∧ A ≈ a ( t, s )) → B.2 Burrieza and Yuste-Ginel Preference between deductive arguments

  36. Introduction An analysis of preference between arguments The logical apparatus References B) Second filter: epistemic attitudes Relevant cases: ( ρ 1 ∧ A > P > B.1 a ( t, s )) → a ( t, s, ϕ ) ( ρ 1 ∧ A ≈ a ( t, s )) → B.2 ? Burrieza and Yuste-Ginel Preference between deductive arguments

  37. Introduction An analysis of preference between arguments The logical apparatus References B) Second filter: epistemic attitudes Relevant cases: ( ρ 1 ∧ A > P > B.1 a ( t, s )) → a ( t, s, ϕ ) ( ρ 1 ∧ A ≈ a ( t, s )) → B.2 ? ρ 2 := ρ 1 ∧ A ≈ a ( t, s ) Burrieza and Yuste-Ginel Preference between deductive arguments

  38. 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

  39. Introduction An analysis of preference between arguments The logical apparatus References C) Third filter: looking for the best source D ab ϕ : ≈ “ b is the best advisor or the best source for a regarding topic ϕ ” [9] Burrieza and Yuste-Ginel Preference between deductive arguments

  40. Introduction An analysis of preference between arguments The logical apparatus References C) Third filter: looking for the best source D ab ϕ : ≈ “ 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

  41. Introduction An analysis of preference between arguments The logical apparatus References C) Third filter: looking for the best source D ab ϕ : ≈ “ b is the best advisor or the best source for a regarding topic ϕ ” [9] Relevant cases (basic version): ( ρ 2 ∧ D ab ϕ ∧ A > C.1 b ( t, s )) → Burrieza and Yuste-Ginel Preference between deductive arguments

  42. Introduction An analysis of preference between arguments The logical apparatus References C) Third filter: looking for the best source D ab ϕ : ≈ “ b is the best advisor or the best source for a regarding topic ϕ ” [9] Relevant cases (basic version): ( ρ 2 ∧ D ab ϕ ∧ A > b ( t, s )) → P > C.1 a ( t, s, ϕ ) Burrieza and Yuste-Ginel Preference between deductive arguments

  43. Introduction An analysis of preference between arguments The logical apparatus References C) Third filter: looking for the best source D ab ϕ : ≈ “ b is the best advisor or the best source for a regarding topic ϕ ” [9] Relevant cases (basic version): ( ρ 2 ∧ D ab ϕ ∧ A > b ( t, s )) → P > C.1 a ( t, s, ϕ ) ( ρ 2 ∧ D ab ϕ ∧ A ≈ C.2 b ( t, s )) → Burrieza and Yuste-Ginel Preference between deductive arguments

  44. Introduction An analysis of preference between arguments The logical apparatus References C) Third filter: looking for the best source D ab ϕ : ≈ “ b is the best advisor or the best source for a regarding topic ϕ ” [9] Relevant cases (basic version): ( ρ 2 ∧ D ab ϕ ∧ A > b ( t, s )) → P > C.1 a ( t, s, ϕ ) ( ρ 2 ∧ D ab ϕ ∧ A ≈ C.2 b ( t, s )) → ? Burrieza and Yuste-Ginel Preference between deductive arguments

  45. Introduction An analysis of preference between arguments The logical apparatus References C) Third filter: looking for the best source D ab ϕ : ≈ “ b is the best advisor or the best source for a regarding topic ϕ ” [9] Relevant cases (basic version): ( ρ 2 ∧ D ab ϕ ∧ A > b ( t, s )) → P > C.1 a ( t, s, ϕ ) ( ρ 2 ∧ D ab ϕ ∧ A ≈ C.2 b ( t, s )) → ? ρ 3 := ρ 2 ∧ D ab ϕ ∧ A ≈ b ( t, s ) Burrieza and Yuste-Ginel Preference between deductive arguments

  46. 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

  47. 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

  48. 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

  49. 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

  50. 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 ρ 3 ∧ Len < ( t, s ) → P > a ( t, s, ϕ ) Burrieza and Yuste-Ginel Preference between deductive arguments

  51. 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 ρ 3 ∧ Len < ( t, s ) → P > a ( t, s, ϕ ) D.2 ρ 3 ∧ Len ≈ ( t, s ) → Burrieza and Yuste-Ginel Preference between deductive arguments

  52. 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 ρ 3 ∧ Len < ( t, s ) → P > a ( t, s, ϕ ) D.2 ρ 3 ∧ Len ≈ ( t, s ) → ? Burrieza and Yuste-Ginel Preference between deductive arguments

  53. 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 ρ 3 ∧ Len < ( t, s ) → P > a ( t, s, ϕ ) D.2 ρ 3 ∧ Len ≈ ( t, s ) → ? ρ 4 := ρ 3 ∧ Len ≈ ( t, s ) Burrieza and Yuste-Ginel Preference between deductive arguments

  54. 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

  55. 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

  56. 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

  57. 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

  58. 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

  59. 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 ( ρ 4 ∧ Inf < ( t, s )) → Burrieza and Yuste-Ginel Preference between deductive arguments

  60. 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 ( ρ 4 ∧ Inf < ( t, s )) → P > a ( t, s, ϕ ) Burrieza and Yuste-Ginel Preference between deductive arguments

  61. 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 ( ρ 4 ∧ Inf < ( t, s )) → P > a ( t, s, ϕ ) E.2 ( ρ 4 ∧ Inf ≈ ( t, s )) → Burrieza and Yuste-Ginel Preference between deductive arguments

  62. 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 ( ρ 4 ∧ Inf < ( t, s )) → P > a ( t, s, ϕ ) E.2 ( ρ 4 ∧ Inf ≈ ( t, s )) → P ≈ a ( t, s, ϕ ) Burrieza and Yuste-Ginel Preference between deductive arguments

  63. Introduction An analysis of preference between arguments The logical apparatus References The logic: syntax I Burrieza and Yuste-Ginel Preference between deductive arguments

  64. 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 L JBPref = �F , T � , where F and T are defined by double recursion as follows: Burrieza and Yuste-Ginel Preference between deductive arguments

  65. 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 L JBPref = �F , T � , where F and T are defined by double recursion as follows: ϕ := p |⊥| ϕ → ϕ | t ≫ ϕ | B a ϕ | D ab ϕ | Inf ( t, s ) | Len ( t, s ) where p ∈ Φ 1 , a ∈ A , t, s ∈ T Burrieza and Yuste-Ginel Preference between deductive arguments

  66. 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 L JBPref = �F , T � , where F and T are defined by double recursion as follows: ϕ := p |⊥| ϕ → ϕ | t ≫ ϕ | B a ϕ | D ab ϕ | Inf ( t, s ) | Len ( t, s ) where p ∈ Φ 1 , a ∈ A , t, s ∈ T t := c ϕ | t + s | t · s where ϕ ∈ F [4] Burrieza and Yuste-Ginel Preference between deductive arguments

  67. Introduction An analysis of preference between arguments The logical apparatus References The logic: syntax II Burrieza and Yuste-Ginel Preference between deductive arguments

  68. Introduction An analysis of preference between arguments The logical apparatus References The logic: syntax II DEF (Doxastic acceptance/rejection) A a t := � c ϕ ∈ sub ( t ) B a ϕ R a t := � c ϕ ∈ sub ( t ) B a ¬ ϕ Burrieza and Yuste-Ginel Preference between deductive arguments

  69. Introduction An analysis of preference between arguments The logical apparatus References The logic: syntax II DEF (Doxastic acceptance/rejection) A a t := � c ϕ ∈ sub ( t ) B a ϕ R a t := � c ϕ ∈ sub ( t ) B a ¬ ϕ Note: it is not generally true that ¬ A a t → R a t , i.e., non-acceptance does not imply rejection. Burrieza and Yuste-Ginel Preference between deductive arguments

  70. 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

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