In Section 3.4 we saw that Carneades is an argumentation model defining burdens and standards of proof. Carneades already has been translated to ASPIC+ [77, 76, 69], so it might look like the translation of these definitions
of burdens and standards of proof to ASPIC+ would be sufficient. How-
ever, Prakken and Sartor [154] argue that the treatment of the burdens and standard of proof by the Carneades argumentation model is not satisfactory, because the shifts of the burden of persuasion are not handled correctly. Therefore Prakken and Sartor define a more specific version of ASPIC+ by instantiating the ASPIC+ model as described in Section 3.3, together with
an adapted definition of defeat to formalise the burden of persuasion more accurately.
3.5.1
Basic definitions
We will first treat definitions which are special cases of the standard ASPIC+ argumentation model.
An argumentation system in ASPIC+ with proof standards/burdens is a special case of Definition 3.30, with instead a language closed under classical negation and a symmetric contrariness relation (classical negation).
Definition 3.65 (Argumentation system (Def 2.1 of [154])). An argumen- tation system is a tuple AS =hL,¯, R, 6i where
• L is a logical language closed under classical negation, • − is a symmetric contrariness relation on
L (p and −p are said to be each other’s contradictories),
• R = Rs∪ Rdis a set of strict (Rs) and defeasible inference rules (Rd)
such thatRs∩ Rd = ∅,
• 6 is a partial preorder on Rd.
Knowledge bases are restricted to contain only necessary axioms and or- dinary premises.
Definition 3.66 (Knowledge base (Def 2.2 of [154])). A knowledge base in an argumentation systemhL,¯, R, 6i is a pair hK, 60i where K ⊆ L and 60 is a partial preorder onKp. HereK is partitioned into two subsets Kp (ordinary
premises) andKa (the assumptions).
Arguments and argumentation theories are constructed in a similar way to before, but now instead depend on the above definitions of knowledge base and argumentation system. For ease of reading we restate the definitions. Definition 3.67 (Arguments (Def 2.3 of [154])). An argument A on the basis of a knowledge basehK, 60i in an argumentation system hL,¯, R, 6i is:
1. ϕ if ϕ∈ K with: P rem(A) ={ϕ}, Conc(A) = ϕ, Sub(A) ={ϕ}.
2. A1, . . . , An → ψ if A1, . . . , An are arguments such that there exists a
strict rule Conc(A1), . . . , Conc(An)→ ψ in Rs,
P rem(A) = P rem(A1)∪ . . . ∪ P rem(An),
Conc(A) = ϕ. Sub(A) = Sub(A1)∪ . . . ∪ Sub(An)∪ {A},
3. A1, . . . , An ⇒ ψ if A1, . . . , An are arguments such that there exists a
defeasible ruleConc(A1), . . . , Conc(An)⇒ ψ in Rd,
P rem(A) = P rem(A1)∪ . . . ∪ P rem(An),
Conc(A) = ϕ,
Sub(A) = Sub(A1)∪ . . . ∪ Sub(An)∪ {A}.
Definition 3.68 (Argumentation theories (Def. 2.4 of [154])). An argumen- tation theory is a triple AT = hAS, KB, i where AS is an argumentation system, KB is a knowledge base in AS and is an argument ordering on the set of all arguments that can be constructed from KB in AS.
Since a symmetric contrariness relation is assumed and furthermore issue premises nor necessary axioms are present in the knowledge base, the defini- tions for attacks and defeats between arguments can be simplified. Further- more, attacks on ordinary premises are disallowed.
Definition 3.69 (Attacks (Def 2.5 of [154])). LetA and B be two arguments. • A undercuts B (on B0) iff Conc(A) =
−B0 for some B0 ∈ Sub(B) of the form B00 1, . . . , B 00 n ⇒ ψ.
• A rebuts B (on B0) iff Conc(A) = −ϕ for some B0 ∈ Sub(B) of the
formB00
1, . . . , B 00 n⇒ ψ.
• A undermines B (on ϕ) iff Conc(A) = −ϕ for some ϕ ∈ P rem(B)∩Ka.
Definition 3.70 (Successful rebuttal and defeat (Def 2.6 of [154])). • A successfully rebuts B if A rebuts B on B0 and A6≺ B0.
• A defeats B iff A undermines, undercut or successfully rebuts B. Again an argumentation framework can be built up by using the corre- sponding definitions of the argument set and defeat relation.
Definition 3.71 (Argumentation framework (Def. 2.7 of [154])). An abstract argumentation framework (AF) corresponding to an argumentation theory AT is a pair hArgs, defeatsi such that:
• Args is the set AAT as defined by Definition 3.67,
• defeats is the relation on Args given by Definition 3.70.
3.5.2
Burden of persuasion and standards of proof
We will now focus on the formalisation of the burden of persuasion and the standards of proof. The standards of proof used are those given in the Carneades model (see Section 3.4, Definition 3.57). The standards of proof are formalised in ASPIC+ by adapting the definition of defeat. This
is achieved by changing the definition of successful rebuttal to a definition that relies on the weights of the arguments and an assigned proof standard. Handling of (inverted) proof burdens is done by extending the definition of an argumentation theory to contain a set of propositions (from the logical language L) which have an explicitly assigned proof burden. From this set inverted (implicit) proof burdens are to be derived.
Definition 3.72 (General bop-argumentation theories (Adapted Def. 5.1 of [154])). A general bop argumentation theory is a tupleAT =hAS, KB, t, B, w, α, β, γi where AS is an argumentation system, KB is a knowledge base inAS as before, and
• t ∈ L (the main topic of the AT )
• I ⊆ B determines a set of inverted proof burdens (we write ibop(ϕ) iff ϕ∈ I),
• w : AAT → R+∪ {0}:
• α, β, γ ∈ R+∪ {0}.
Definition 3.72 has been adapted to explicitly contain inverted proof bur- dens. It is assumed that B was meant to define the default burden of proof dbop, instead of ebop.
For any A ∈ AAT such that Conc(A) has the proof standard beyond
reasonable doubt Rs is assumed to contain rules → ¬A if w(A) < α, and
rules of the forum B1, . . . , Bn → ¬A for any B = B1, . . . , Bn → ¬Conc(A)
such that w(B) > γ. Note that the weight function, w, is assumed to only depend on the content of A, avoiding circularity.
The conditions on which an argument A successfully rebuts an argument B (on B0) depends on the assignment of an explicit or implicit proof burden.
The original definition in Prakken and Sartor is not strictly correct, e.g. allowing arguments with an inverted proof burden to still satisfy the second or last rule.
Definition 3.73 (Original definition of successful rebuttal under burden of persuasion (Def. 5.2 of [154])). Argument A successfully rebuts argument B if A rebuts B on B0 and
1. ibop(Conc(A)) and w(A) > w(B0) +β; or else
2. dbop(Conc(A)) and w(A)6< w(B0); or else
3. w(A) + β 6< w(B0).
My corrected definition is the following:
Definition 3.74 (Successful rebuttal under burden of persuasion (Adapted Def. 5.2 of [154])). Argument A successfully rebuts argument B if A rebuts B on B0 and
1. ibop(Conc(A)) and w(A) > w(B0) +β; or else
2. ¬ibop(Conc(A)) and dbop(Conc(A)) and w(A) 6< w(B0); or else 3. ¬ibop(Conc(A)) and ¬dbop(Conc(A)) and w(A) + β 6< w(B0).