Bipolar Argumentation

Origin. Cayrol and Lagasquie-Schiex (2005) added a second, positive relation to Dung's frameworks, so that one argument may support another as well as attack it, on the ground that pro-and-con reasoning is not reducible to attack alone. What support means was left open, and later work fixed it in incompatible ways: deductive support (Boella, Gabbay, van der Torre and Villata 2010), necessary support (Nouioua and Risch 2010), evidential support (Oren and Norman 2008). Each reading yields its own notion of complex attack and its own semantics.

Models. Two relations over one set of arguments. Attack is Dung's; support is positive and transmits acceptance, or necessity, along a chain. It is the interaction of the two that does the work, generating derived attacks — supported, secondary, mediated, extended — which the semantics then act on. Two strategies compete: flatten the bipolar graph into an ordinary Dung framework by adding the derived attacks and discarding support, or define extensions directly on the bipolar graph.

Formalism.

Bipolar AF: ⟨A, R⁻, R⁺⟩ where: A: arguments. R⁻ ⊆ A × A: attack. R⁺ ⊆ A × A: support.

Attack (R⁻): (a, b) ∈ R⁻: a attacks b. Negative evidence. Dung-style.

Support (R⁺): (a, b) ∈ R⁺: a supports b. Positive evidence. Strengthens b.

Supported attack: a supported-attacks b if: Path a R⁺* c R⁻ b. Support chain then attack.

Secondary attack: a secondary-attacks b if: c supports b (c R⁺ b). a attacks c (a R⁻ c). Undercut support.

Defense types: Direct defense: attack attacker. Supported defense: via support chains. Multiple modes.

Semantics approaches: Reduce to Dung AF. Direct bipolar semantics. Deductive vs necessary support. Various proposals.

Deductive support: If a accepted and a R⁺ b, then b acceptable. Support transfers acceptance. But controversial.

Necessary support: a supports b: b needs a. a rejected implies b rejected. Support as prerequisite.

Mediated attack: a mediated-attacks b if: Path b R⁺* c. a R⁻ c. Attack what b supports.

Example: A = {a, b, c}, R⁻ = {(a,b)}, R⁺ = {(b,c)}. b supports c; a attacks b. Secondary attack: a → c. Flattening adds (a,c) to R⁻. Deductive reading: c falls with b.

Symbols.

Symbol	Unicode	Name	Meaning
⟨	U+27E8	Mathematical left angle bracket	Opens the framework triple
⟩	U+27E9	Mathematical right angle bracket	Closes the framework triple
⁻	U+207B	Superscript minus	Marks the attack relation
⁺	U+207A	Superscript plus	Marks the support relation
⊆	U+2286	Subset	Each relation lies inside A × A
×	U+00D7	Multiplication sign / Cartesian product	Relations are sets of ordered pairs
∈	U+2208	Membership	Ordered pair belongs to a relation
→	U+2192	Rightwards arrow	Edge of the graph, attack or support
*	U+002A	Asterisk / Reflexive transitive closure	R⁺* is a chain of zero or more supports
A	—	Arguments	Nodes of the framework
R⁻	—	Attack relation	Negative edges
R⁺	—	Support relation	Positive edges
BAF	—	Bipolar argumentation framework	The triple ⟨A, R⁻, R⁺⟩

Metatheory. Cayrol and Lagasquie-Schiex (2005) supplied both the derived attacks — supported and secondary — and the reduction strategy that has dominated since: close the two relations into a set of derived attacks, discard support, and evaluate the resulting Dung framework. Which derived attacks are correct depends on what support is taken to mean, and the readings are formally related rather than merely rival, necessary support being the converse of deductive support; the deductive reading flattens by supported and mediated attacks, the necessary reading by secondary and extended attacks. The reduction carries a consequence usually left implicit. Closure is computable in polynomial time and leaves the argument set unchanged, so the acceptance problems stay at the levels Dunne and Bench-Capon established for Dung frameworks, and a bipolar framework expresses nothing an ordinary framework over the same arguments cannot. Evidential support is the exception, since requiring that support trace back to a distinguished evidence node is a condition on sets rather than a further attack.

Applies to. Evidence weighing. Pro/con reasoning. Scientific argumentation. Debates.

Limitations. Support has no fixed meaning: the deductive, necessary and evidential readings each generate a different set of derived attacks, so one graph has different extensions according to a choice the formalism does not record. Where the semantics proceed by flattening, the second relation adds modelling convenience rather than expressive power, and the flattening erases the distinction between an original attack and a derived one, so the result does not show why an argument was defeated. Support is not graded, so a conclusion backed by many arguments fares no better than one backed by a single argument. Cycles in the support relation are left undefined by most proposals rather than resolved. There is no accepted set of postulates for bipolar semantics, so competing proposals cannot be adjudicated as Dung's can by the fixpoint and hierarchy results.