In this paper we explore the relation between an agent's doxastic attitude and her arguments in support of a given claim. Formally, we build further on Dung's argumentation framework in Dung (1995). We start by introducing a logic to reason about binary arguments which are either in favor or against a certain claim. Next we explore a number of notions from standard argumentation theory in our system, including the attack of an argument, the acceptability of an argument, the conflict-freeness of a set of arguments and its admissibility. Our setting will allow us to define new concepts, indicating when an argument perfectly defends a given claim P or when an argument only strategically defends a given claim P . The concept of strategic defensibility is then used to link an agent's arguments to her doxastic attitude. This setting offers a formal characterization of “argument”-based beliefs. As such we address an issue which was raised but not worked out in Dung (1995).
2. s1 ↢P s2 if and only if s1 ↢P s2;
3. if s1 ↢P s2 and f (s1) ⊆ Q ⊆ P , then s1 ↢Q s2;
Explanation of Notation. Taking subsets of W as propositions, our notation f (s) ⊆ P expresses that argument
s supports proposition P . The condition that f (s) ≠ ∅, which we impose on f , requires that there should not be
any argument supporting a contradiction. We use the notation s1 ↢P s2 to express that argument s1 is attacked
by s2 on whether P is the case. The idea of working with accessibility relations labelled by propositions is used
in the literature on conditional logic and in particular in the context of logics for belief revision
Explanation of Frame Conditions. Whether the attack relation holds between two arguments has something to do with the contents that are supported by the two arguments, as is indicated by the frame conditions in definition 1.1. The first condition says that the attack relation ↢Pw only exists between two arguments which both take a different stands with regard to the claim P . The second condition captures the intuition that a debate about P is the same as a debate about P . The third condition also specifies the relation between attack relations with respect to different topics. In particular, it says that if one argument is attacked on its claim that P is the case by certain argument, then it would be attacked by the same argument on its stronger claim.
Given the above frame, we build a model by adding a valuation-map V : Definition 1.2 (Argumentation-Support model). An argumentation-support model is a structure M = ⟨W, AR, {↢P }P ⊆W , f, V ⟩ where • ⟨W, AR, {↢P }P ⊆W , f } is an argumentation-support frame. • V ∶ Prop → ℘(W) assigns a set of possible worlds to each atomic sentence from a given set of atomic propositions Prop.
Our models captures a dual perspective, we can either look at the set of arguments and the claims each of them supports or at the set of possible worlds and the arguments that support the propositions they satisfy. For the purpose of illustration, we formalize the following example in our setting: Example 1.1. In front of a vague picture of an animal, some people are arguing: • s1: The animal in the picture has wings, so it is a bird; • s2: The animal looks like a bat, so it is not a bird, it is a mammal; • s3: The animal looks like a pterosaur, so it is neither a bird nor a mammal, it is a reptile; Let Prop = {b, m, r} where b says that the animal in the picture is a bird, m says that the animal in the picture is a mammal and r says that the animal in the picture is a reptile. Define a model M = ⟨W, AR, {↢P }P ⊆W , f, V ⟩ where • W = {w, v, u} and AR = {s1, s2, s3}; • V (b) = {w}, V (m) = {v}, V (r) = {u}; • ↢{Wv}==↢↢∅{=w,∅u},=↢{{(ws}2=, ↢s3{)v,,(us}3=,{s(2s)1,(,ss12,)s,(2s)1},,s3)},
↢↢{u}=↢{w,v}= {(s2, s3), (s3, s2), (s1, s3)}; • f (s1) = {w}, f (s2) = {v}, f (s3) = {u}.
Readers can check that this model is an argumentation-support model. (Also see the graph in Figure 1.1 for a more intuitive representation.) Note that the attack relation on m forms a cycle between s2 and s3, which is also the case for attack relation on r.
From Arguments to Beliefs. Given the above semantics, we can address the issue raised in
This question is in part also addressed in the work by
Without going into all the details of our formal setting we offer the main definitions and propositions of the argumentation theoretical notions that play a central role. 2
The following definitions are directly based on the related concepts in
X is conflict free for P if X ⊆ nP (X); X is admissible for P if X ⊆ nP (X) and X ⊆ dP (X). Definition 2.2 (Defenders and attackers of an argument). Given an ASF F and an argument s ∈ AR, • Def P (s) = {sn ∈ AR ∣ ∃s0, s1, . . . , sn ∶ ⋀in=0 si ↢P si+1 where s = s0 and n ≠ 0 is an even number } • AttP (s) = {sn ∈ AR ∣ ∃s0, s1, . . . , sn ∶ ⋀in=0 si ↢P si+1 where s = s0 and n is an odd number } Based on these notions, we define the main notion in this paper, which focuses on the status of one single argument rather than on a set of arguments.
Definition 2.3 (Defensibility). Given an ASF F and an argument s ∈ AR, 1. s perfectly defends P if and only if f (s) ⊆ P and {s} ∪ Def P (s) is admissible for P ; 2. s strategically defends P if and only if f (s) ⊆ P and there is a subset S of Def P (s) such that {s} ∪ S is admissible for P .
Having assigned a content to each argument, we can ask how the status of one argument changes according to the change of debated claim.
Proposition 2.1. Given an ASF F and an argument s ∈ AR, • if s strategically defends P , then s strategically defends Q where P ⊆ Q; • even if s perfectly defends P , s may not perfectly defend Q where P ⊆ Q; • even if s perfectly/strategically defends P and also perfectly/strategically defends Q, s may not perfectly/strategically defend P ∩ Q
Readers can check that in the graph on the right-hand side of Figure 2.1, s1 perfectly defends P = {w} bQu)t. does not perfectly defend Q = {w, v} (the solid arrow represents ↢P and the dashed arrow represents ↢ However, s1 can strategically defend Q. And in the graph of the left-hand side of Figure 2.1, s1 perfectly defends P = {w, u} (dotted arrow for ↢P ) and Q = {w, v} (solid arrow for ↢Q), but does not strategically defend P ∩ Q (dashed arrow for P ∩ Q).
We say that P perfectly/strategically dominates a debate about P, if the claim P is perfectly/strategically defended by some argument s while P is not.
Definition 2.4 (Domination). Given an ASF F, P perfectly/strategically dominates a debate about P if and only if there exists at least one argument which perfectly/strategically defends P but no argument which perfectly/strategically defends P .
Although perfect defensibility implies strategic defensibility, strategic domination implies perfect domination: Proposition 2.2. Given an ASF F, if P strategically dominates, then P perfectly dominates, but not the other way around.
Corresponding to Proposition 2.1, we have the following result for domination: Proposition 2.3. Given ASF F, • if P strategically dominates, then Q strategically dominates where P ⊆ Q; however, this does not hold for perfect domination; • even if both P and Q perfectly/strategically dominates, P ∩ Q may not perfectly/strategically dominate.
Proposition 2.2 and Proposition 2.3 indicate that the strategic notions are better behaved. In order to
characterize an agent’s doxastic attitude on the basis of her arguments, we will work further with the notions of
strategic defensibility and domination. Let us first offer the reader an important alternative characterization of
the notion of ‘strategic defensibility’, based on the use of fixed point (in line with the use of greatest fixed point
in
( ) ⊆ P and s ∈ GF P.dP . In the next section, the syntax and
So s strategically defends P if and only if f s
its truth conditions are proposed to reason about these strategic notions, which is inspired by the use of the
two-dimensional logic and modal µ-calculus for studying argumentation theory in
Definition 3.1. Let Prop = {p, q, r, . . . } be a non-empty set of atomic propositions. L is the language generated by the following grammar: α ∶∶=⊺ ∣ p ∣ ¬α ∣ α ∧ α ∣ ⊟α ∣ -β β ∶∶=⊺ ∣ ◻α ∣ ¬β ∣ β ∧ β ∣ [↢α]β ∣ Gfpα where p ∈ Prop. The duals of the operators are defined as standard, such as / for ¬ - ¬ and <↢α> for ¬[↢α]¬.
The language is divided into two parts α and β. The α part is used to state facts about possible worlds, while β part is dedicated to the description of arguments. For example, the intuitive reading of ◻α says that the current argument supports α; [↢α]β says that with respect to the debate about the proposition α: for all arguments directly attacking the current argument, β is the case; Gfpα indicates that the current argument is acceptable in the sense of being in the the greatest fixed point of the function d with respect to α. - is a universal operator ranging over the whole set of arguments, which is used to express whether arguments satisfying certain properties exist. And ⊟ is the universal operator ranging over the whole set of possible worlds, which can be interpreted as a knowledge operator in this setting. We will call formulas that belong to the α part (β part) of this language α-formula (β-formula). When there is no need to make a distinction, ϕ is used to denote formulas in the whole language L. The truth conditions follow below:
Let M be an argumentation-support model, !α"M = {w ∈ W ∣ M, (w, s) ⊧ α}. We omit the subscript M whenever its use is clear from the context.. The truth of ϕ ∈ L is defined as follows: Definition 3.2. Given an argumentation-claim model M, • M, (w, s) ⊧ ⊺ • M, (w, s) ⊧ p iff w ∈ V (p) • M, (w, s) ⊧ ¬α iff M, (w, s) ⊧/ α • M, (w, s) ⊧ α ∧ α′ iff M, (w, s) ⊧ α and M, (w, s) ⊧ α′ • M, (w, s) ⊧ ⊟α iff for all w′ ∈ AR, M, (w′, s) ⊧ α • M, (w, s) ⊧ -β iff for all s′ ∈ AR, M, (w, s′) ⊧ β • M, (w, s) ⊧ ◻α iff f (s) ⊆ !α" • M, (w, s) ⊧ ¬β iff M, (w, s) ⊧/ β • M, (w, s) ⊧ β ∧ β′ iff M, (w, s) ⊧ β and M, (w, s) ⊧ β′ • M, (w, s) ⊧ [↢α]β iff for any s′ ∈ AR such that s ↢!α" s′, M, (w, s′) ⊧ β.
• M, (w, s) ⊧ Gfpα iff s ∈ GF P.d!α" Characterization of Strategic Defensibility and Domination. In this setting, strategic defensibility and domination can be defined:
Str α ∶= ◻ α ∧ Gfpα Dom α ∶= / Str α ∧ ¬ / Str ¬α
Bα ∶= Dom α.
Bϕ → B(α ∨ α′) is valid in the class of argumentation-support frames, but it is not closed under conjunction. Moreover, in this setting,
Bα → BBα and ¬Bα → B¬Bα are valid, since if Bα (¬Bα) is true somewhere in W then !Bα" = W (!¬Bα" = W). Recall that ↢W is empty, so the greatest fixed point of dW is W. Together with the fact that !◻⊺" = W, it implies that - Str Bα (- Str ¬Bα).
We have shown that in the above syntax, the strategic notions can be expressed and and we have given an
epistemic interpretation to these notions. The way we define belief based on arguments is closely related to the
way van Benthem et al. (2012) define belief based on evidence. However, the logical properties of belief defined
in this setting turn out to be different from those in van Benthem et al. (2012), where belief is not only closed
upwards but also closed under conjunction. Moreover, ¬B1 is not valid in the evidence logic
Future Outlook. In the full version of this paper, we provide an axiomatization for our logical system and show how this setting can be extended to a more general multi-agent system, in which we can reason about debates between agents and the role of agents’ epistemic states in these debates.