An Awareness Epistemic Framework for
Belief, Argumentation and Their Dynamics

Belief and argumentation are two central dimensions of humans’ cognitive architecture. They have received attention from antiquity to nowadays, and from a broad range of disciplines. It is then unsurprising that formal researchers have undertaken the task of modelling both phenomena. Regarding beliefs, there is an important amount of options for capturing some of its formal aspects [23]. These models usually capture what kind of things are believed (typically, propositions or sentences); who believes them (intelligent agents); and, only sometimes, how strong or safe these beliefs are (for instance, in probabilistic models of belief or in plausibility structures [8]). However, most of them fail to capture why agents do believe certain things. This lack motivates the recent trial within the epistemic logic community of capturing the missing justification component. This enterprise has been approached from a variety of methods: justification logic [4, 5, 2, 3], evidence logics based on neighbourhood semantics [13, 12], and its further topological development [6], amongst others. Yet another natural candidate to model justification consists in using conceptual and technical tools coming from argumentation theory (as done, e.g. in [24, 37, 29, 16]).

As to argumentation theory, it is a well-established, interdisciplinary field of research [20]. Since the last few decades, formal argumentation has gained more and more attention within the field of artificial intelligence, and its general advantages have been highlighted several times [11, 33]. Within formal approaches to argumentation, it is frequent to distinguish between abstract approaches (those that consider arguments as primitive, atomic entities) and structured approaches (those that explicitly account for the structure of arguments). For expository purposes, we just mention the popular Dung’s approach to abstract argumentation [19], based on so-called abstract argumentation frameworks, and the ASPIC family of formalisms for structured argumentation, e.g., ASPIC⁺ [31, 32], that will be the main argumentative resources used in this paper.

Recently, some works have taken the first steps to explore and exploit the relations among the two different traditions (epistemic logic and formal argumentation). These can be divided in two groups. On the one hand, there are works using epistemic logic tools to reason about argumentation frameworks [36, 35, 34]. On the other hand, there are works using argumentation tools to provide an (argumentatively inspired) notion of justified belief (the already mentioned [24, 37, 29, 16]). The current paper is inserted in the latter group, and it follows the ideas of [16] that, contrarily to [24, 37, 29], and according to more standard ideas in structured argumentation, decides to model arguments as syntactic entities.

We start by pointing out that the informal relation between argumentation and belief is itself problematic. Arguably, there is a tension between two intuitive principles governing belief formation and argument evaluation. These principles are:

$\mathsf{P1}$

Beliefs are an input for argument evaluation, meaning that arguments with believed premisses are better to those with contingent or even rejected premisses.¹¹1We use the term contingent in its doxastic sense, that is, a sentence is said to be contingent iff it is neither believed nor believed to be false.

$\mathsf{P2}$

Argumentation is an input for belief formation, meaning that rational agents should believe sentences that are ground in good arguments.

The mentioned tension arises when one tries to embrace both principles without any restriction, leading to an infinite regress. A very similar problem can be found in the root of a long-standing debate about the structure of epistemic justification within contemporary epistemology. Foundationalist solutions to such a tension, to which we adhere here, consists in distinguishing between basic (non-inferred) beliefs and non-basic (inferred) beliefs, where the latter inherit the justification from the former [27]. This implies accepting qualified version of both principles, but giving some sort of priority to Introduction over Introduction. Curiously enough, an analogous distinction can be found as one of the basis of the recent argumentative theory of reason advocated by Mercier and Sperber [30]. In this context, basic beliefs are called intuitive beliefs while inferred beliefs are called reflective beliefs (see [38] for a detailed exposition of the distinction).

In the rest of this paper, we follow up the work made in [16], by extending it in three different directions. First, and after recalling the logic introduced there, whose language allows talking about basic beliefs and structured arguments, we provide its sound and complete axiomatisation (Section 2). We then explain how to use this logic for reasoning about explicit basic beliefs and argument-based belief, discussing some of the design choices, as well as depicting some alternatives (Section 3). Finally, we extend the basic fragment of the logic so as to capture different kinds of informational dynamics, illustrating their effects on both types of beliefs (Section 4).

Let us start by recalling the logic introduced in [16]. We follow the traditional order for presentation: syntax, semantics, and proof theory. We assume a countable set of propositional letters $\mathsf{At}$ as fixed from now on. The language $\mathcal{L}$ is defined as the the pair $(\mathsf{F},\mathsf{A})$ of formulas and arguments which are respectively generated by the following grammars:

$\varphi::=p\mid\lnot\varphi\mid(\varphi\land\varphi)\mid\square\varphi\mid\mathsf{aware}(\alpha)\mid\mathsf{conc}(\alpha)=\varphi\mid$

$\mid\mathsf{strict}(\alpha)\mid\mathsf{undercuts}(\alpha,\alpha)\mid\mathsf{wellshap}(\alpha)\qquad p\in\mathsf{At},\alpha\in\mathsf{A}\text{.}$

$\alpha::=\langle\varphi\rangle\mid\langle\alpha_{\_}1,...,\alpha_{\_}n\mathsf{\twoheadrightarrow}\varphi\rangle\mid\langle\alpha_{\_}1,...,\alpha_{\_}n\Rightarrow\varphi\rangle\qquad\varphi\in\mathsf{F},n\geq 1\text{.}$

The rest of Boolean operators ($\to,\lor,\leftrightarrow$) and constants ($\top,\perp$), as well as the dual of $\square$ (noted $\lozenge$), are defined as usual. Arguments of $\mathcal{L}$ have the following informal readings. $\langle\varphi\rangle$ is an atomic argument. Note that this kind of arguments are rather strange in real-life examples, since they have one sole premise and conclusion, and there is not a proper inference step. Mathematically, they can be understood as a one-line proof from $\varphi$ to $\varphi$. As for $\langle\alpha_{1},...,\alpha_{n}\mathsf{\twoheadrightarrow}\varphi\rangle$ (resp. $\langle\alpha_{1},...,\alpha_{n}\Rightarrow\varphi\rangle$), it represents an argument claiming that $\varphi$ follows deductively (resp. defeasibly) from the conclusions of arguments $\alpha_{\_}1,...,\alpha_{\_}n$. As an example of a complex argument consider $\langle\langle\langle\mathsf{Bird}\rangle,\langle\mathsf{Bird}\to\mathsf{Wings}\rangle\mathsf{\twoheadrightarrow}\mathsf{Wings}\rangle\Rightarrow\mathsf{Flies}\rangle$ that informally reads “This has wings, because it is a bird and all birds have wings. Moreover, since it has wings, it presumably (defeasibly) flies”.

Regarding formulas, elements of $\mathsf{At}$ represent factual, atomic propositions. $\square\varphi$ means that the agent implicitly (ideally) believes that $\varphi$. $\mathsf{aware}(\alpha)$ reads “the agent is aware of $\alpha$”. $\mathsf{conc}(\alpha)=\varphi$ reads the “conclusion of $\alpha$ is $\varphi$”. $\mathsf{strict}(\alpha)$ means that $\alpha$ does not contain defeasible inference steps. $\mathsf{undercuts}(\alpha,\beta)$ means that $\alpha$ undercuts $\beta$, that is, $\alpha$ attacks some defeasible inference link of $\beta$. Finally, $\mathsf{wellshap}(\alpha)$ means that $\alpha$ has been constructed properly, that is, all its deductive inference steps are valid and all its defeasible inference steps are accepted by the agent.

We use $\mathsf{SEQ}(\mathsf{F})$ to denote the set of all finite sequences over $\mathsf{F}$. We denote an arbitrary sequence of $n$+1 elements over $\mathsf{F}$ as $((\varphi_{\_}1,...,\varphi_{\_}n),\varphi)$. Sequences of formulas are useful to represent inference steps in the meta-language. Although strongly connected from a conceptual point of view, the sequence $((\varphi_{\_}1,...,\varphi_{\_}n),\varphi)$ is not the same object as, for instance, the object language argument $\langle\langle\varphi_{\_}1\rangle,...,\langle\varphi_{\_}n\rangle\Rightarrow\varphi\rangle$. Let $R=((\varphi_{\_}1,...,\varphi_{\_}n),\varphi)\in\mathsf{SEQ}(\mathsf{F})$ we use $\alpha^{R}$ as a shorthand for $\langle\langle\varphi_{\_}1\rangle,...,\langle\varphi_{\_}n\rangle\Rightarrow\varphi\rangle$. We can see $\alpha^{R}$ as the simplest argument using $R$. As an example, consider the rule $R_{\_}1=((\mathsf{Wings}),\mathsf{Flies})$, we have $\alpha^{R_{\_}1}=\langle\langle\mathsf{Wings}\rangle\Rightarrow\mathsf{Flies}\rangle$, but note that the are infinitely many other arguments using $R_{\_}1$, for instance $\langle\langle\langle\mathsf{Bird}\rangle,\langle\mathsf{Bird}\to\mathsf{Wings}\rangle\mathsf{\twoheadrightarrow}\mathsf{Wings}\rangle\Rightarrow\mathsf{Flies}\rangle$.

Let us define the following meta-syntactic functions for analysing an argument’s structure, taken form ASPIC⁺ [31]:

$\mathsf{Prem}(\alpha)$ returns the premisses of $\alpha$ and it is defined as follows: $\mathsf{Prem}(\langle\varphi\rangle):=\{\varphi\}$,  $\mathsf{Prem}(\langle\alpha_{\_}1,...,\alpha_{\_}n\hookrightarrow\varphi\rangle):=\mathsf{Prem}(\alpha_{\_}1)\cup...\cup\mathsf{Prem}(\alpha_{\_}n)$ where $\hookrightarrow\in\{\mathsf{\twoheadrightarrow},\Rightarrow\}$.

$\mathsf{Conc}(\alpha)$ returns the conclusion of $\alpha$ and it is defined as follows $\mathsf{Conc}(\langle\varphi\rangle):=\{\varphi\}$ and $\mathsf{Conc}(\langle\alpha_{\_}1,...,\alpha_{\_}n\hookrightarrow\varphi\rangle):=\{\varphi\}$ where $\hookrightarrow\in\{\mathsf{\twoheadrightarrow},\Rightarrow\}$.

$\mathsf{sub_{A}}(\alpha)$ returns the subarguments of $\alpha$ and it is defined as follows: $\mathsf{sub_{A}}(\langle\varphi\rangle):=\{\langle\varphi\rangle\}$ and $\mathsf{sub_{A}}(\langle\alpha_{\_}1,...,\alpha_{\_}n\hookrightarrow\varphi\rangle):=\{\langle\alpha_{\_}1,...,\alpha_{\_}n\hookrightarrow\varphi\rangle\}\cup\mathsf{sub_{A}}(\alpha_{\_}1)\cup...\cup\mathsf{sub_{A}}(\alpha_{\_}n)$ where $\hookrightarrow\in\{\mathsf{\twoheadrightarrow},\Rightarrow\}$.

$\mathsf{TopRule}(\alpha)$ returns the top rule of $\alpha$, i.e. the last rule applied in the formation of $\alpha$. It is defined as follows: $\mathsf{TopRule}(\langle\varphi\rangle)$ is left undefined, $\mathsf{TopRule}(\langle\alpha_{\_}1,...,\alpha_{\_}n\mathsf{\twoheadrightarrow}\varphi\rangle)=\mathsf{TopRule}(\langle\alpha_{\_}1,...,\alpha_{\_}n\Rightarrow\varphi\rangle):=$ $((\mathsf{Conc}(\alpha_{\_}1),...,\mathsf{Conc}(\alpha_{\_}n)),\varphi)$.

$\mathsf{DefRule}(\alpha)$ returns the set of defeasible rules of $\alpha$ and it is defined as $\mathsf{DefRule}(\langle\varphi\rangle):=\emptyset$, $\mathsf{DefRule}(\langle\alpha_{\_}1,...,\alpha_{\_}n\mathsf{\twoheadrightarrow}\varphi\rangle):=\mathsf{DefRule}(\alpha_{\_}1)\cup...\cup\mathsf{DefRule}(\alpha_{\_}n)$ and $\mathsf{DefRule}(\langle\alpha_{\_}1,...,\alpha_{\_}n\Rightarrow\varphi\rangle):=\{((\mathsf{Conc}(\alpha_{\_}1),...,\mathsf{Conc}(\alpha_{\_}n)),\varphi)\}\cup\mathsf{DefRule}(\alpha_{\_}1)\cup...\cup\mathsf{DefRule}(\alpha_{\_}n)$.

Let us also define semantic propositional negations, for any $\varphi,\psi\in\mathsf{F}$: $\varphi=\sim\psi$ abbreviates $\mathsf{wellshap}(\langle\langle\varphi\rangle\mathsf{\twoheadrightarrow}\lnot\psi\rangle)\land\mathsf{wellshap}(\langle\langle\psi\rangle\mathsf{\twoheadrightarrow}\lnot\varphi\rangle)$.

Let us now move to semantics. A model for $\mathcal{L}$ is a tuple $M=(W,\mathcal{B},\mathcal{O},\mathcal{D},\mathfrak{n},||\cdot||)$ where:

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  $W\neq\emptyset$ is a set of possible worlds.

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  $\mathcal{B}\subseteq W$ and $\mathcal{B}\neq\emptyset$ is the set of worlds that are doxastically indistinguishable for the agent.

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  $\mathcal{O}\subseteq\mathsf{A}$ is the set of available arguments, also called the awareness set of the agent.

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  $\mathcal{D}\subseteq\mathsf{SEQ}(\mathsf{F})$ is a set of accepted defeasible rules. Moreover, for every $((\varphi_{\_}1,...,\varphi_{\_}n),\varphi)\in\mathcal{D}$ we require that:

  • –

    $\{\varphi_{\_}1,...,\varphi_{\_}n,\varphi\}\nvdash_{\_}0\perp$ (defeasible rules are consistent), where $\vdash_{\_}{0}$ denotes the consequence relation of classical propositional logic, and

  • –

    $\{\varphi_{\_}1,...,\varphi_{\_}n\}\nvdash_{\_}0\varphi$ (defeasible rules are not deductively valid).

• •

  $\mathfrak{n}:\mathsf{SEQ}(\mathsf{F})\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\to\mathsf{At}$ is a (possibly partial) naming function for rules, where $\mathfrak{n}(R)$ informally means “the rule $R$ is applicable”.

• •

  $||\cdot||$ is and an atomic valuation, i.e. a function $||\cdot||:\mathsf{At}\to\wp(W)$.

Let $M=(W,\mathcal{B},\mathcal{O},\mathcal{D},\mathfrak{n},||\cdot||)$ be a model for $\mathcal{L}=(\mathsf{F},\mathsf{A})$. The set of well-shaped arguments $WS^{M}\subseteq\mathsf{A}$ (depending on $\mathcal{D}$ in $M$) is the smallest set fulfilling the following conditions:

1. 1.

   $\langle\varphi\rangle\in WS^{M}$ for any $\varphi\in\mathsf{F}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}$.

2. 2.

   $\langle\alpha_{\_}1,...,\alpha_{\_}n\mathsf{\twoheadrightarrow}\varphi\rangle\in WS^{M}$ iff both $\alpha_{\_}i\in WS^{M}$ for every $1\leq i\leq n$ and $\{\mathsf{Conc}(\alpha_{\_}1),...,\mathsf{Conc}(\alpha_{\_}n)\}\vdash_{\_}0\varphi$.

3. 3.

   $\langle\alpha_{\_}1,...,\alpha_{\_}n\Rightarrow\varphi\rangle\in WS^{M}$ iff both $\alpha_{\_}i\in WS^{M}$ for every $1\leq i\leq n$ and $((\mathsf{Conc}(\alpha_{\_}1),...,\mathsf{Conc}(\alpha_{\_}n)),\varphi)\in\mathcal{D}$.

We drop the superscript $M$ from $WS^{M}$ whenever there is no danger of confusion.

Let $(M,w)$ be a pointed model for $\mathcal{L}$, that is, $M=(W,\mathcal{B},\mathcal{O},\mathcal{D},\mathfrak{n},||\cdot||)$ is a model and $w\in W$. The truth relation, relating pointed models and formulas, is given by:²²2Note that we do not need to consider $\mathsf{undercuts}$ as a primitive operator, since it could be defined through a (simpler) operator that captures the meaning of $\mathfrak{n}$. We make this choice for the sake of succinctness, as well as for studying the axiomatic behaviour of $\mathsf{undercuts}$.

A formula $\varphi$ is said to be valid (noted $\models\varphi$) iff it is true at all pointed models. We use $||\varphi||_{\_}{M}$ to denote the truth-set of $\varphi$, i.e., the set of worlds of $M$ where $\varphi$ is true, and $\mathcal{M}$ to denote the class of all models.

We now present a sound and complete axiomatisation of $\mathcal{L}$ w.r.t. $\mathcal{M}$, a topic that was left out in [16], and that constitutes one of the main technical contributions of the current paper. Although our models provide a compact representation of the needed components for reasoning about basic and argument-based beliefs in a single-agent context, they are rather non-standard from a technical point of view. Besides the strongly syntactic character of some of their elements, their modal components are not defined as usual, therefore the definition of its canonical model cannot be extrapolated straightforwardly. Nevertheless, we can provide an indirect completeness proof (see Appendix A1 for details).

The logic introduced above can be used to study a rich repertoire of doxastic attitudes. We start by discussing basic beliefs, informally representing those that are not grounded on inferential processes. As mentioned, they can also be understood in terms of intuitive beliefs, i.e., those that the agent extracts from a sort of data-base, seen by her as completely trustworthy [38]. As usual in awareness epistemic logic, we have two versions of this notion. On the one hand, we have the implicit (ideal) version of basic beliefs, modelled through $\square\varphi$, that suffers from the extensively discussed problem of logical omniscience (see e.g. [22, Chapter 9]). On the other hand, we have its explicit counterpart, for which we have chosen $\square^{e}\varphi:=\square\varphi\land\mathsf{aware}(\langle\varphi\rangle)$. Note that, like in other logics for implicit and explicit belief, it holds that $\models\square^{e}\varphi\to\square\varphi$. Moreover, under the current semantics, $\square^{e}\varphi$ is equivalent to a schema that resembles another usual option for modelling explicit beliefs (e.g. [40]): $\models\square^{e}\varphi\leftrightarrow\square(\varphi\land\mathsf{aware}(\langle\varphi\rangle))$.

Besides basic beliefs, we can also capture in $\mathcal{L}$ a sort of deductive-explicit belief. Deductive-explicit beliefs are those rooted in a deductive argument s.t. the agent has a basic belief that all its premisses are true. Formally, and following [7], we define doxastic argument acceptance as

$\mathsf{accept}(\alpha):=\bigwedge_{\_}{\varphi\in\mathsf{Prem}(\alpha)}\square\varphi$,

and deductive-explicit belief as

$\mathsf{B}^{\mathsf{D}}(\alpha,\varphi):=\mathsf{accept}(\alpha)\land\mathsf{aware}(\alpha)\land\mathsf{conc}(\alpha)=\varphi\land\mathsf{strict}(\alpha)\land\mathsf{wellshap}(\alpha)$.

Note that $\models\mathsf{B}^{\mathsf{D}}(\alpha,\varphi)\to\square\varphi$ and $\models\square^{e}\varphi\leftrightarrow\mathsf{B}^{\mathsf{D}}(\langle\varphi\rangle,\varphi)$. The first validity shows that deductive-explicit beliefs are a subset of basic-implicit beliefs. The second one shows that basic-explicit beliefs are an extreme case of deductive-explicit beliefs (those that are rooted in the trivial deduction that goes from $\varphi$ to $\varphi$, i.e., in the atomic argument $\langle\varphi\rangle$).

Up to now, we have not gone far from the kind of attitudes that are usually discussed in the awareness logic literature (e.g. in [21, 14, 25, 26]). We now take a small detour through argumentation theory in order to define argument-based beliefs. Roughly speaking, argument-based beliefs are grounded in arguments that may involve non-deductive steps. They can be understood, at least to some extent, in terms of the reflective beliefs of [38]. Recall that we are after formalising the principle Introduction presented in the introduction: the beliefs of a rational agent should be grounded in good arguments. But, what does it means good in this context? Following [10], the very notion of argument strength can be analysed in three different layers or dimensions: the support dimension (how strong is the reason given by an argument to accept its conclusion), the dialectic dimension (how arguments attack and defeat each other), and the evaluative dimension (how the former conflicts are to be solved).

Hence, the first step is to set up a notion of argument strength regarding the support dimension. Formally, we seek to define a preference relation among the arguments of $\mathcal{L}$ that takes into account Introduction (arguments with believed premisses are to be preferred over arguments with premisses that are not believed). In [16], we showed how to do this by splitting all arguments in three preference classes that were based on the basic doxastic attitude of the agent toward the premisses of the arguments. Here, we take a much simpler view, for the sake of brevity, and directly exclude arguments whose premisses are not believed. Both options makes Introduction dependent on Introduction, since in the process of grounding arguments in beliefs and these in turn in new arguments, we arrive at good arguments that are good just because the agent has a basic belief that all their premisses hold. However, inference links must still play a role when determining the relative strength of two arguments, the simplest principle that can be adopted in this regard is captured by the following binary relation among arguments of $\mathcal{L}$: $\alpha\geq\beta:=\mathsf{strict}(\alpha)\lor\lnot\mathsf{strict}(\beta)$. This relation informally corresponds to the idea that, ceteris paribus, deductive arguments are to be preferred to non-deductive ones.

Regarding the dialectic dimension of argument strength, we capture two forms of argumentative defeat, namely, undercutting (attacking a defeasible inference step of any subargument) and successful rebuttal (attacking the conclusion of a less or equally preferred subargument). Formally,

• •

  Undercutting a subargument $\mathsf{undercuts}^{\ast}(\alpha,\beta):=\bigvee_{\_}{\beta^{\prime}\in\mathsf{sub_{A}}(\beta)}\mathsf{undercuts}(\alpha,\beta^{\prime})$.

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  Unrestricted successful rebuttal
  $\mathsf{U}\mathsf{rebuts}(\alpha,\beta):=\lnot\mathsf{strict}(\beta)\land\bigvee_{\_}{\beta^{\prime}\in\mathsf{sub_{A}}(\beta)}(\mathsf{conc}(\alpha)=\varphi\land\mathsf{conc}(\beta^{\prime})=\psi\land\varphi=\sim\psi\land\alpha\geq\beta^{\prime})$.

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  Defeat $\mathsf{defeat}(\alpha,\beta):=\mathsf{undercuts}^{\ast}(\alpha,\beta)\lor\mathsf{U}\mathsf{rebuts}(\alpha,\beta)$.

As discussed in the formal argumentation literature, there is a more restrictive alternative for the notion of rebuttal, requiring the top rule of the attacked subargument to be defeasible³³3See [42] for a discussion about the two possible design choices. Note moreover that the other customary type of attack, i.e. undermining (attacking a premise), makes sense only when non-believed premisses are taken into account.

• •

  Restricted successful rebuttal
  $\mathsf{R}\mathsf{rebuts}(\alpha,\beta):=\lnot\mathsf{strict}(\beta)\land\bigvee_{\_}{\langle\beta_{\_}1,...,\beta_{\_}n\Rightarrow\varphi\rangle\in\mathsf{sub_{A}}(\beta)}(\mathsf{conc}(\alpha)=\psi\land\varphi=\sim\psi)$.

Argumentation frameworks and their semantics [19] are the most studied tool to capture the evaluative dimension of argument strength. We now explain how to incorporate them in the current approach. Let $(M,w)$ be a pointed model for $\mathcal{L}=(\mathsf{F},\mathsf{A})$, we define its associated argumentation framework as $AF^{M}:=(\mathsf{A}^{M},\rightsquigarrow)$, where $\mathsf{A}^{M}:=\{\alpha\in\mathsf{A}\mid M,w\models\mathsf{aware}(\alpha)\land\mathsf{wellshap}(\alpha)\land\mathsf{accept}(\alpha)\}$ and $\rightsquigarrow\subseteq\mathsf{A}^{M}\times\mathsf{A}^{M}$ is given by $\alpha\rightsquigarrow\beta$ iff $M,w\models\mathsf{defeat}(\alpha,\beta)$. We stress the fact that in the domain of our frameworks (i.e., in $\mathsf{A}^{M}$), basic beliefs act as a filter (in the clause $\mathsf{accept}(\alpha)$), instantiating a qualified, unproblematic version of Introduction, namely $\mathsf{P1}^{\prime}$: basic beliefs are an input for argument evaluation. Given a set of possibly conflicting arguments (an argumentation framework), we need a mechanism for the agent to decide which of the arguments are to be selected (an argumentation semantics). We say that a set $B\subseteq\mathsf{A}^{M}$ is conflict-free iff there are no $\alpha,\beta\in B$ s.t. $\alpha\rightsquigarrow\beta$. Moreover, we say that $B\subseteq\mathsf{A}^{M}$ defends $\alpha\in\mathsf{A}^{M}$ iff for every $\gamma\in\mathsf{A}^{M}$, $\gamma\rightsquigarrow\alpha$ implies that there is $\beta\in B$ s.t. $\beta\rightsquigarrow\gamma$. We say that $B\subseteq\mathsf{A}^{M}$ is a complete extension iff it is conflict-free and it contains precisely the elements of $\mathsf{A}^{M}$ that it defends. We say that $B\subseteq\mathsf{A}^{M}$ is the grounded extension of $AF^{M}=(\mathsf{A}^{M},\rightsquigarrow)$ iff it is the smallest (w.r.t. set inclusion) complete extension. We use $GE(AF^{M})$ to denote the grounded extension of $AF^{M}$. As it is well-known, the grounded extension of an argumentation framework always exists and it is moreover unique [19]. The unfamiliar reader is referred to [9] for an extensive discussion on argumentation semantics.

Finally, we use the grounded extension to define the argument-based beliefs of the agent. First, let us extend $\mathcal{L}=(\mathsf{F},\mathsf{A})$ to $\mathcal{L}^{\mathsf{AB}}=(\mathsf{F}^{\mathsf{AB}},\mathsf{A})$ by adding a new kind of formulas $\mathsf{B}(\alpha,\varphi)$ where $\alpha\in\mathsf{A}$ and $\varphi\in\mathsf{F}$. $\mathsf{B}(\alpha,\varphi)$ means that the agent believes that $\varphi$ based on argument $\alpha$. We interpret the new language in the same class of models, by adding the truth clause:

Note that $\models\mathsf{B}^{\mathsf{D}}(\alpha,\varphi)\to\mathsf{B}(\alpha,\varphi)$ and $\models\square^{e}\varphi\to\mathsf{B}(\langle\varphi\rangle,\varphi)$.

We close this section by analysing our notion of argument-based belief under the view of [17]’s rationality postulates. In a nutshell, if no restrictions are imposed, our agent behaves according to a kind of minimal rationality (i.e. she does not explicitly believe in inconsistencies). If, however, we add some ideal assumptions, then she satisfies all [17]’s postulates.

The current framework can throw some light on the relations between dynamics of information, argumentation and doxastic attitudes. We can distinguish several kinds of actions, that have different potential effects on basic and argument-based beliefs. The framework naturally allows for the use of tools imported from dynamic epistemic logic (DEL) [18]. In particular, we can describe these actions using dynamic modalities, for which complete axiomatisations can be then provided by finding a full list of reduction axioms [28, 18, 41]. In order to do so, one first need to show that the rule of replacement of proved equivalents is sound (it preserves validity) in the extended language (see [28] for details). Although this is not the case in $\mathcal{L}$, as it happens with other languages containing awareness operators [21, 25], we can restrict the domain of application of the rule, and it still does the job for axiomatizing certain dynamic extensions. More precisely, we will work with the rule:

(RE)

From $\varphi\leftrightarrow\psi$, infer $\delta\leftrightarrow\delta[\varphi/\psi]$,

with $\delta[\varphi/\psi]$ the result of replacing one or more non-$\star$ occurrences of $\psi$ in $\delta$ by $\varphi$.⁴⁴4A non-$\star$ of $\psi$ in $\delta$ is just an occurrence of $\psi$ in $\delta$ where $\psi$ is not inside the scope of $\star\in\{\mathsf{aware},\mathsf{conc},\mathsf{wellshap},\mathsf{undercuts}\}$. Note that we assume that $\varphi$ is inside the scope of $\mathsf{conc}$ in the formula $\mathsf{conc}(\alpha)=\varphi$. Semantically, this amounts to showing that each of the actions $\mathsf{act}$ we are about to discuss is well defined, in the sense that whenever we compute $M^{\mathsf{act}}$ (the result of executing action $\mathsf{act}$ in model $M$), we stay in the intended class of models. When this does not happen, as it is the case with many DEL actions (e.g. public announcements [18]), one needs to find a set of preconditions for the action. Preconditions works as sufficient conditions for the action to be “safe” i.e., to secure that after executing it, we stay in the intended class of models.

Let us start by defining four different actions. Let $\mathcal{L}=(\mathsf{F},\mathsf{A})$ be given, let $M=(W,\mathcal{B},\mathcal{O},\mathcal{D},\mathfrak{n},||\cdot||)$ be an $\mathcal{L}$-model, let $\alpha\in\mathsf{A}$, let $R\in\mathsf{SEQ}(\mathsf{F})$, and let $\varphi\in\mathsf{F}$. We define:

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  The act of acquiring argument $\alpha$ (resp. forgetting argument $\alpha$) produces the model $M^{\alpha+!}:=(W,\mathcal{B},\mathcal{O}^{\alpha+!},\mathcal{D},\mathfrak{n},||\cdot||)$, where $\mathcal{O}^{\alpha+!}:=\mathcal{O}\cup\{\alpha\}$ (resp. $M^{\alpha-!}:=(W,\mathcal{B},\mathcal{O}^{\alpha-!},\mathcal{D},\mathfrak{n},||\cdot||)$, where $\mathcal{O}^{\alpha-!}:=\mathcal{O}\setminus\{\alpha\}$).

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  The act of accepting the defeasible rule $R$ produces the model $M^{R+!}:=(W,\mathcal{B},\mathcal{O},\mathcal{D}^{R+!},\mathfrak{n},||\cdot||)$, where $\mathcal{D}^{R+!}:=\mathcal{D}\cup\{R\}$.

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  The act of publicly announcing $\varphi$ produces the model $M^{\varphi!}:=(W^{\varphi!},\mathcal{B}^{\varphi!},\mathcal{O},\mathcal{D},\mathfrak{n},||\cdot||^{\varphi!})$, where $W^{\varphi!}:=W\cap||\varphi||_{\_}{M}$; $\mathcal{B}^{\varphi!}:=\mathcal{B}\cap||\varphi||_{\_}{M}$; and $||p||_{\_}{M}^{\varphi!}:=||p||_{\_}{M}\cap||\varphi||_{\_}{M}$ for every atom $p$.