A logic of defeasible argumentation: Constructing arguments in justification logic

2.1.The logic of non-defeasible reasons JT

Justification logics with modal semantics opened up a possibility to study formal systems for non-defeasible epistemic reasons. These systems include an explicit counterpart to the modal Truth axiom: □F→F, read as “If F is known, then F”.99 In order to introduce our system of default reasons, we build upon the existing systems for non-defeasible reasons. In this respect, one can see our strategy as being analogous to the standard default logic approach [3,72] where agents reason from known or certain information. This section gives preliminaries on one of the logics of non-defeasible reasons.

What are the formal ingredients of justification logics? The language of justifications builds on the language of propositional logic, which is augmented by formulas labelled with reason terms (t:F) and a grammar of operations on such terms. Reason terms are constructed from constants and variables, combined with the use of operations on terms. Intuitively, constants justify logical postulates and variables justify contingent facts or inputs outside the structure. The basic operation of standard justification logics is application. Intuitively, application produces a reason term (u·t) for a formula G which is a syntactic “imprint” of the modus ponens step from F→G and F to G for some labelled formulas u:(F→G) and t:F. We say that the term u has been applied to the term t to obtain the term (u·t). As a distinctive feature of justification terms, the history of reasoning steps taken in producing such terms is recorded in their structure.

Another common operation on justification terms is sum. Intuitively, if one takes that a reason term t justifies some formula F, then one is allowed to affix any other reason u term by the use of sum so that the new reason term (t+u) still justifies F. On an epistemic interpretation, this operation can be informally motivated as follows [9, Section 2.2]: t and u might be thought of as two volumes of an encyclopedia that are used as evidence for some statement F. If one volume justifies F, then adding the other volume to the corpus of evidence does not compromise the justification for F. The axioms regulating the sum and application operations are formally described in this section, following the definition of the language.

Since we assume in the next section that an agent starts to reason from indefeasible information, we want our underlying logic to represent “factive” or “truth-inducing” reasons. However, additional constraints on the system are not necessarily needed to introduce the system of default reasons. For the sake of formal clarity, we leave out standard axioms and operations that ensure positive or negative introspection, although these can be easily added. Accordingly, an adequate logical account of factive justifications is the logic JT, a justification logic with the axiom schemes that are explicit analogues of the axiom schemes for the modal logic T.1010 Intuitively, a reader can think of the JT logic as modelling an idealized arguer whose arguments fully exhaust all the possible information regarding claims and who, therefore, gives indisputable reasons for those claims. Note that there are also weaker variants of justification logic that do not assume factivity of reasons. These systems are not adequate for our purposes since we want to build defeasible arguments from a base of fact-inducing reasons — just as in standard default logic where reasoning starts from non-defeasible information or facts [3, p. 19]. After we define the underlying logic that represents non-defeasible argumentation, we develop our novel non-monotonic approach to reasons and provide this logic with the semantics for defeasible argumentation.

3.1.Operational semantics of default justifications

The logic of default justifications we develop here relies on the idea of operational semantics for standard default logics presented in [3]. Let us informally describe the role of the steps of operational semantics in determining acceptable reasons. First, in the operational part of the semantics, default reasons are taken into consideration at face value. Then we check dependencies among default reasons in order to find out what are the non-defeated reasons. Finally, a rational agent includes in its knowledge base only acceptable pieces of information that are based on those reasons that are ultimately non-defeated. An important part of the latter step is an acceptance semantics analogous to the argument acceptance semantics of formal argumentation frameworks.

The basis of operational semantics for a default theory T=(W,D) is the procedure of collecting new, reason-based information from the available defaults. A sequence of default rules Π=(δ0,δ1,…) is a possible order in which a list of default rules without multiple occurrences from D is applied (Π is possibly empty). Applicability of defaults is determined in the following way:

Reasons are brought together in the set of JTCS formulas that represents the current evidence base:

The set In(Π) collects reason-based information that is yet to be determined as acceptable or unacceptable depending on the acceptability of reasons and counter-reasons for formulas.

We need to further specify sequences of defaults that are significant for a default theory T: default processes. For a sequence Π, the initial segment of the sequence is denoted as Π[k], where k stands for the number of elements contained in that segment of the sequence and where k is a minimal number of defaults for the sequence Π. Any segment Π[k] is also a sequence. Intuitively, the set of formulas In(Π) represents an update of the incomplete evidence base W where the new information is not yet taken to be granted. Using the notions defined above, we can now get clear on what a default process is:

The kind of process that we are focusing on here is called closed process and we say that a process Π is closed iff every δ∈D that is applicable to In(Π) is already in Π. For default theories with a finite number of defaults, closure for any process Π is obviously guaranteed by the applicability conditions. However, if a set of defaults is infinite, then this is less-obvious.

To illustrate how the basic notions of the operational semantics work, Fig. 2 shows the process tree for the default theory T0 from our running Example 8.

Fig. 2.

The process tree of T0 from Example 8.

The figure shows that T0 has four closed processes: Π1=(δa,δb), Π2=(δb,δa), Π3=(δb,δc) and Π4=(δc,δb). The In-sets In(Π1) and In(Π2) are equal and JTCS-inconsistent with In(Π3) and In(Π4), which are also equal. Whenever two sets In(Π) and In(Π′) are not equal, they are JTCS-inconsistent. We can already see that JTCS-inconsistent In-sets capture the idea of rebuttal in our semantics, as introduced informally in Example 8. For example, JTCS-inconsistent In-sets In(Π1) and In(Π4) reflect the opposition between the reasons (ua·ta) and (uc·tc). At the level of process trees, however, we are not yet able to explain the attack on (ua·ta) by the undercutting reason (ub·tb). To do so, we need to move further from the semantics of collecting new information.

We have already discussed the key components of our operational semantics that bear some similarity to standard default theories. Now we develop our new argument semantics that builds on the expressivity of the justification logic language. We show that the default variant of the application operation is essential to the way in which we represent arguments and their mutual attacks in justification logic.

3.2.Argumentative schemes and argumentative attacks in justification logic

In a complete specification of I, acceptability of reasons for a formula is determined ex officio by assigning formulas to reasons through the function ∗(·). In contrast, in reasoning from an incomplete evidence base W, a closure ThJTCS(W) is typically underspecified as to whether a reason t is acceptable for a formula F. In “guessing” what a true interpretation is, every default rule introduces a reason term whose structure codifies an application operation step from an unknown justified conditional. For example, in rule δ above, we rely on the justified conditional u:(F→G). Even though this justified conditional is not a part of the rule δ itself, it is the underlying assumption on the basis of which we are able to extend an incomplete evidence base. The propositions of this kind are in one sense taken as rules allowing for default steps, but they are also specific justification logic formulas. They will be referred to as “warrants”, because their twofold role in our system corresponds to Toulmin’s concept of argument warrants.1919 Justification logic defaults give a formal meaning to Toulmin’s philosophical idea that warrants are formulated as statements, even though they function as rules of inference within arguments. Each underlying formula of this kind can be made explicit by means of a function warrant assignment: #(·):D→Fm. The function maps each default rule to a specific justified conditional as follows:

A set of all such underlying warrants of default rules is called Warrant Specification (WS) set.

We will use warrant specification sets that are relativized to default processes:

The extension of the application operation to its defeasible variant opens new possibilities for a semantics of justifications. In particular, it enables reasoning that is not regimented by the standard axioms A1 and A2 of basic justification logic [7, p. 482]. For instance, if a set of JTCS formulas contains both a prima facie reason t and its defeater u, then the set containing a conflict of justifications does not support concatenation of reasons by which t:F→(t+u):F holds for any two terms t and u. In other words, the possibility of a conflict between reasons eliminates the monotonicity property of justifications assumed in the sum axioms (A2).

In explaining the basics of the operational semantics, we qualified the semantics of rebutting attacks as being straightforward. Rebuttal is already captured in the mechanism of multiple extensions known from standard default theories. What requires additional explanation is the semantics of undercutting defeaters. Notice that each formula #(δ) has the format of a justified material conditional. This formula is not a part of a default inference δ itself, but the default application described by δ depends on a conjecture that the conditional holds and the justification assertion cons(δ) encodes this conjecture in the internal structure of the resulting reason term. This brings to attention the following possibility: an evidence base may at the same time contain justified formulas of the type t:F, (u·t):G and v:¬[u:(F→G)], without the evidence base being JTCS-inconsistent.

Although the application axiom A1 does not say that t:F and (u·t):G together entail the formula u:(F→G), there is, intuitively, something wrong with the reason (u·t) justifying the formula G, taken together with t justifying F and v justifying ¬u:(F→G). This new type of opposition among reasons explains why we need to refer to warrant formulas. The co-occurrence of the formulas t:F, (u·t):G and v:¬[u:(F→G)] together is not significant in standard justification logic where reasoning is exclusively regulated by the standard axioms for idealized reasons, such as the axioms of the basic JTCS logic. It only becomes significant with default application.2020 We will now use the presented “reverse engineering” of axiom A1 to model undercut.2121

We have already discussed why the semantics of undercut cannot be reduced to the existence of multiple inconsistent extensions. Nevertheless, JTCS inconsistency is important for undercutting attacks.2222 Notice that adding arbitrary warrants from WST to an evidence base In(Π) could lead to an inconsistent set of JTCS formulas. In Example 8, if we start from any evidence base of T0 and add the warrant ua:(R→T) of δa to it, the union becomes JTCS-inconsistent with both the warrant ub:(L→¬[ua:(R→T)]) of δb and the warrant uc:(W→¬T) of δc. This means that the three warrants are jointly incompatible in the context of default reasoning defined by T0. An agent needs to find out which warrants and, thereby, which reasons prevail in a conflicting set of warrants. This procedure relies on the following definition that captures the above-discussed intuition behind undercut:

We will also specify the way in which sets of JTCS formulas undercut some default reason. This definition will be used in defining different variants of default theory extensions. Sets of justification logic formulas are said to undercut reasons according to the following definition:

One can think of Γ as a set of reasons against which the reason t is tested as a reason that justifies the formula F. This is further elaborated in the semantics of acceptability of reasons. By introducing default reasons through default application and considering rebuttal and undercut among such reasons, it is possible to take an argumentation perspective to justification logic formulas. For example, Fig. 3 provides an intuitive Toulminian interpretation of the default reasoning steps with justification formulas in Example 8, where each step can be associated with a corresponding step in the Toulminian argument scheme.2323

Fig. 3.

Toulminian layout of arguments in Example 8.

Note that the formula (uc·tc):¬T is captioned as a rebuttal of the formula (ua·ta):T, but (ua·ta):T also rebuts (uc·tc):¬T. Their rebuttal relation is symmetric because the two conclusions T and ¬T of the default reasons (ua·ta) and (uc·tc) are contradictory, which means that applying either of the default rules δa and δc blocks the application of the other default rule. Moreover, in Toulmin’s scheme of argumentation, backing is understood as a certification or evidence for the use of a warrant to introduce some conclusion. In justification logic, backing naturally translates into a JTCS derivation (with undischarged assumptions) of a default conditional, and the steps of that derivation are codified in the reason term ua that justifies the conditional R→T. Consider a simple backing for δa from Example 8:

A reader may notice here that the self-referential mechanism in which the language of justification logic treats its own reasoning steps within the language gives a three-layered understanding of arguments. The first layer is an argument seen as a pair of reason terms and formulas, e.g. the formula (ua·ta):T, resulting from the default δa=ta:R::(ua·ta):T(ua·ta):T. In argumentative terms, this layer includes the formula ta:R that represents Toulminian grounds or data. Since the term (ua·ta) formally realizes the default application step of δa, the formula (ua·ta):T will always be explicitly featured in the semantic treatment of the acceptability of the reasoning steps codified by the term (ua·ta). Argumentation semantics for such formulas will be presented in the next section. The second layer gives a wider understanding of the argument. It includes the rule δa together with its warrant formula ua:(R→T). This layer explains the reasoning step from the grounds ta:R to the claim T. It provides an answer to Toulmin’s question [77, p. 90] “How did you get there?”, that is, how to justify that some claim follows from the available data or grounds. Finally, the third layer of the argument for T additionally includes the backing or the unfolded formal structure of the reasoning steps represented by ua that are given in support of the use of the warrant ua:(R→T). Analogously to Toulmin’s argument scheme [77, p. 92], the warrant makes explicit the connection between the grounds and the claim, while the backing explains why the warrant counts as a justified one. Argument warrant can themselves become a part of the reasoning process, especially upon questioning their authority. This is illustrated by the default rule δb in the running example.

3.3.Argument acceptance in justification logic

By introducing default reasons in justification logic it becomes possible not only to use argumentation terminology in talking about formulas of the type t:F but also to give standard abstract argumentation theory conditions of argument acceptance of such formulas. The idea of conflicting default reasons overlaps with abstract argumentation frameworks that treat conflicts between arguments. This section shows that all the formal conditions of argument acceptance as defined in Dung’s framework [30] can be defined for default justifications introduced here. In Section 4, this is used to prove that the logic of default justifications generalizes Dung’s frameworks.

The semantics of reason acceptance starts from characterizing conflict-free sets of JTCS formulas. Note that by introducing default justification, conflicts are not only defined in terms of JTCS inconsistency, but also in terms of undercut from Def 14. The following definition gives conditions for conflict-free sets with respect to undercut:

Note that, if a set of formulas In(Π) for any process Π is conflict-free according to Def. 16, then it is also free from rebuttal for a consistent set of formulas W. To see why, first consider that rebuttal occurs between formulas that are contained in inconsistent evidence bases. Since we know that the conditions under which a default can be applied to an evidence base preserve consistency of each segment In(Π[k]) of In(Π), we also know that In(Π) is rebuttal-free. Consistency preservation of extended evidence bases is established in the following theorem:

The theorem ensures that, for any non-empty process Π, a set of conflict-free formulas In(Π) that an agent could eventually accept is free from any possible conflict.

As stated before, the set W contains certain information and this means that any information from W is always acceptable regardless of what has been collected later on. Therefore, any set of formulas Γ that extends the initial information contains W. To decide whether a consequent of a default δ is acceptable, an agent looks at those sets of reasons that can be defended against all the available counter-reasons. For any set of JTCS formulas Γ, we define the notion of acceptability of a justified formula t:F:

An agent looks at finding a defensible set of arguments in the space of all possible arguments defined by all certain information taken together with the consequents of applicable defaults. Accordingly, for a default theory T=(W,D), an agent considers potential extension sets of JTCS formulas that meet the following conditions:

After considering all the available reasons, an agent accepts only those defeasible statements that can be defended against all the available reasons against these statements.

The two latter definitions introduce the idea of “external stability” of knowledge bases [30, p. 323] into default logic by taking into account that only those reasons that are able to defend themselves against the reasons that question their plausibility eventually become accepted. In addition to that, our operational semantics prompts an implicit revision procedure. Any new default rule that is applicable to the set of formulas In(Π[k]) potentially makes changes to what an agent considered to be acceptable relying on the set of formulas In(Π[k−1]). Before we show this on the formalized example from the beginning of this section, we introduce the idea of default extension for a default theory T. Extension is the fundamental concept in defining logical consequence in standard default theories. We think of preferred extensions as maximal plausible world views based on the acceptability of reasons:

In other words, JTCS-preferred extensions are maximal JTCS-admissible extensions with respect to set inclusion. The existence of JTCS-preferred extensions is universally defined for default theories. To ensure that this result also holds for the case of an infinite number of default rules and infinite closed processes, we make use of Zorn’s lemma and restate it as follows:

The semantics of defeasible reasons enables us to define additional types of extensions that are not necessarily based on the admissibility of reasons. One of them is the stable extension familiar from formal argumentation theory [30]:

The intuition behind the definition is that every reason left outside the accepted set of reasons is attacked. To understand the process semantics workings of the stable extension definition, we can parse this definition into two components. First, it is clear that a stable extension ThJTCS(Γ) undercuts each default reason t for every cons(δ)=t:F such that t:F is not contained in Γ, but δ occurs in a closed process Π of T for which it holds that Γ is a subset of the evidence base In(Π). Intuitively, from those reasons that are applicable within a closed default process, only the reasons that are undercut by ThJTCS(Γ) are left outside. But notice that, secondly, for each default reason u and a formula cons(δ′)=u:G such that Γ and u:G do not co-occur in any potential extension of T, but u:G is included in some potential extension of T, it holds that u also has to be undercut. This means that if δ′ cannot be applied to the default process Π and δ′ occurs in some other closed process Π′, then Γ undercuts u. To see why, take for example the justification assertions t:F=cons(δ) and v:¬F=cons(δ″). For any potential extension Γ⊂In(Π) such that δ∈Γ and δ″ is not applicable to Π due to the inconsistency of the formula req(δ″), ThJTCS(Γ) contains an undercutter for the reason v. In fact, if t:F∈Γ, then ThJTCS(Γ) entails a formula ¬r:(J→¬F) for any formula J∈In(Π) and any reason r. Therefore, it also contains some reason term s that undercuts the warrant of the default rule cons(δ″). This means that inconsistent justification assertions responsible for rebuttal indirectly undercut rebutted reasons. This undercut is further inherited by all the potential default reasons that are inferred from inconsistent default reasons, even if these are not involved in any rebuttal induced by JTCS inconsistency. The following lemma generalizes this observation on the dependence between rebuttal and undercut:

If a potential extension Γ of T undercuts all the formulas left outside, then Γ also has to maximize admissibility with respect to set inclusion. This straightforwardly leads to the following lemma:

We can check that in the red-looking-table example, JTCS-stable and JTCS-preferred extension coincide. Formally, theory T0 has a unique JTCS-stable and JTCS-preferred extension ThJTCS(W0∪{cons(δb),cons(δc)}). Moreover, note that the process Π1=(δa,δb) includes revising the resulting set of acceptable reasons, since the reason (ub·tb) undercuts (ua·ta) being a reason for formula T.

However, JTCS-stable extensions are not universally defined for any default theory T. To show this, we will formalize Pollock’s “pink elephant” example [64, pp. 119–120, 66, pp. 181–182]. This example is an instance of defeasible reasoning with a self-defeating argument. The concept of self-defeat is notorious in argumentation theory. Firstly, suppose that Robert says that the elephant beside him looks pink. Normally, we would take Robert’s testimony to support the conclusion that the elephant is pink. However, Robert suffers from what is known as “pink-elephant phobia”. People in this condition “become strangely disoriented so that their statements about their surroundings cease to be reliable” [66, p. 181]. Therefore, it seems that “if it were true that the elephant beside Robert is pink, we could not rely upon his report to conclude that it is” [66, p. 181].

The theory T1 has a JTCS-preferred extension ThJTCS(W1). However, it has no JTCS-stable extension, because the available reasons cannot form a conflict-free set that attacks all the reasons outside that set. This result conforms to similar results about preferred and stable semantics in abstract argumentation frameworks [30, p. 328]. By the end of the section, we define the theory T3 that shows the same type of a self-defeating argument alongside other arguments. In our default theories, self-defeating arguments do not influence other independent arguments, except in the above-illustrated sense of affecting the existence of stable semantics.

In addition, we can easily define other significant notions of extensions in formal argumentation. In particular, we can define variants of Dung’s [30, p. 329] complete and grounded extension:

Unsurprisingly, the results for different types of extensions from [30] are valid for our default theory extensions.

It does not hold, however, that every JTCS-complete extension is also JTCS-preferred. The following theory T2 is a counterexample. Let the theory be defined as T2=(W2,D2), where W2={p:K,q:L} and D2 consists of the default rules

Considering some proposition as justified might be seen as a function of interacting reasons. Each of the presented JTCS extensions is a method to compute extensions with justified formulas. Moreover, each of the JTCS extension definitions can be used as a way to define a corresponding characterization of logical consequence. Given a particular JTCS extension of a theory T, the formulas contained in that extension are valid formulas for T under that specific JTCS semantics. There are some analogies with the traditional notions of non-monotonic consequence relations. For example, JTCS-grounded extensions correspond to cautious consequence relations describing what a skeptical reasoner would accept for some default theory. In a similar way, JTCS-preferred semantics describes a credulous inference relation. The consequence relation defined by JTCS-stable extension is an interesting case in this context. Although for many default theories JTCS-stable and JTCS-preferred semantics coincides, there are some intuitive grounds to consider JTCS-stable extensions as skeptical in nature. This specifically relates to the demand that the existence of JTCS-stable extensions depends on whether a set of JTCS formulas is able to defeat all other reasons outside that set or not. Such excessive demands on the validity of formulas do not comply to our ordinary intuitions about credulous consequence relations.

To illustrate the differences among the above defined semantics, we will elaborate on an example of a single default theory whose JTCS-grounded, JTCS-complete, JTCS-preferred and JTCS-stable extensions do not coincide, although each of them exists. We define the default theory T3=(W3,D3) with W3={t1:F,t2:H,t3:I} and D3={δ5,δ6,δ7,δ8}, where δ5, δ6, δ7 and δ8 are defined as follows:

Fig. 4.

The process tree of T3.

In total, the theory T3 has six closed processes, as shown in the process tree of T3 displayed in Fig. 4. Building a process tree for our default theories proceeds in the following way: each node of the process tree is labeled with an In-set after a default rule (connecting edges) has been applied. Note that, for each node of the process tree in Fig. 4 and a closed process Π of T3, if a node corresponds to some segment Π[k] of Π we indicate only the formula that has been added to In(Π[k]) as a result of applying an available default rule to In(Π([k−1]). The process tree helps us to check the status of JTCS extensions for T3. The theory has two preferred extensions, namely ThJTCS(W3∪{cons(δ5),cons(δ7)}) and ThJTCS(W3∪{cons(δ8)}. Of the two JTCS-preferred extensions, only ThJTCS(W3∪{cons(δ5),cons(δ7)}) is also JTCS-stable. A skeptical reasoner will only accept ThJTCS(W3), the unique JTCS-grounded extension of T3. Finally, all the three mentioned JTCS closures are JTCS-complete for T3.