We address the issue of defining a semantics for deontic argumentation that supports weak permission. Some recent results show that grounded semantics do not support weak permission when there is a conflict between two obligations. We provide a definition of Deontic Argumentation Theory accounting for weak permission, and we recall the result about grounded semantics. Then, we propose a new semantics that supports weak permission.
Suppose there is a norm 1 prescribing under condition , with a sanction 1 for not doing . Concurrently, there is a second norm 2, independent of the first, forbidding when holds, and the violation of this norm is sanctioned by 2. Both norms are at the same hierarchical level in the normative system, meaning there are no clear criteria to determine if one norm overrides the other. This situation constitutes a genuine deontic conflict; specifically, the deontic conclusion that “ is obligatory” is ambiguous due to two conflicting reasoning chains, one supporting this conclusion and another opposing it.
While such scenarios are familiar in deontic reasoning, there is less exploration of how this type of conflict can impact subsequent deontic reasoning chains. Now, consider a third norm 3, independent of 1 and 2, which requires if is permitted. The sanction for violating the third norm is 3.
Now, consider the situation where and are true, but and are not. What sanctions, if any, apply in this case? Additionally, how does the outcome change if 3 requires to be not forbidden or explicitly authorised, rather than merely permitted? Is handling this scenario independent of the argumentation semantics adopted for modelling deontic reasoning?
Deontic logic ofers a powerful framework for analysing and resolving such ambiguities. However, most Deontic Logic systems are not well-suited to deal with conflicting norms. Formal argumentation systems provide the tools to address these challenges. They ofer a structured way to represent deontic statements, reason about their implications, and evaluate the validity of arguments. By explicitly stating premises, rules of inference, and conclusions, formal methods promote transparency, clarity, and consistency in deontic reasoning. This rigorous approach helps mitigate the risks of implicit bias, ad hoc judgments, and inconsistencies that plague informal reasoning. The next sections will explore how formal argumentation frameworks can enhance our ability to navigate the complexities of deontic logic in diverse contexts, from legal judgments to ethical dilemmas.
The layout of the paper is as follows. Section 2 addresses the challenges of deontic argumentation,
particularly focusing on weak permission and its theoretical nuances. In Section 3, we introduce the
notion of Deontic Argumentation Theory and recall the results of [
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Deontic argumentation presents unique challenges. For example, dealing with weak permission within
this context highlights both theoretical nuances and practical implications. As is well known, deontic
logic distinguishes between weak and strong permissions (for an outline [
The concept of weak permission is rooted in an action being permitted merely due to a lack of explicit prohibition. When neither prohibition nor obligation is decisively concluded (as, for example, when the relative strength of conflicting rules is undetermined), both conflicting actions might be weakly permitted.
The implications of weak permission are significant in deontic reasoning systems like Defeasible
Deontic Logic [
In many real-world cases, like judicial decision-making, the role of weak permission becomes critical. Judges and legal practitioners frequently encounter scenarios where multiple, potentially conflicting obligations and permissions must be weighed and interpreted. The arguments must consider all potential influences and outcomes, requiring a robust framework for integrating diverse types of permissions and obligations.
Our research identifies and addresses these challenges through the design of a sophisticated argumentation framework that integrates deontic reasoning with formal argumentation techniques. This approach looks to provide clarity in situations where weak permissions exist alongside conflicting obligations.
Assume we work with a Defeasible Deontic Logic in which deontic rules have the form : ⇒ obl()
: obl(), ⇒ perm().
If is applicable, we derive obl(); if we know that obl() and are the case, then we conclude that is permitted, i.e., that perm(). Suppose that we also have the following rule:
: , ⇒ obl(¬).
If both and are applicable and we do not have additional information, we cannot conclude either obl() or obl(¬) but that both and ¬ are weakly permitted, i.e., perm() and perm(¬).
Now, consider judicial adversarial scenarios where the lack of the prohibition is derived during a legal proceeding. Multiple scenarios can be identified.
Article 520 of the Italian Code of Criminal Procedure states that a judge must issue a sentence of acquittal if (i) the fact does not constitute a crime or (ii) is not provided by law as a crime. In this case, the absence of a prohibition is derived from the absence of a crime. In the first case, there is an obligation or prohibition, but the facts upon which the proceeding is based are not covered by the obligation or prohibition. Thus, we see that obl() is in force, but (the fact of the case) holds but ̸≡ , something which is established via argumentation.
For instance, in the Italian legal system, a public oficer who intentionally signs a forged document is punished with imprisonment. This means that the person is obligated not to sign a document knowing it is forged. However, if the person accidentally signs the document or signs it without knowing it is forged, the fact is not covered by the prohibition. In this case, we can say that the action of signing is weakly permitted because it does not result in punishment.
A more complicated example is the following.
Example 1. In 2019, the Sea Watch 3, an NGO vessel, was on a mission in the Mediterranean Sea to rescue migrants. The Italian Government issued a decree that banned the Sea Watch 3 from rescue operations (for migrants) in the Italian Contiguous Zone waters.
1 : distress , proximity ⇒ obl(assistance) 2 : SeaWatch, migrants , ItalianContiguousZone ⇒ obl(¬assistance) Maritime Law stipulates that a vessel in proximity to a distress vessel must assist the vessel in distress (rule 1). On the other hand, 2 encodes the prohibition on rescuing migrants in Italian Contiguous Zone waters. Moreover, Maritime Law requires vessels permitted to refrain from assisting vessels in distress to alert the closest relevant authorities and keep clear of the rescue area.
3 : perm(¬assistance) ⇒ obl(alertAuthorities )& obl(keepClear ) Suppose a migrant vessel is in distress in the Italian Contiguous Zone, and Sea Watch 3 is nearby. Does Sea Watch 3 have the obligation to alert the authorities and keep clear of the rescue area? Here, we have a conflict over two opposite norms, and it is unclear which one takes precedence over the other. 1 Also, suppose that Sea Watch 3 alerted the competent authorities but remained in the operation area (and eventually assisted with the rescue operation). Do the Sea Watch 3 actions contravene the norms? As we said, there are two options. The permission does not follow from the conflict. In this case, the Sea Watch is not required to leave the area, and it does not contravene the norms. In the second approach, the permission from the conflict follows (as a weak permission). Leaving the area is a legal requirement, and Sea Watch 3 does not comply with the norms.
What about if weak permission does not follow from a conflict? We can consider an alternative formulation of the last provision. Assume that it stipulates that vessels for which the obligation to assist does not hold must keep clear of the rescue area.
3′ : not obl(assistance) ⇒ obl(alertAuthorities )& obl(keepClear ) Given the conflict, we can argue that the obligation to ofer assistance does not hold, and the Sea Watch 3 has to follow the obligations given by 3′.
The primary motivation for our research is the limitations in existing argumentation semantics as applied to deontic logic, such as grounded and stable semantics, which inadequately handle weak permissions under conflict. These methods often prevent justifications for weak permission due to their inherent limitations in conflict resolution, notably when multiple deontic rules apply simultaneously.
To address these limitations, we propose an argumentation semantics that efectively incorporates weak permission by allowing a more nuanced evaluation of arguments. Our approach ensures that both obligations and permissions are considered robustly within argumentative frameworks, promoting a balance between legal precision and interpretative flexibility.
We recall the results of [
The language of a deontic argument is built from a set of literals (Lit), where a literal is either an atom () or its negation (¬). In addition, we extend the language with deontic literals. The set of deontic literals is defined as
{dop(), not dop()| ∈ Lit} where dop is a deontic operator, more precisely dop ∈ {obl, perm, perm}. Given a literal , we use ∼ to denote the complement of ; more precisely, if is an atomic proposition, then ∼ = ¬ . If is a negated atomic proposition = ¬, then ∼ = .
Arguments are built from rules, where a rule has the following format: where {1, . . . , } is a (possibly empty) set of literals and deontic literals, and is either a literal or deontic literal but not a weak permission (i.e., ̸= perm(), otherwise it would be an explicit permission).
1, . . . ⇒
(, ) where is a (finite and possibly empty) set of literals (the fact or assumption of the theory), and is a (finite) set of rules.
The key concept of an argumentation theory is the notion of an argument. Definition 2 below defines what an argument is. Each argument has associated with it, its conclusion () and its set of sub-arguments Sub().
Definition 2. Given a Deontic Argumentation Theory (, ), is an argument if has one of the following forms: 1. = perm() for any literal ∈ ℒ, the conclusion of the argument () = perm(), and
Sub() = {}. 2. = for ∈ ; () = and Sub() = {}. 3. = 1, . . . , ⇒ , if there is a rule 1, . . . , ⇒ in such that for all ∈ {1, . . . , } there is an argument such that () = ; () = and Sub() = {}∪Sub(1)∪· · ·∪Sub( ). 4. = ⇒ perm(), if is an argument such that () = obl(); () = perm(), and
Sub() = {} ∪ Sub().
We call the weak permission arguments (arguments defined by clause (1) above) imaginary arguments. Arguments defined by the other clauses are called natural arguments. The set of all arguments for a Deontic Argumentation Theory is Args.
Condition (1) encodes the idea that weak permission is the failure to derive an obligation to the contrary (more on this when we discuss the notion of attack between arguments). Thus, by default, every literal is potentially weakly permitted, and we form an argument for this type of conclusion. Condition (2) gives the simplest form of an argument. We have an argument for if is one of the assumptions/facts of a case/theory. Condition (3) allows us to form arguments by forward chaining rules. Thus, we can form an argument from a rule, if we have arguments for all the elements of the body of the rule. The way the condition is written allows us to create arguments from rules with an empty body. Finally, condition (4) corresponds to the D axiom of Standard Deontic Logic.
Definition 3. Let and be arguments. attacks ( > ) if 1. = perm() and () = obl(∼); 2. ∃′ ∈ Sub(), (′) = and () = ∼; 3. ∃′ ∈ Sub(), (′) ∈ {obl(), perm()} and () = obl(∼); 4. ∃′ ∈ Sub(), (′) = obl(), and () = perm(∼).
As we alluded to in Section 1, the idea of weak permission is the negation as failure of the obligation to the contrary. Thus, in Definition 2, we create an argument for the weak permission for any literal ; however, this argument is attacked by any argument for obl(∼) (condition (1) above and notice this is the only case where the attack is not symmetrical). The rest of the conditions define an attack when the two arguments have opposite conclusions: condition (2) covers the case of plain literals, while conditions (3) and (4) are reserved for deontic literals. Specifically, we have opposite deontic conclusions when one of the two is an obligation for a literal, and the other is either an obligation or a permission for the opposite literal. Furthermore, an argument attacks another argument when the conflict is on the conclusion of the second argument (this corresponds to the notion of rebuttal) or when there is a conflict with one of the sub-arguments of the attacked argument (known as undercutting attack). Example 2. Let us consider the Deontic Argumentation Theory (, ) where = {} and contains the following rules: 1 : ⇒ obl() 2 : ⇒ obl(¬) 3 : obl() ⇒ obl() 4 : ⇒ obl(¬) 5 : perm() ⇒ 6 : perm() ⇒ The set of natural arguments contains 0 : (given that is a fact) and the following arguments built from the above rules: 1 : ⇒ obl() 4 : ⇒ obl(¬) 7 : 3 ⇒ perm()
2 : ⇒ obl(¬) 5 : 1, ⇒ perm() 8 : 4 ⇒ perm(¬) 10 : 7 ⇒
3 : 1 ⇒ obl() 6 : 2 ⇒ perm(¬) 9 : 0, 1 ⇒ Moreover, for each propositional letter , , , , we have two imaginary arguments, one for the letter and one for its negation. Specifically, we consider the following imaginary arguments: 1 : perm() 2 : perm(¬) 3 : perm() 4 : perm(¬) Arguments 0, 1, and 4 do not have proper subarguments, and have only themselves as their subarguments.
Sub(3) = {1, 3}
Sub(5) = {1, 5}
Sub(6) = {2, 6} Sub(7) = {1, 3, 7}
Sub(8) = {4, 8} Sub(9) = {0, 1, 9}
Sub(10) = {1, 3, 7, 10} The attack relation is as follows (for each argument , we list the arguments attacks): 1 : 2, 6, 2 3 : 4, 8, 4 5 : 2, 6 7 : 4, 8 2 : 1, 3, 5, 1, 10 4 : 3, 7, 3, 10 6 : 1, 3, 5, 7, 10 8 : 3, 7, 10.
We are ready to recall the standard definitions of Dung argumentation semantics [
• is a stable extension if is conflict free and ∀ ∈/ , ∃ ∈ such that > .
Definition 5. Let = (, ) be a Deontic Argumentation Theory, an argument is sceptically justified under a semantic if ∈ for all sets of arguments that are an extension under . Definition 6. A literal or a deontic literal ∈ ℒ is a Justified conclusion under a semantics if for every extension under , there is an argument such that ∈ and () = .
Example 3. Let us consider a Deontic Argumentation Theory where = ∅ and contains the two rules 1 : ⇒ obl() 2 : ⇒ obl(¬) This theory has the following arguments: 1 : perm() 2 : perm(¬) 3 : ⇒ obl() 4 : ⇒ obl(¬) 5 : 3 ⇒ perm() 6 : 4 ⇒ perm(¬) For the attack relation, we have the following instances 3 > 2 4 > 1 3 > 4 4 > 3 3 > 6 4 > 5 5 > 4 6 > 3 5 > 6 6 > 5 It is easy to verify that {} is a complete extension (and trivially, it is the minimal complete extension w.r.t. set inclusion). Thus, it is the grounded extension of the theory. Accordingly, there is no argument in the grounded extension such that its conclusion is either perm() or perm(¬). Hence, perm() and perm(¬) are not justified conclusions under the grounded semantics.
When we consider the stable semantics, we have the following two extensions: {1, 3, 5} {2, 4, 6} Clearly, perm() is a conclusion of the first extension but not of the second one; conversely, perm(¬) is a conclusion of the second extension but not of the first one. Consequently, perm() and perm(¬) are not justified conclusions under the stable semantics.
The above example should sufice to show that weak permission is not supported by grounded and stable semantics when a deontic conflict exists: we have a scenario where we fail to conclude that an obligation, but at the same time, we cannot conclude the weak permission of the opposite. However, we can generalise the result; the result holds for any theory with a conflict between two applicable obligation rules. This is formalised by the following definition.
Definition 7. A Deontic Argumentation Theory is conflictual when it contains a pair of rules 1, . . . , ⇒ obl() and 1, . . . ⇒ obl(¬), such that there are arguments with conclusion , 1 ≤ ≤ , and with conclusion , 1 ≤ ≤ . We will call the conflicted literal.
Theorem 1. Let = (, ) be a conflictual theory. For any conflicted literal are not justified conclusions under grounded and stable semantics.
Proof. Let be a conflicted literal. By the definition of argument, the theory theory is conflicted, we have the following arguments: , perm(), perm(¬) and the fact that the 1 : perm() 2 : perm(¬) 3 : 1, . . . , ⇒ obl() 4 : 1, . . . , ⇒ obl(¬) such that 3 > 2 and 4 > 1. Given that there are arguments attacking 1 and 2, these two arguments are not in the minimal complete extension. Accordingly, perm() and perm(¬) are not justified conclusions under the grounded semantics.
Given the attack relationship among 1, 2, 3 and 4, we can conclude that there are at least two extensions 1 and 2 such that 1, 3 ∈ 1 and 2, 4 ∈/ 1, and 2, 4 ∈ 2 and 1, 3 ∈/ 2. Hence, there is an extension where no argument has perm() as its conclusion, and there is an extension where no argument has perm(¬) as its conclusion. Therefore, perm() and perm(¬) are not justified conclusions under the stable semantics.
We take the notions defined in the previous section and extend it to a semantics that can deal with weak
permission. The new semantics is based on and extends the argumentation semantics for Defeasible
Logic [
The first step is to use the notion of attack between arguments to define the notions of support and undercut. Support means that all proper subarguments of an argument are in the set of arguments that supports it; undercut means that the set of arguments that undercuts an argument supports an argument that attacks a proper subargument of the argument. This means that the premises of the undercut argument do not hold.
Definition 8. A set of arguments supports an argument if every proper subargument of is in . Definition 9. A set of arguments undercuts an argument if supports an argument that attacks a proper natural subargument of .
Example 4. Consider a Deontic Argumentation Theory where contains the following rules: 1 : ⇒ 2 : ⇒ obl() 3 : perm(¬) ⇒ 4 : perm(¬) ⇒
At this stage, we can define the notion of wp-acceptable and wp-rejected arguments. The main diference is that we have to distinguish between natural and imaginary arguments. An imaginary argument is an argument for a weak permission, and weak permission means that it is impossible to have a valid argument for the obligation to the contrary. In other words, weak permission succeeds when we fail to derive the obligation to the contrary. Accordingly, we must be able to tell when an argument is acceptable or when an argument is rejected. Given that weak permission requires that we reject the obligation of the contrary, an imaginary argument is acceptable when all arguments for the prohibition are rejected. A natural argument is acceptable when we can show that the arguments that attack it cannot fire (their premises do not hold).
Definition 10. An argument is wp-acceptable by the sets of arguments and if 1. is an imaginary argument and ∀ ∈ Args, > , is wp-rejected by and ; or 2. is a natural argument, supports , and ∀ ∈ Args, > , either undercuts or ∈ .
The next definition specifies when an argument is rejected. Specifically, an imaginary argument is rejected when there is an argument for the obligation to the contrary that is acceptable (thus we did not fail to establish the obligation to the contrary). A natural argument is rejected when one of its subarguments is rejected or when there is an argument attacking it that is supported by the already accepted arguments. Thus, there is an applicable rule for the opposite of the conclusion of the (rejected) argument Definition 11. An argument is wp-rejected by the sets of arguments and if 1. is an imaginary argument and ∃ ∈ Args, > , ∈ ; or 2. is a natural argument and a) ∃ ∈ Sub(), ̸= , and ∈ ; or b) ∃ ∈ Args, > , and supports .
The wp-semantics (or wp-extension) is then defined by the following fixed-point construction, where we iteratively build the sets of wp-acceptable and wp-rejected arguments. Notice that since we have to capture both what arguments are acceptable and what arguments are rejected, the extension is given by the pair of such sets of arguments.
Definition 12. The wp-extension of a Deontic Argumentation Theory is the pair
(JArgs, RArgs) such that where
∞ JArgs = ⋃︁ =1
∞ RArgs = ⋃︁ =1 • 0 = ∅; 0 = ∅; • +1 = { ∈ Args : is wp-acceptable by and }; • +1 = { ∈ Args : is wp-rejected by and }.
Whenever clear from the context, we will drop the references to the sets and when speaking of wp-accepted and wp-rejected arguments.
Example 5. Let us consider again the Deontic Argumentation Theory of Example 2. The arguments and the instances of the attack relation are the same for the grounded semantics and the wp-semantics.
Let us go step-by-step through the construction of the wp-extension. The first step is to determine the arguments that are wp-acceptable and wp-rejected by 0 and 0 , namely the empty set. To this end, we start by listing what arguments are supported and undercut by 0 and 0 . It is easy to verify that the only arguments supported by 0 and 0 are
0, 1, 2, 4.
These are the arguments that do not have any proper subarguments (we ignored all imaginary arguments since they are not involved in any attack, and the condition to ). The next step is to determine the arguments that are undercut, namely:
3, 5, 6, 7, 10. 6 is undercut because it is attacked by 1; the arguments 3, 5, 7 are undercut because they are attacked by 2, and, finally, 10 because it is attacked by 4.
Then, we can compute 1 and 1 : 1 = {0, 4}
1 = {1, 2, 3, 5, 6, 7, 10} Argument 0 is trivially wp-acceptable because no argument attacks it, and it does not have any proper subarguments. 4 is attacked by 3 and 7, but these two arguments are undercut by 0 . Arguments 1, 5, 7, and 10 are wp-rejected because they are attacked by 2, which is supported by 0 . Arguments 2 and 6 are wp-rejected because they are attacked by 1, which is supported by 0 . 3 is wp-rejected because it is attacked by 4, which is supported by 0 .
We can now move to the next step, where we compute 2 and 2 . In addition to the arguments supported by 0 , 1 supports 8 (4 ∈ 1 ).
2 = 1 ∪ {1, 2, 4, 8} 2 = 1 ∪ {3} The imaginary arguments 1, 2 and 4 are wp-acceptable because they are attacked respectively by 2, 1 and 3, and these arguments are wp-rejected by 1 . Argument 8 is wp-acceptable because it is now supported by 1 and it is attacked by 3 and 7. 3 and 7 are undercut by 1 . 3 is wp-rejected because it is attacked by 4, which is wp-acceptable by 1 .
For the next and final step, we compute 3 and 3 . This time, 2 adds support for argument 9 (because 1 ∈ 2 ), and then 9 is wp-acceptable, because it is attacked by 2, but 1 attacks 2 and it is supported by 2 .
Theorem 2. For any Deontic Argumentation Theory = (, ) the wp-extension is unique. Proof. The proof that the wp-extension is unique is by induction on the steps of the construction of the wp-extension, that if an argument is wp-justified/ wp-rejected by and , then the argument is wpjustified/wp-rejected by +1 and +1. The induction proves that the extension of the wp-extension is monotonically increasing, and then we can apply the Knaster-Tarski Theorem to conclude that the wp-extension is the unique least/greatest fixed-point.
Let us start by the case when an argument ∈ 1 . We have two sub-cases.
1) is an imaginary argument. Thus, it does not have any proper subargument (so no argument can be wp-rejected), and every argument attacking it is rejected by ∅ and ∅ (0 and 0 ). Let be an argument attacking . Since is wp-acceptable, then is wp-rejected; therefore there is an argument , > and is supported by ∅. This means that does not have any proper subarguments, which in turn means that any supports rejecting in any iteration of the construction. Hence, ∈ 1 .
2) is a natural argument. By construction is wp-acceptable by 0 and 0 ; by definition, is supported by 0 , namely ∅, thus, again, does not have any proper subarguments. Accordingly, is supported by any . The arguments attacking it are undercut by 0 . This means that for every argument attacking , there is an argument supported by 0 . But as we have just seen for , the argument is supported by any . So, ∈ 1 .
We can now move to the case of a wp-rejected argument. Thus, ∈ 1 . Again, we have two sub-cases.
1) is an imaginary argument. is wp-rejected by 0 and 0 , if there is an argument attacking that is supported by 0 . We have already seen that an argument supported by 0 is supported by any . Accordingly, is wp-rejected at any step of the construction of the wp-extension; hence ∈ 1 .
2) is a natural argument. Given that 0 = ∅, cannot be in 0 because one of its proper subarguments is in 0 . Thus, there is an argument attacking in that is supported by 0 . We can argue, as in the other case, that is supported by any , and thus, ∈ 1 .
For the inductive step, as usual, we assume that the property holds for arguments in and . Hence, for the justified arguments we can conclude that if an argument is wp-accepted for some set of arguments ( < ), then the argument is supported by ; at this stage we can replicate the reasoning of the inductive base to conclude that if an argument is wp-acceptable by and , then it is wp-acceptable by +1 and +1.
For the case when an argument is in +1 in addition to the cases discussed in the inductive base that carry over by the inductive hypothesis, we have to examine the situation that there a proper subargument ′ of is wp-rejected by and . This means that ′ ∈ , and that argument has been added at a step before . Thus, we can appeal to the inductive hypothesis to conclude that ′ would be in all , ≥ . Thus, ∈ +1.
We have just proved that the construction of the wp-extension is monotonically increasing. Then, by the Knaster-Tarski Theorem, the construction has a least fixed point, and the least fixed point is unique.
The next step is to show that the wp-semantics is correct. For correctness we have to consider two properties. We have to show that the wp-extension is conflict-free, thus the set of wp-acceptable arguments does not contain arguments for and against a given conclusion and that no argument is both wp-justified and wp-rejected.
Theorem 3. For any Deontic Argumentation Theory = (, ): • JArgs ∩ RArgs = ∅ (no argument is both wp-justified and wp-rejected); • the wp-extension is conflict-free.
Proof. We start by proving that the intersection of the wp-acceptable argument and the wp-rejected sets of arguments is empty. The proof is by contradiction and induction. Assume that there is an argument such that ∈ JArgs and ∈ RArgs. This means that there is some , ∈ and ∈ . We prove by induction that this is impossible.
For the inductive base, suppose that ∈ 0 and ∈ 0 . We have two cases: is an imaginary argument or is a natural argument.
is an imaginary argument. ∈ 1 means that there is an argument , such that > and ∈ 0 . But 0 = ∅, so there is no such argument . Hence, ∈/ 1 . Thus, cannot be in the intersection of 1 and 1 .
is a natural argument. ∈ 1 means that 0 = ∅ supports , thus Sub() = {} and does not have any proper subarguments. Moreover, 0 undercuts all arguments attacking . Suppose > , then 0 undercuts , thus 0 supports an argument that attacks a proper subargument of of , ∈ Sub(), ̸= . ∈ 1 means that there is an argument , such that > and 0 = ∅ supports . Thus, Sub() = {}, and we get a contradiction since cannot attack a proper subargument of .
For the inductive step, we assume that the property holds for ( ∩ = ∅) and we show that it holds for + 1. Assume that ∈ and ∈ . Again, we have two cases: is an imaginary argument or is a natural argument.
is an imaginary argument. ∈ +1 means that every argument attacking is wp-rejected by and . Thus, any argument , such that > , is in . ∈ +1 means that there is an argument , such that > and ∈ . Hence, ∈ ∩ , against the inductive hypothesis.
is a natural argument. ∈ +1 means that supports , and every argument attacking is undercut by . Thus, for every argument ′ ∈ Sub(), ′ ̸= , ′ is in . Moreover, for an argument such that > , supports an argument that attacks a proper subargument of ′ of (′ ∈ Sub(), ′ ̸= ). ′ ∈ Sub(), ′ ̸= such that ′ ∈ (and this is against the inductive hypothesis) or (2) there is an argument such that > and supports . Therefore, ∀′ ∈ Sub(), ′ ̸= , ′ ∈ . But, ′ is undercut by , thus there is an argument supported by that attacks ′. Thus, ′ ∈ against the inductive hypothesis.
For the wp-semantics, conflict-free means that we do not have pairs of arguments and such that > and , ∈ JArgs.
Let us suppose that an extension is not conflict-free; this means that there is some such that there are two arguments and such that , ∈ +1, and > . By construction of the attack relation, we have that is a natural argument. Thus, ∈ +1 means that supports , and every argument attacking is undercut by .
is either an imaginary argument or a natural argument. If is an imaginary argument, ∈ +1 means that every argument attacking is wp-rejected by and . But, attacks , thus is wp-rejected by and , so ∈ +1; but we just proved that this is impossible, no argument is both wp-justified and wp-rejected.
If is a natural argument, attacks , and supports , thus, ∈ +1. But, as we have just seen, this is impossible.
We extend the notion of wp-justified conclusions to literals and deontic literals. A literal is wp-justified if it is the conclusion of a wp-justified argument.
Definition 13. Given a Defeasible Argumentation Theory , an argument is justified under the wp-semantics (or wp-justified) if ∈ JArgs.
A literal or a deontic literal is justified under the wp-semantics (or wp-justified) if it is the conclusion of a justified argument under the wp-semantics.
Example 6. When we consider the Deontic Argumentation Theory of Example 2, and the construction of the extension in Example 5, we can see that the arguments 0, 4, 8, 9, 1, 2 and 4 are wp-justified. Thus, the wp-justified conclusions are
, obl(¬), perm(¬), , perm(), perm(¬), perm(¬)
The final step is to show that weak permission is supported by the wp-semantics. This means that we can infer that a weak permission is in force even when a conflict between two obligations exists. Accordingly, we refine the definition of a conflictual theory to be more aligned with the wp-semantics. Definition 14. A Deontic Argumentation Theory is wp-conflictual it contains a pair of rules 0, . . . , ⇒ obl() 0, . . . , ⇒ obl(¬) such that the arguments (0 ≤ ≤ ) and (0 ≤ ≤ ) have conclusions () = and ( ) = , and the and are wp-justified. We also say that is a wp-conflictual literal. Theorem 4. Let be a wp-conflictual theory and a wp-conflictual literal such that the proper subarguments of the arguments for obl() and obl(¬) are wp-justified. Then perm() and perm(¬) are wp-justified.
Proof. We can assume, without any loss of generality, that the arguments and have no attacks. Hence, the set of attacks for the theory is 1 > 2 2 > 1 1 > perm(¬) 2 > perm() By definition all subarguments of 1 and 2 are wp-justified. 1 is the only argument attacking perm(¬), 2 attacks 1 and 2 is supported by the wp-extension. Hence, 1 is wp-rejected. 1 is the only argument attacking perm(¬). Hence, perm(¬) is a justified argument. We can repeat the same argument for perm().
To conclude we show that the wp-semantics is an extension of the grounded semantics. To this end we need the following lemma.
Lemma 1. All subarguments of a justified argument are justified.
Theorem 5. Given a deontic argumentation theory, every argument justified under the grounded semantics is also justified under the wp-semantics.
Proof. It follows immediately from Lemma 1, since an argument that is justified by a set of arguments is also supported by the same set.
Formal argumentation has been investigated for a long time in the field of Artificial Intelligence and
Law (for a comprehensive though a bit outdated survey, see [
van der Torre and Villata [
The scope of [13] is formal argumentation for handling norms in multi-agent systems, presenting methods for representing and interpreting hierarchical normative systems, and addressing open texture in legal interpretation using fuzzy logic. No focus is devoted on deontic reasoning and weak permissions: deontic assertions are simply conclusion in ASPIC+ setting.
The aim of [14] is to propose an argumentation framework to address diferent types of normative conflicts. Arguments are built by triples (, , ) where is a formula for the brute facts of the arguments, represents the institutional facts, and finally is the deontic conclusion of the argument, which is either an obligation or a permission. While the work considers various notions of conflicts, it does not consider weak permission. Moreover, the argumentation framework adopts Dung’s semantics, including grounded semantics. Thus, it cannot deal with weak permission when unresolved conflicts exist.
Riveret, Rotolo and Sartor [15] presented a modular rule-based argumentation system designed to represent and reason about conditional norms, including obligations, prohibitions, and weak and strong permissions. However, the work treated such notions as labels of literals without exploring their argumentative nature. Similar considerations apply to [16], in which deontic argumentation and probabilistic argumentation were merged. Riveret, Rotolo and Sartor [15] presented a modular rule-based argumentation system designed to represent and reason about conditional norms, including obligations, prohibitions, and weak and strong permissions. However, the work treated such notions as labels of literals without exploring their argumentative nature. Similar considerations apply to [16], in which deontic argumentation and probabilistic argumentation were merged.
The work by Straßer, Arieli and Berkel [17, 18] focuses on the relationships between formal
argumentation and sequent style and Input/Output inference systems. However, in general, they do not consider
permission (and weak permission) [18] or Standard Deontic Logic [17]. While [18] restricts its focus to
grounded semantics, [19, 20] consider also other semantics (preferred and stable) but no permission,
and given the results of [
The proposal in [21] builds an argumentation framework in ASPIC+ style for deontic reasoning based on Standard Deontic Logic extended with a preference relation. The approach considers permission as the dual of obligation, and then evaluates arguments based on the complete semantics. Notice that in case of a conflict, the complete semantics will provide two extensions, and the proof of Theorem 1 shows the two extensions contain opposite obligations, and weak permission is not sceptically justified.
The approach closer to our work is Defeasible Deontic Logic (DDL) [
Grounded semantics and the semantics proposed in [
While the wp-semantics covers the case of weak permission, it fails to explain the deontic operators (apart from weak permission being the failure to conclude the prohibition of the opposite). To fill this gap, we plan to extend the approach we proposed in [25] to account for the meaning of the deontic operators in a possible world (neighbourhood) semantics for Defeasible Deontic Logic [26].
In this paper, we have proposed a new semantics for deontic argumentation that can deal with weak permission. The semantics is based on the argumentation semantics for Defeasible Logic. We have shown that the semantics is unique, conflict-free, and that no argument is both justified and rejected. We have also shown that the semantics is able to characterise weak permission when there are unresolved conflicts addressing thus the issue with grounded semantics.
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