In this paper we propose a general approach to define a many-valued preferential interpretation of gradual argumentation semantics. The approach allows for conditional reasoning over arguments and boolean combination of arguments, with respect to some chosen gradual semantics, through the verification of graded (strict or defeasible) implications over a preferential interpretation. The paper also develops and discusses a probabilistic semantics for gradual argumentation, which builds on the many-valued conditional semantics.
variables, and a typicality operator is allowed, inspired by
the typicality operator proposed in the Propositional
TypiArgumentation is one of the major fields in non-monotonic cality Logic [
In a companion paper [18], we have proposed an ASP |∼ in the KLM approach [21, 3]. More precisely, in
approach for conditional reasoning over weighted argu- this paper we consider graded conditionals of the form
mentation graphs in a specific gradual semantics (the - T( ) → ≥ , meaning that “normally argument
coherent semantics), through the verification of graded implies argument with degree at least ", where and
conditional implications over arguments and over boolean can be boolean combination of arguments. They are
incombinations of arguments. In this paper, we show that spired by graded inclusion axioms in fuzzy DLs [22] and
the proposal can be generalized to a larger class of gradual in weighted defeasible knowledge bases in DLs [
The paper proposes a general approach to define a pref- interpretation of an argumentation graph (wrt. a
erential interpretation of an argumentation graph under a given semantics ), exploits multiple preference relations
gradual semantics (provided weak conditions on the do- < over labellings, each one associated with an
argumain of argument interpretation are satisfied), to allow for ment .
conditional reasoning over the argumentation graph, by We reformulate the KLM postulates of a preferential
formalizing conditional properties of the graph in a many- consequence relation for graded conditionals and prove
valued logic with typicality: a many-valued propositional that they are satisfied by the conditionals which hold in
logic in which arguments play the role of propositional the multi-preferential interpretation , for some choice
of combination functions. We also prove that the
satisfia21st International Workshop on Nonmonotonic Reasoning, bility of a graded conditional T( ) → ≥ in a finite
*SeCpotrermesbpeorn2d–in4g,2a0u2th3o,rR. hodes, Greece preferential interpretation can be decided in
polyno$ mario.alviano@unical.it (M. Alviano); laura.giordano@uniupo.it mial time in the size of the interpretation times the size
(L. Giordano); dtd@uniupo.it (D. Theseider Dupré) of the conditional formula.
https://alviano.net/ (M. Alviano); The definition of a preferential interpretation
ashttps://people.unipmn.it/laura.giordano/ (L. Giordano); sociated with an argumentation graph and a gradual
https://people.unipmn.it/dtd/ (D. Theseider Dupré) semantics also sets the ground for the definition of a
(L.0G0i0o0r-d0a0n0o2)-;20005020--20006031-(6M7.98A-l4v3ia8n0o()D;0.0T0h0e-s0e0id0e1r-9D4u4p5r-é7)770 probabilistic interpretation of gradual semantics . For
© 2023 Copyright for this paper by its authors. Use permitted under Creative Commons License the gradual semantics with domain of argument valuation
CPWrEooUrckReshdoinpgs IhStpN:/c1e6u1r3-w-0s.o7r3g ACttEribUutiRon 4W.0Iontrekrnsathioonapl(CPCrBoYc4e.0e)d.ings (CEUR-WS.org)
in the unit real interval [
4–13
ment to argument when the weight (, ) is
positive and as an attack of argument to argument when
Given an argumentation graph and some gradual argu- (, ) is negative. In case the graph is non-weighted,
mentation semantics , we define a preferential
(manyvalued) interpretation of the argumentation graph , with we let (, ) = − 1 mean that argument attacks
arrespect to the gradual semantics . We generalize the sguupmpeonrttsar,gaunmdent(,. ) = +1 mean that argument
approach proposed in [18] for weighted argumentation Bipolar argumentation has been studied in the literature
graphs, without assuming a specific gradual semantics. In [
We follow Baroni, Rago and Toni [16, 26] (in their ifcation and the properties of gradual semantics, when
definition of a Quantitative Bipolar Argumentation Frame- the argumentation graph is non-weighted, and to Potyka’s
work, QBAF) in the choice of the domain of argument
work [
For the definition of an argumentation graph, we con- nition) the -coherent semantics [
We let a (weighted) argumentation graph to be a quadru- bellings. For instance, = (0, 4/5, 3/5, 2/5, 2/5, 3/5) ple = ⟨, ℛ, 0, ⟩, where is a set of arguments, (meaning that (1) = 0, (2) = 4/5, and so on) is a ℛ ⊆ × a set of edges, 0 : → assigns a base score of arguments, and : ℛ → R is a weight function assigning a positive or negative weight to edges. An example of weighted argumentation graph is in Figure 1, where the base score (i.e., the initial valuation of arguments) is not represented.
A pair (, ) ∈ ℛ is regarded as a support of
argu1Clearly, not any mapping qualifies as a labelling of a gradual
semantics , as a gradual semantics is intended to satisfy some principles,
such as those identified in the different frameworks mentioned above
[
4–13 , whose set of propo- In the following, we will restrict our consideration to set assume our language ℒ contains the connectives ∧, ∨, founded for all arguments in Σ . We will call such a set labelling for = 5.
In the following, we introduce a propositional language to represent boolean combination of arguments and a many-valued semantics for it over the domain of argument valuation. Then, we extend the language with a typicality operator, to introduce defeasible implications over boolean combinations of arguments and define a (multi-)preferential interpretation associated with the argumentation graph and a set of labellings Σ .
Given an argumentation graph = ⟨, ℛ, 0, ⟩, we introduce a propositional language ℒ sitional variables is the set of arguments . We ¬ and →, and that formulas are defined inductively, as usual. Formulas built from the propositional variables in correspond to a boolean combination of arguments (denoted , ,
), which are considered, for instance, by Hunter, Polberg and Thimm in [31]. and 1 otherwise. ▷ and ⊖ bination of arguments, with as the truth degree set. Let ⊗ , ⊕ , ▷ and ⊖ be the truth degree functions in for the connectives ∧
, ∨, ¬ and → (respectively). When is
[
can be chosen as a t-norm, s-norm, implication function, and negation function in some system of manyvalued logic [32]; for instance, in Gödel logic, that we will consider later, ⊗ = {, }, ⊕ = {, }, ▷ = 1 if ≤ and otherwise; and ⊖ = 1 if = 0
A labelling : → of graph , assigning to each argument ∈ a truth degree in , can be regarded as a many-valued valuation. A valuation can be inductively extended to all propositional formulas of ℒ as follows: ( ∧ ( ) = ( )
⊗ ( ) → ) = ( ) ▷ ( ) (¬ ) = ⊖ ( ) ( ∨
) = ( ) ⊕ ( ) Based on the choice of the combination functions, a labelling uniquely assigns a truth degree to any boolean combination of arguments. We will assume that the false argument ⊥ and the true argument ⊤ are formulas of and that (⊥) = 0 and (⊤) = 1, for all labellings
< Σ ′ iff ′() < (), (or is more plausible than ′ for argument ), when the degree of truth of in is greater than the degree of truth of in ′. The preference relation <Σ is a strict partial order relation on Σ . We will be simply write < , omitting Σ , when it is clear from the context.
The definition of preference over arguments which is induced by a set of labellings Σ also extends to the boolean combination of arguments in the obvious way, based on the choice of combination functions. A set of labellings Σ induces a preference relation < on Σ , for each boolean combination of arguments , as follows: for all ,
′ ∈ Σ , < ′ iff ′( ) < ( ).
When the set Σ of labellings of a graph in an argumentation semantics is infinite, the preference relations < (and < ) are not guaranteed to be well-founded, as there may be infinitely-descending chains of labellings. of labellings Σ such that both < and <¬ are wellof labellings Σ a well-founded set of labellings. From the monotonicity properties of t-norms and s-norms, and the antitonicity property of negation functions, it follows that, for any well-founded set of labellings Σ , < is also well-founded for any boolean combinations of arguments
Given an argumentation graph , a gradual semantics with domain of argument valuation , and the set of labellings Σ of wrt , we let the preferential interpretation of wrt to be the pair = (, Σ ).
pretation are left implicit, as they are induced by the labellings in Σ (according to Definition 1). Often, we will simply write or , rather than .
Language ℒ
T is obtained by extending language ℒ with
a unary typicality operator T. Intuitively, “a sentence
of the form T( ) is understood to refer to the typical
situations in which holds" [
(T( )) = ︂{ ( ) 0 if otherwise ∈ < (Σ) (1) .
as follows: for ,
′ ∈ Σ ,
Given a set of labellings Σ , for each argu- the typicality operator, an implication ment ∈ , we define a preference relation < on Σ , strict. In the language ℒT, we allow general implications → is called
Labelling is preferred to ′ with respect to argument fined with respect to a preferential interpretation =
∈ Σ and ∄ ′ ∈ Σ s.t. in the obvious way, based on the semantics of classical where < (Σ) = ′ < }
.
When (T()) > 0, is a labelling assigning a maximal degree of acceptability to argument in , i.e., it maximizes the acceptability of argument , among all the labellings in .
Given a preferential interpretation = (, Σ) , we can now define the satisfiability in
of a graded implication, having form → ≥ or
→ ≤ a preferential interpretation as follows: ifrst define the truth degree of an implication and and boolean combination of arguments. We
→ wrt , with and in
→ ) = ∈Σ( ( ) ▷ ( )).
that: (T( ) → ) = ∈Σ( (T( )) ▷ ( )).
graded implication in an interpretation = (, Σ) .
idate (under the semantics ) properties of interest of an argumentation graph , expressed by graded implications (including strict or defeasible implications or their boolean combination) based on the semantics . When the preferential interpretation is finite (contains a finite set of labellings), the satisfiability of graded implications (or their boolean combinations) can be verified by model checking over the preferential interpretation (see section 4). In case there are infinitely many labellings of the graph in the semantics , which may give rise to non well-founded preference relations, approximations of the semantics over a finite domain can be considered for proving properties of the argumentation graph. In [18] we have developed an ASP approach for defeasible reasoning over an argumentation graph under the -coherent semantics in the finitely-valued case. ↔ and
a preferential consequence relation in the many-valued setting and prove that, for the choice of combination functions as in Gödel logic, they are satisfied by the set of hgoralddeidn caognivdeitnioinnatelsrporfettahteiofnorm T=((), →Σ ). ≥
relations [21, 3, 33] can be reformulated by replacing
with the conditional implication a conditional
|∼ T( ) → ≥ 1, as follows: (Reflexivity) T( ) → ≥ 1 (LeftLogicalEquivalence) If |= T( ) → ≥
1, then T( ) → ≥ 1 (RightWeakening) If |= → and T( ) → 6 → 1 ∧ 2 ≥ 1/5 does not hold.
On the other hand, for instance, the strict implication ≥ 1, then T( ) → ≥ 1
For instance, this means that the strict implication 6
→ 1 ∧ 2 has a very low degree but, in the situations (labellings) which maximize the acceptability of argument 6, implication 6 → 1 ∧ 2 holds with a degree not lower that 4/5, i.e., roughly speaking, in the labellings maximizing the acceptability of argument 6, arguments 1 and 2 are likely to hold.
(And) If T( ) → ≥ T( ) → ∧ ≥ 1 T( (Or) If T( ) → ≥ ∨ ) → ≥ 1 1 and T( ) → ≥ 1 and T( ) → ≥
T( ) → ≥ 1, then T( ∧ ) → ≥ 1.
(CautiousMonotonicity) If T( ) → ≥ 1 and that
→ ≥ Here, we also reinterpret |=
→ as the requirement 1 is satisfied in all interpretations = Notice that the valuation of a graded implication (e.g., (, Σ) , that is, ( ) ▷ ( ) ≥ 1 holds for any labelling → ≥
) in a preferential interpretation is two- valued, that is, either the graded implication is satisfied in (i.e., |= → ≥ ) or it is not (i.e., ̸|= → ≥ interpreted as |= → and |=
→ ∈ Σ , in any interpretation = (, Σ) . |= ↔ is Concerning the meaning of the postulates in this con). Hence, it is natural to consider boolean combinations text, for instance, the meaning of (And ) is that, if of graded implications, such as (T(1) → 2 ∧ 3 ≤ 0.7) ∧ (T(3) → 4) ≥
0.6) → (T(1) → 4) ≥ 0.6), and define their satisfiability in an interpretation
T( ) → ≥
1 and T( ) → ≥ ifed in , then T( ) → ∧ ≥
1 is also satisfied in .
that |= → (i.e., ( ) ▷ ( ) ≥ 1 for any labelling in any interpretation ), and T( ) → ≥ 1 is satisfied in then T( ) → ≥ 1 is also satisfied in .
Given the interpretation = (, Σ ), associated with an argumentation semantics of a graph , we can prove the following result.
Proposition 1. Under the choice of combination functions as in Gödel logic, interpretation satisfies the KLM postulates of a preferential consequence relation given above.
Proof (Sketch). Let = (, Σ ) be the interpretation associated with an argumentation semantics of a graph where the t-norm, s-norm, implication function and negation functions are as in Gödel logic (i.e., ⊗ = {, }, ⊕ = {, }, ▷ = 1 if ≤ and otherwise; and ⊖ = 1 if = 0 and 1 otherwise). We proceed by cases. (Reflexivity) To prove that T( ) → ≥ 1 is satisfied in , we have to prove that ∈Σ (T( ))▷ ( ) ≥ 1. Let us prove that for all ∈ Σ , (T( ))▷ ( ) ≥ 1.
We consider two cases: (T( )) = 0, and (T( )) > 0.
If (T( )) = 0, (T( )) ▷ ( ) = 0 ▷ ( ) = 1, and the thesis holds trivially.
If (T( )) > 0, by definition (T( )) = ( ). Hence, (T( )) ▷ ( ) = 1, and the thesis holds.
Given that (T( ∨ )) = ( ∨ ) = { ( ), ( )}, it follows that there is no ′ ∈ Σ such that { ′( ), ′( )} > { ( ), ( )}.
Let us assume, without loss of generality, that { ( ), ( )} = ( ). Then, there cannot be a ′ ∈ Σ such that ′( ) > ( ), that is, maximizes the acceptability degree for .
Furthermore, (T( )) = ( ) = (T( ∨ )) From the hypothesis, we know that T( ) → ≥ 1 is satisfied in and, hence, (T( )) ≤ ( ) holds. It follows that (T( ∨ )) = (T( )) ≤ ( ), and then (T( ∨ )) ▷ ( ) ≥ 1.
The case where { (), ()} = () is similar, and this concludes the case for (T( ∨ )) > 0.
For the other postulates the proof is similar.
on the choice of the negation function. The same properties also hold for Zadeh’s logic. However, some of the properties above might not hold depending on other choices of combination functions. Note that whether the KLM properties are intended or not, may depend on the kind of conditionals and on the kind of reasoning one aims at, and it is still a matter of debate [34, 35, 36, 37].
(Or) Let us assume that T( ) → ≥ 1 and T( ) → ≥ 1 are satisfied in . Then, ∈Σ (T( )) ▷ ( ) ≥ 1 and ∈ ∈Σ Σ, ((TT(( )))) ▷ ▷ (( )) ≥ ≥ 1 hold.
Hence, for all 1 and (T( )) ▷ ( ) ≥ 1 also hold.
As we have seen above, this implies that: for all ∈ Σ , (T( )) ≤ ( ) and (T( )) ≤ ( ).
To prove that T( ∨ ) → ≥ 1 is satisfied in , we prove that for all ∈ Σ , (T( ∨ )) ≤ ( ).
If (T( ∨ )) = 0, the thesis follows trivially.
If (T( ∨ )) > 0, maximizes the acceptability degree for ∨ , and there is no ′ ∈ Σ such that ′(T( ∨ )) > (T( ∨ )). (RightWeakening) Assume |= → holds, i.e., In this section we show that, for a finite interpretation ( ) ▷ ( ) ≥ 1 holds for all labellings in any prefer- = (, Σ ) associated with an argumentation semanential interpretation = (, Σ) . Hence, ∈Σ ( ) ▷ tics of an argumentation graph , the satisfiability of ( ) ≥ 1 and, for all ∈ Σ , ( ) ▷ ( ) ≥ 1. This a conditional T( ) → ≥ in can be decided in implies that, for all = (, Σ) , and for all ∈ Σ , polynomial time in the size of Σ times the size of the ( ) ≤ ( ) (in particular, this must hold for all in formula T( ) → . Σ ). To verify the satisfiability of a graded conditional
Let us assume that T( ) → ≥ 1 is satisfied in , T( ) → ≥ in a preferential interpretation = i.e., ∈Σ (T( )) ▷ ( ) ≥ 1 holds. Hence, for (, Σ ) of the graph , one has to check that for all all ∈ Σ , (T( )) ▷ ( ) ≥ 1 holds. Thus, for labellings ∈ Σ , it holds that T( ( )) ▷ ( ) ≥ all ∈ Σ , (T( )) ≤ ( ) and, then, (T( )) ≤ . In particular, one has to identify all the labellings ( ) ≤ ( ). From this it follows that T( ) → ≥ 1 ∈ Σ which maximize the acceptability degree of the is satisfied in . boolean combination of arguments (i.e., those such that (T( )) = > 0) as, for all other labellings, T( ( )) = 0 and 0 ▷ ( ) ≥ holds trivially.
Let |Σ | be the size of Σ . Identifying the labellings which maximize the acceptability degree of , requires to evaluate the acceptability degree ′( ) of , for any ′ ∈ Σ . For a given labelling ′ this evaluation is polynomial in | |, the size of (the number of subformulas of is polynomial in | |). Then, determinimg the acceptability degree of for all labellings in Σ , requires a polynomial number of steps in |Σ | × | |. In particular, a single scan of the list of the labellings in Σ , also allows to identify the labellings maximizing the acceptability degree of (call them 1, . . . , ℎ), the value = 1( ) = . . . = ℎ( ), and the acceptability degree of in each labelling. We restrict to a -compatible t-norm ⊗ [41], with Overall this requires a polynomial number of steps in associated t-conorm ⊕ and the negation function ⊖ = |Σ | × (| | + | |). 1 − . For instance, one can take the minimum t-norm,
Considering that = T( 1( )) = . . . = T( ℎ( )), product t-norm, or Lukasiewicz t-norm. Given =
the verification that ▷ ( ) ≥ holds, for all = ⟨:,Σ Σ ⟩, we assume a discrete probability distribution
1, . . . , ℎ, may require in the worst case a polynomial → [
(2) ∈Σ Proposition 2. Given a finite interpretation = (, Σ ), associated with an argumentation semantics of a graph . The satisfiability of a graded conditional T( ) → ≥ in can be decided in (|Σ | × (| | + | |)).
For a single argument ∈ , when labellings are twovalued (that is, () is 0 or 1), the definition above becomes the following: () = ∑︀ ∈Σ ∧ ()=1 ( ),
Of course, whether the interpretation is finite or not which relates to the probability of an argument in the
depends on the argumentation semantics under consid- probabilistic semantics by Thimm in [25]. Indeed, in [25]
eration. For the finitely-valued -coherent semantics, an the probability of an argument in is “the degree
ASP based approach for the verification of graded condi- of belief that is in an extension", defined as the sum of
tionals has been considered in [18], through a mapping the probabilities of all possible extensions that contain
of an argumentation graph to a weighted knowledge base, argument , i.e., () = ∑︀∈⊆ (), where an
for which ASP encodings have been developed [38]. extension ∈ 2 is a set of arguments in , and ()
is the probability that is an extension.
5. Towards a probabilistic In Thimm’s semantics [25] a notion of p-justifiable
probability function is introduced to restrict to the
“probasemantics for gradual bility functions that agree with our intuition of
argumenargumentation tation", so that relationships with classical argumentation
semantics can be established. Here, instead, for a given
When the domain of argument valuation is the inter- gradual semantics with labellings Σ , we only consider
val [
Let Σ be the set of labellings of in a gradual argumen- (|{ }) = ().
tation semantics with domain of argument valuation The result holds when the t-norm is chosen as in Gödel,
in [
The notion of probability defined by equation (2) works by instantiating the arguments and characterizing satisfies Kolmogorov’s axioms for any -compatible t- the attacks with suitable sets of conditionals describing norm, with associated t-conorm, and the negation function constraints over ranking models”. In doing this, he has ⊖ = 1 − [41]. However, there are properties of exploited the JZ-evaluation semantics, which is based on classical probability which do not hold (depending on the system JZ [46]. Our approach does not commit to any choice of t-norm), as a consequence of the fact that not all specific gradual argumentation semantics, and aims at proclassical logic equivalences hold in a fuzzy logic. viding a preferential and conditional interpretation for a
For instance, the truth degree of ∧ ¬ in a labelling large class of gradual argumentation semantics. may be different from 0 depending on the t-norm (e.g., For Abstract Dialectical Frameworks (ADFs) [8], the with Gödel and Product t-norms). Hence, it may be the correspondence between ADFs and Nonmonotonic Concase that ( ∧ ¬) is different from 0. Similarly, it may ditional Logics has been studied in [9] with respect to the be the case that ( ∨ ¬) is different from 1 (e.g., with two-valued models, the stable, the preferred semantics Göedel t-norm) and that (|) = ( ∧ )/ () and the grounded semantics of ADFs. is different from 1 (e.g., with Product t-norm). While In [10] Ordinal Conditional Functions (OCFs) are in () + (¬) = 1 holds (due to the choice of negation terpreted and formalized for Abstract Argumentation, by function), (|) + (¬|) may be different from 1. developing a framework that allows to rank sets of argu
While this approach can be regarded as a simple gener- ments with respect to their plausibility. An attack from alization of probabilistic semantics to the gradual case, on argument a to argument b is interpreted as the conditional the negative side, some properties of classical probability relationship, “if a is acceptable then b should not be acare lost. Hence, we can consider this approach only as ceptable". Based on this interpretation, an OCF inspired a first step towards a probabilistic semantics for gradual by System Z ranking function is defined. In this paper we argumentation. focus on the gradual case, based on a many-valued logic.
In [