Reasoning about preferences is a topic that has received a lot of attention in Artificial Intelligence since many years, see for instance [HGY12, DHKP11, Kac11]. Two main approaches to representing and handling preferences have been developed: the relational and the logic-based approaches.
In the classical setting, a (weak) preference binary relation on a set of alternatives is usually modeled as a preorder, i.e. a reflexive and transitive relation, where is understood as is at least as preferred as . In the fuzzy or graded setting, preference relations can be attached degrees (usually belonging to the unit interval ) of fulfillment or strength, so they become fuzzy relations. A weak fuzzy preference relation on a set will be now a fuzzy preorder , where is interpreted as the degree in which is at least as preferred as . Given a t-norm , a fuzzy relation is a -preorder if it satisfies
reflexivity: for each , and
-transitivity: for each .
The most influential reference is the book by Fodor and Roubens [FR94], that was followed by many other works like, for example [DBM07, DBM10, DMB04, DBM08, DGLM08]. In this setting, many questions have been discussed, like e.g. the definition of the strict fuzzy order associated to a fuzzy preorder (see for example [Bod08a, Bod08b, BD08, EGV18]).
The basic assumption in logical-based approaches is that preferences have structural properties that can be suitably described in a formalized language. This is the main goal of the so-called preference logics, see e.g. [HGY12]. The first logical systems to reason about preferences go back to S. Halldén [Hal57] and to von Wright [vW63, vW72, Liu10]. Others related works are [EP06, vBvOR05]. More recently van Benthem et al. in [vBGR09] have presented a modal logic-based formalization of representing and reasoning with preferences. In that paper the authors first define a basic modal logic with two unary modal operators and , together with the universal and existential modalities, and respectively, and axiomatize them. Using these primitive modalities, they consider several (definable) binary modalities to capture different notions of preference relations on classical propositions, and show completeness with respect to the intended preference semantics. Finally they discuss their systems in relation to von Wright axioms for ceteris paribus preferences [vW63]. On the other hand, with the motivation of formalizing a comparative notion of likelihood, Halpern studies in [Hal97] different ways to extend preorders on a set to preorders on subsets of and their associated strict orders. He studies their properties and relations among them, and he also provides an axiomatic system for a logic of relative likelihood, that is proved to be complete with respect to what he calls preferential structures, i.e. Kripke models with preorders as accessibility relations. All these works relate to the classical (modal) logic and crisp preference (accessibility) relations.
In the fuzzy setting, as far as the authors are aware, there are not many formal logic-based approaches to reasoning with fuzzy preference relations, see e.g. [BEFG01]. More recently, in the first part of [EGV18] we studied and characterized different forms to define fuzzy relations on the set of subsets of , from a fuzzy preorder on , in a similar way to the one followed in [Hal97, vBGR09] for classical preorders, while in the second part we have semantically defined and axiomatized several two-tiered graded modal logics to reason about different notions of preferences on crisp propositions, see also [EGV17]. On the other hand, in [VEG17a] we considered a modal framework over a many-valued logic with the aim of generalizing Van Benthem et al.’s modal approach to the case of both fuzzy preference accessibility relations and fuzzy propositions. To do that, we first extended the many-valued modal framework of [BEGR11] for only a necessity operator by defining an axiomatic system with both necessity and possibility operators and over the same class of models. Unfortunately, in the last part of that paper, there is a mistake in the proof of Theorem 3 (particularly, equation (4)). This left open the question of properly axiomatizing the logic of graded preferences defined there.
In this paper we address this problem, extending the work developed in [VEG18]. We propose an alternative approach to provide a complete axiomatic system for a logic of fuzzy preferences, studying first the logic with reflexive graded preference relations (as in [VEG18]) an later, extending this system with the corresponding strict (irreflexive) preferences. Namely, given a finite MTL-chain (i.e. a finite totally ordered residuated lattice) as set of truth values, and given an -valued preference Kripke model , with a fuzzy preorder valued on , we consider the -cuts of the relation for every , and for each -cut , we consider the corresponding modal operators . These operators are easier to be axiomatized than the original , since the relations are not fuzzy any longer, but a nested set of classical (crisp) relations.
The good news is that, in the our rich (multi-modal) logical framework, we can show that the original modal operators and are definable, and vice-versa if we expand the logic with Monteiro-Baaz’s operator. Thus, we define and axiomatize a conservative extension of the logic where the original operators can be defined using the new graded operators.
The paper is structured as follows. After this introduction, Section 2 deals with basic facts on fuzzy preference relations. In Section 3 we present the many-valued modal logics in the more general context (over arbitrary finite bounded commutative integral residuated lattices), and the intended semantics given by valued Kripke models. We close an open problem existing in this setting, namely, whether the operations and are interdefinable, proving this is the case, and providing the explicit definition of each operator in terms of the other one. This strongly simplifies the symbolic approach to the logic, since it is only necessary axiomatize one of the modal operators to obtain a logic referring to both. In Section 4 we show how to adapt the previous general setting to model graded preference relations: we restrict the evaluations to some arbitrary MTL-chain , introduce auxiliary crisp modalities and exhibit a complete axiomatization of a conservative extension111Namely, while the modal language used is larger, the restriction of the logic to the , fragment coincides with the original one. of the preference logic studied in [VEG17a]. In Section 5 we study the extension of the previous logic with the strict preference modality , corresponding to the irreflexive restriction of the preference relation associated to the original . We propose a a complete axiomatization of a conservative extension of this logic (relying again in the crisp modalities). In Section 6 we observe how, by the addition to the logic of the so-called Monteiro-Baaz operation, we can also provide an axiomatization of the original logic of graded preference models pursued in [VEG17a], without the necessity of additional modal operations. Lastly, in Section 7, we discuss different possibilities to formalize notions of preferences on fuzzy propositions in preference Kripke models. We finish with some conclusions and open problems.
2 Preliminaries on fuzzy preference relations
In the classical setting, a (weak) preference relation on a set of alternatives is usually modeled as preorder relation (i.e. a reflexive and transitive relation) by interpreting as denoting is at least as preferred as . From one can define three disjoint relations:
the strict preference ,
the indifference relation , and
the incomparability relation .
where , and . It is clear that is a strict order (irreflexive, antisymmetric and transitive), is an equivalence relation (reflexive, symmetric and transitive) and is irreflexive and symmetric. The triple is called a preference structure, where the initial weak preference relation can be recovered as .
In the fuzzy setting, preference relations can be attached degrees (usually belonging to the unit interval ) of fulfillment or strength, so they become fuzzy relations. In this paper we will assume preference degrees are the domain of a finite and linearly ordered scale , with and being its bottom and top elements respectively. Sometimes we will write also to emphasize the lattice operations.
In this paper, we will assume that a weak -valued preference relation on a set will be now a fuzzy -preorder , where is interpreted as the degree in which is at least as preferred as , that is, satisfying:
reflexivity: for each
-transitivity: for each
As in the classical case, from it is easy to define two other -valued relations corresponding to graded counterparts of the strict and indifference relations associated to .
First, we can define the indifference degree between two states, from the preferential point of view, by , providing the degree to which both is preferred to and, vice-versa, is preferred to . This is a -similarity relation, i.e. a reflexive, symmetric and -transitive -valued relation.
This allows for defining a second preference relation corresponding to the strict counterpart of by, roughly speaking, “removing” the indifferent pairs of worlds from the relation . This amounts to consider as the minimum -valued relation such that . Taking the point-wise smallest solution of this equation leads to the following definition:
It can be checked that if is -transitive, then so is (see e.g. [EGV18]), and thus it can be considered to be a fuzzy strict order, in the sense that the following counterpart of anti-symmetry property holds for : if then .
In the next sections we define and axiomatize a modal preference logic where the initial preorder together with its corresponding indifference relation and strict preference can be dealt with. To do so, we need to resort the level-cuts of the preference relations and to observe the following facts:
Given the initial fuzzy -preorder , we can define, for each , its corresponding -cut , which is a classical preorder.
Analogously, from the corresponding fuzzy strict order , we can also define, for each , the corresponding -cut . In this case, the relations are classical orders.
For each level-cut relation we can also define the corresponding strict order . By definition it is
An equivalent expression for is .
Finally, one can also check that is always included in , i.e. .
In general, and do not coincide, as the following example shows.
Let be the scale where and . Let be the -valued preorder on the universe defined by and . Then it is obvious that,
is defined as, if and , and otherwise. Then .
As usual, one can recover the fuzzy relations and from their crisp level-cut relations:
Moreover, even if the relations and do not generally coincide, can also be recovered from the crisp relations .
Let be an -valued -preorder on a universe . Then for all ,
If then it is easy to check that for all .
If , then . Then:
For , it is obvious that and for all .
By definition, for we have and . Then and it is also obvious that for all .
Thus the claim is proved. ∎
3 Many-valued modal logics: language and semantics
A suitable formalism over which we can construct a graded preference framework is that of many-valued modal logics. In particular, we take as starting point the modal logic introduced in [BEGR11] and further studied in [VEG17a]: finitely-valued (propositional) fuzzy logics enriched with modal-like operations.
Let us begin by defining the formal language of our underlying many-valued propositional setting. Let be a finite (bounded, integral, commutative) residuated lattice [GJKO07], and consider its canonical expansion by adding a new constant for every element (canonical in the sense that the interpretation of in is itself). A negation operation can always be defined as .
The logic associated with will be denoted by , and the set of propositional formulas of its language is defined as in the usual way from a set of propositional variables in the language of residuated lattices (we will use the same symbol to denote connectives and operations), including constants . The corresponding logical consequence relation is defined as follows: for any set ,
if, and only if,
if then ,
where denotes the set of evaluations (homomorphisms) of formulas on .
Lifting to the modal level, we can expand the propositional language by modal operators in different ways. The most general way to do so is consider a pair of unary operators , and build the corresponding set of modal formulas, again defined as usual from a set of propositional variables, residuated lattice operations , truth constants , and modal operators .
We are now ready to introduce -valued Kripke models, a generalization to of classical Kripke models.
An -model is a triple such that
is a set of worlds,
is an -valued binary relation between worlds, and
is a world-wise -evaluation of variables.
The evaluation is uniquely extended to formulas of by using the operations in for what concerns propositional connectives, and letting
We will denote by the class of all -models. Given an -model and , we write whenever for any , if for all , then too. Analogously, for , we write whenever for any .
In [BEGR11], the - fragment of the previous logic was axiomatized, but it was left as an open question how to axiomatize the logic with both and operations. That question was addressed in [VEG17a], where an axiomatic system was proposed and proved complete. Nevertheless, we propose below a new solution to the problem, that also closes an open question: namely, that of the inter-definability of the modal operators in the above valued setting. While it is well known that in classical modal logic both modal operators are inter definable and ), it was not known if something similar happened in valued cases. In particular, since the negation might fail to be involutive (for instance, it is involutive in Łukasiewicz logic, but not in other well known fuzzy logics), the classical interdefinition fails.
Nevertheless, we can prove different equalities, that will serve us to work with the axiomatic systems presented in [BEGR11] plus a simple definition of the dual operation.
Given two formulas , we will write if and only if for any -model and any it holds .
Let be a finite (bounded, integral, commutative) residuated lattice. Then for any it holds
On the one hand, by residuation, for each , since which is always true. Thus, .
On the other hand, , so
Lemma 3.3 (Interdefinability).
Let be a finite (bounded, integral, commutative) residuated lattice. Then, the following equalities hold:
Is easy to prove that for any ,the following equalities hold:
The first one follows from the definition of the evaluation of as a conjunction. The second one follows from a general property of any residuated lattice (see eg. [JipTsi02]), that states that for any set of elements of the universe and any other element
Concerning the definition of from , the previous properties imply that, for any ,
From the previous lemma, we also know that , so the two formulas evaluate equally in any world of any model. Thus, in particular, for any -model, and any , we can conclude
For what concerns the definability of from , we can use Lemma 3.2 again to get that
From the second property of -models above, we can conclude
After the previous results, it turns out that an axiomatic system addressing both and operators with their intended semantics for can be easily given by adding to the logic presented in [BEGR11] the abbreviation
We will denote this axiomatic system by . See Appendix A for the details on its definition.
4 Multi-modal preference logic
Our objective is that of formalizing using the previously defined modal setting a framework for addressing graded preferences between settings. Thus, several particularities arise in respect to the previous general case. First, not all -models are consistent with the notion of graded preference from Section 2, but only those where is -transitive and reflexive (to capture the transitive and reflexive characteristics of a preference relation). On the other hand, it is important to be able to address global preferences (namely, preferences not in relation to the current world of evaluation, but over the whole model), for instance in the style of von Wright’s treatment of preferences [vW63].
While the restriction to transitive and reflexive models can be dealt with in a systematic way, the addition operations (and its dual through lemma 3.3, ) that behave as global modalities makes it necessary to pass through the unfolding the modality in a family of cut-modalities . Moreover, we also need to restrict the kind of propositional algebras of evaluation to linearly ordered ones. Thus, from this point on, assume is a linearly ordered finite (bounded, integral, commutative) residuated lattice, or equivalently, a finite MTL-chain. These modifications are due to technical reasons in the completeness proof, resulting from the difficulties posed to axiomatize many-valued modal logics with a crisp accessibility relation (necessary in order to get the desired modality) over non-linearly ordered algebras.
Thus, let us define by of multi-modal formulas, again defined as usual from a set of propositional variables, residuated lattice operations , truth constants , and modal operators .
We are now ready to introduce -valued preference Kripke models.
An -preference model is a triple such that
is a set of worlds,
is an -valued -pre-order, i.e. a reflexive and -transitive -valued binary relation between worlds, and
is a world-wise -evaluation of variables.
The evaluation is uniquely extended to formulas of by using the operations in for what concerns propositional connectives, and letting for each ,
Sometimes we will also write for , or even for .
We will denote by the class of -preference models. Given an -preference model and , we write whenever for any , if for all , then too. Analogously, we write whenever for any .
We will denote by differentiated names some particular definable modal operators that enjoy a special meaning in our models. Namely:
Simple computations show that, as expected (from Lemma 3.3), .
It is easy to check that the evaluation of these operators in a preference model as defined here, coincides with the usual one for fuzzy Kripke models, i.e.,
These operators are in fact global necessity and possibility modal operators respectively, i.e.,
4.1 Axiomatizing fuzzy (weak) preference models
In this section we axiomatize the logic whose semantics is given by the class of -preference models, and based on the use of the graded modalities (and in some cases, also the abbreviation ), with , introduced above.
We define the fuzzy multi-modal logic by the following axioms and rules:
Logic (Appendix A) for each with . (This is the axiomatic system of the minimal modal logic over crisp -models ([BEGR11], see Appendix A for details).
For each such that , nestedness axioms
For each , reflexivity and transitivity axioms, namely
Symmetry axiom for , namely ;
Modus Ponens rule and the necessitation rule for each ,222Observe that in , due to the inclusion axioms, the necessitation rules for for are derivable from the one for . namely
from derive .
It will be also useful later to consider the system obtained from by dropping the following axioms:
the reflexivity axioms , for
any axiom involving the subindex
Note that , for , (and so ) are graded counterparts of S4 modalities, while (and so ) is an S5 modality.
Let be the notion of proof for the previous axiomatic system, defined as usual. We can now show that it is indeed complete with respect to our intended semantics given by the class of preference structures .
For any ,
if and only if .
Soundness (left to right direction) is easy to check. For what concerns completeness (right to left direction), we can define a canonical model as in [BEGR11], with a set of crisp accessibility relations as follows, where denotes the set of theorems of :
if and only if for all ,
, for any propositional variable .
To proceed with the completeness proof, it is necessary to prove the so-called Truth Lemma, which states that the evaluation of modal formulas in the model is compatible with the intended semantics. Namely, to show that for any and any . This is proven in [BEGR11], see the appendix A for details on the axiomatic system and the particular references.
Next we show that the set is a nested set of reflexive and transitive relations. That if directly follows from the nestedness axioms, and that each relation is reflexive and transitive follows from axioms and .
Now, from the (crisp) relations , let us define the fuzzy relation as follows:
It is clear that if and only if . Then, the Truth Lemma for the previous Canonical Model immediately implies
It follows from axioms that each is reflexive, and so, is a reflexive relation as well. Moreover, from axioms , we get that is -transitive.
The structure is almost an -preference model: might be a proper subset of , and not the total relation. Indeed, observe that, thanks to axioms and , can be proven to be an equivalence relation, even though it is not necessarily the case that . Hence, the only remaining step is to show that we can obtain an equivalent model (in the sense of preserving the truth-values of formulas) in which is the total relation, and thus to really get that and are global modalities). Nevertheless, since for all , for any we can define a restricted model where , is the restriction of to , and, for any and any formula ,
Now, this model is indeed an -preference model and thus it belongs to the class .
To conclude the proof, observe that, if , then there is such that and (because the only modal inference rules affect only theorems of the logic).
Then, if , there is such that and . All the previous considerations allow us to prove that, in the model , we have and . Hence, and thus as well, and this concludes the completeness proof. ∎
5 Adding strict preferences
As it has been mentioned before, in order to provide a framework allowing a finer handling of preference relations, it would be desirable to have a richer language able to also represent strict preference relations between states.
Within the setting developed in the previous sections, this amounts to consider in the language new modalities and in the models, besides -valued (weak) preference relations on worlds, their strict counterpart. Namely, given an -preference model , recall the relation , the fuzzy strict counterpart of defined in Section 2:
Then, a richer set of formulas, including (fuzzy) modalities for strict preferences , can be evaluated in a -preference model relying on the strict preference relation , as it was done for formulas over -preference models, namely:
As in Lemma 3.3, and are inter-definable, so we will mainly work with the modalities, and use the abbreviation
In the previous section, we relied on the level-cut relations , S4 modalities and global modality to get an indirect axiomatization (the logic ) of the graded preference modality ). We follow a similar approach in this section and consider cut strict modalities for . These modalities are to be interpreted by transitive and irreflexive relations. However, the addition to the system of these modalities in such a way that the new system keeps being complete with respect to the intended semantics (that is, models where the relations that evaluate the strict modalities are irreflexive counterparts of the relations that evaluate the S4 modalities) is not immediate. Indeed, it is well known that an irreflexive modality cannot be axiomatized by a usual axiom or rule schemata (in the sense that an axiom or rule closed under arbitrary substitutions) [BdRV01], and some more involved techniques have been developed [Seg71, Gab81]. We will resort here to the bulldozing construction that, in the classical setting, transforms a reflexive and transitive model into a irreflexive and transitive one with an equivalent logical behavior.
We will see in Section 5.3 and in Appendix B how this classical construction keeps working in the finite-valued case. Nevertheless, the full proof of completeness does not directly follow from the one in used in classical case [vBGR09]) since, although the level-cut accessibility relations are are crisp, the values of the formulas at each world are many-valued, posing additional problems to solve.
5.1 Language and semantics
Let be the expanded set of graded preference formulas defined as usual from a set of propositional variables, residuated lattice operations , truth constants , plus modal operators and .
The interpretation of the modalities will be exactly the same as in Section 4, that is, given a an -preference model , we let
Regarding the new modalities, a first decision that must be taken is choosing the evaluation of the modalities. As discussed in Section 4, there are two possible ways to approach the definition of the strict relations starting from the original fuzzy relation : either with the ’s, the strict versions of the -cuts of , or with the ’s, the -cuts of the strict version of . As it is shown in Prop. 2.2, the original can be recovered from both families, which allows to define using either of the two semantics for (see Lemma 5.1 below). We will present in this section an axiomatization of the logic using the family of crisp relations for each .
Therefore, given the model , we define
where, for any and , stands for , that is, and . In terms of the notation introduced from Definition 4.1, this is equivalent to say that if and only if and .
We will keep denoting by the logical consequence relation over the extended language , defined exactly as done for in Section 4.
As it happened in the previous section, the graded modality , and the corresponding with the intended meaning can be defined from the new set of operations. Namely, we consider the following abbreviations in our language:
It follows from Lemma 3.3 that, under the above definition, in any preference model we get .
On the other hand, also the definition of in the above terms holds the intended meaning stated at the beginning of this section, namely.
For any -preference model and , it holds
from Proposition 2.2. This equals understanding as a -valued relation. By properties of residuated lattices, the previous coincides with . Since the infima are independent, we can swap them and get the independent element out of the interior one to get which is exactly . ∎
In this section we axiomatize over using the systems introduced in Definition 4.2. Note that, as we commented above, the modalities and will be omitted, and that is the reason behind removing in the previous definition all axioms concerning the value .
Soundness of with respect to the intended semantics is not hard to check. The inclusion axioms follow immediately from the fact that . Let us show soundness of the other axioms.
Interaction 1 and 2 axioms are valid in .
Assume , and consider any -preference model , and any . Assume and . From -transitivity of we get that , and since ,
If moreover it holds that , by definition it means that . Using reflexivity and transitivity of it follows that . Since , necessarily , so by definition, . This proves the Interaction 1 cases. The proof of soundness of Interaction 2 is analogous. ∎
Interaction 3 axiom is valid in .
Consider a preference model , and any . By definition, This infimum can be naturally divided in
Concerning the first expression, by monotonicity it is greater or equal than .
Similarly, the second expression is greater or equal than . By using the definition and applying some lattice basic results, we get the following chain of (in)equalities:
Then, , proving the lemma.
To prove completeness of with respect to , we define the canonical model putting together the two sets of modalities in a similar way as it was done for only in Section 4.1. That is to say, we let be the model333In order to lighten the notation we omit the subscript in the elements of the canonical model. defined by: