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1 Introduction
Conditionals are at the heart of human everyday reasoning and play an important role in the logical formalisation of reasoning. They can usually be interpreted in many ways: as necessity Kratzer (1979); Bouchon-Meunier et al. (2002), as presumption Kraus et al. (1990); Lehmann and Magidor (1992); Boutilier (1994), normative Hansson (1969); Makinson and Torre (2000), causal Beller and Kuhnmünch (2007); Giordano and Schwind (2004), probabilistic Schurz (1998); Snow (1999); Hawthorne and Makinson (2007), counterfactual Starr (2021); Lewis (1973), and many others. Two very common interpretations, that are also strongly interconnected, are conditionals representing expectations (‘If it is a bird, then presumably it flies’), and conditionals representing counterfactuals (‘If Napoleon had won at Waterloo, the whole of Europe would be speaking French’). Although they are connected by virtue of being conditionals, the types of reasoning they aim to model differ somewhat. For instance, the first example above assumes that the premises of conditionals are consistent with what is believed, while the second example assumes that those premises are inconsistent with an agent’s beliefs. That this point is problematic can be made concrete with an extended version of the (admittedly over-used) penguin example.
Example 1.1.
Suppose we know that birds usually fly, that penguins are birds that usually do not fly, that dodos were birds that usually did not fly, and that dodos do not exist anymore. As outlined in more detail in Example 3.1 later on, the standard preferential semantic approach to representing conditionals Lehmann and Magidor (1992) is limited in that it allows for two forms of representation of an agent’s beliefs. In the one, it would be impossible to distinguish between atypical (exceptional) entities such as penguins, and non-existing entities such as dodos (they are equally exceptional). In the other, it would be possible to draw this type of distinction, but at the expense of being unable to reason coherently about counterfactuals—the agent would be forced to conclude anything and everything from the (nowadays absurd) existence of dodos.
In this work we introduce a logic of situated conditionals to overcome precisely this problem. The central insight is that adding an explicit notion of situation to standard conditionals allows for a refined semantics of this enriched language in which the problems described in Example 1.1 can be dealt with adequately. It also allows us to reason coherently with counterfactual conditionals such as ‘Had Mauritius not been colonised, the dodo would not fly’. That is, counterfactuals can be inconsistent with the premise of a conditional without lapsing into inconsistency. Moreover, it is possible to reason coherently with situated conditionals without needing to know whether their premises are plausible or counterfactual. In the case of penguins and dodos, for example, it allows us to state that penguins usually fly in the situation where penguins exist, and that dodos usually fly in the situation where dodos also exist, while being unaware of whether or not penguins and dodos actually exist. At the same time, it remains possible to make classical statements, as well as statements about what necessarily holds, regardless of any plausible or counterfactual premise.
The remainder of the paper is organised as follows. Section 2 outlines the formal preliminaries of propositional logic and the preferential semantic approach to conditionals on which our work is based. Section 3 is the heart of the paper. It describes the language of situated conditionals, furnishes it with an appropriate and intuitive semantics, and motivates the corresponding logic by way of examples, formal postulates, and a formal representation result. With the basics of the logic in place, Section 4 defines a form of entailment for it that is based on the well-known notion of rational closure Lehmann and Magidor (1992). As such, it plays a role that is similar to the one that rational closure plays for reasoning with conditionals—it is a basic form of entailment on which other forms of entailment can be constructed. Section 5 shows that, from a computational perspective, the version of entailment we propose in the previous section is reducible to classical propositional reasoning. Section 6 reviews related work, while Section 7 concludes and considers future avenues to explore. Longer proofs are presented in the appendix.
2 Formal background
In this paper, we assume a finite set of propositional atoms and use to denote its elements. Sentences of the underlying propositional language are denoted by , and are built up from the atomic propositions and the standard Boolean connectives in the usual way. The set of all propositional sentences is denoted by .
A valuation (alias world) is a function from into . The set of all valuations is denoted , and we use to denote its elements. Whenever it eases presentation, we represent valuations as sequences of atoms (e.g., ) and barred atoms (e.g., ), with the usual understanding. As an example, if , with the atoms standing for, respectively, ‘being a ird’, ‘being a lying creature’, and ‘being a enguin’, then the valuation conveys the idea that is true, is false, and is true.
With we denote the fact that satisfies . Given , with we denote its models. For , . We say (classically) entails , denoted , if . Given a set of valuations , indicates a sentence characterising the set . That is, is a propositional sentence satisfied by all, and only, the valuations in .
2.1 KLM-style rational defeasible consequence
A defeasible consequence relation is a binary relation on . Intuitively, , which is usually represented as the statement , captures the idea that “ is a defeasible consequence of ”, or, in other words, that “if , then usually (alias normally, or typically) ”. The relation is said to be rational Kraus et al. (1990) if it satisfies the well-known KLM postulates below:
The merits of these postulates have been addressed extensively in the literature Kraus et al. (1990); Gabbay (1984) and we shall not repeat them here.
A suitable semantics for rational consequence relations is provided by ordered structures called ranked interpretations.
Definition 2.1 (Ranked Interpretation).
A ranked interpretation is a function from to , satisfying the following convexity property: for every and every , if , then, for every s.t. , there is a for which .
For a given ranked interpretation and valuation , we denote with the rank of . The number indicates the degree of atypicality of . So the valuations judged most typical are those with rank 0, while those with an infinite rank are deemed so atypical as to be implausible. We can therefore partition the set w.r.t. into the set of plausible valuations , and implausible valuations . (Throughout the paper, we shall use the symbol to refer to finiteness.) With , for , we indicate all the valuations with rank in (we omit the subscript whenever it is clear from the context).
Assuming , with the intuitions as above, Figure 1 below shows an example of a ranked interpretation.
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Let be a ranked interpretation and let . Then , and , for all . A defeasible consequence relation can be given an intuitive semantics in terms of ranked interpretations as follows: is satisfied in (denoted ) if , with referred to as a ranked model of . In the example in Figure 1, we have , , , , and . It is easily verified that iff . Hence we frequently abbreviate as . Two defeasible consequences and are said to be rank equivalent iff they have the same ranked models — that is, if for every ranked interpretation , iff .
The correspondence between rational consequence and ranked interpretations is formalised by the following representation result.
Theorem 2.1 (Lehmann & Magidor, 1992; Gärdenfors & Makinson, 1994).
A defeasible consequence is rational iff there is a ranked interpretation such that iff .
2.2 Rational closure
One can also view defeasible consequence as formalising some form of (defeasible) conditional and bring it down to the level of statements. Such was the stance adopted by Lehmann and Magidor Lehmann and Magidor (1992). A conditional knowledge base is thus a finite set of statements of the form , with . As before, in knowledge bases we shall also abbreviate with . As an example, let . Given a conditional knowledge base , a ranked model of is a ranked interpretation satisfying all statements in . As it turns out, the ranked interpretation in Figure 1 is a ranked model of the above . It is not hard to see that, in every ranked model of , the valuations and are deemed implausible—note, however, that they are still logically possible, which is the reason why they feature in all ranked interpretations. Two conditional knowledge bases are rank equivalent iff they have exactly the same ranked models.
An important reasoning task in this setting is that of determining which conditionals follow from a conditional knowledge base. Of course, even when interpreted as a conditional in (and under) a given knowledge base , is expected to adhere to the postulates of Section 2.1. Intuitively, that means whenever appropriate instantiations of the premises in a postulate are sanctioned by , so should the suitable instantiation of its conclusion.
To be more precise, we can take the defeasible conditionals in as the core elements of a defeasible consequence relation . By closing the latter under the preferential rules (in the sense of exhaustively applying them), we get a preferential extension of . Since there can be more than one such extension, the most cautious approach consists in taking their intersection. The resulting set, which also happens to be closed under the preferential rules, is the preferential closure of , which we denote by . It turns out that the preferential closure of contains exactly the conditionals entailed by . (Hence, the notions of closure of and entailment from a conditional knowledge base are two sides of the same coin.) The same process and definitions carry over when one requires the defeasible consequence relations also to be closed under the rule RM, in which case we talk of rational extensions of . Nevertheless, as pointed out by Lehmann and Magidor (Lehmann and Magidor, 1992, Section 4.2), the intersection of all such rational extensions does not, in general, yield a rational consequence relation: it coincides with preferential closure and therefore may fail RM. Among other things, this means that the corresponding entailment relation, which is called rank entailment and defined as if every ranked model of also satisfies , is monotonic and therefore it falls short of being a suitable form of entailment in a defeasible reasoning setting. As a result, several alternative notions of entailment from conditional knowledge bases have been explored in the literature on non-monotonic reasoning Lehmann (1995); Booth and Paris (1998); Weydert (2003); Giordano et al. (2012, 2015); Booth et al. (2019); Casini et al. (2019), with rational closure Lehmann and Magidor (1992) commonly acknowledged as the ‘gold standard’ in the matter.
Rational closure (RC) is a form of inferential closure extending the notion of rank entailment above. It formalises the principle of presumption of typicality (Lehmann, 1995, p. 63), which, informally, specifies that a situation (in our case, a valuation) should be assumed to be as typical as possible (w.r.t. background information in a knowledge base).
Multiple equivalent characterisations of RC have been proposed Lehmann and Magidor (1992); Pearl (1990); Booth and Paris (1998); Hill and Paris (2003); Britz et al. (2020), and here we rely on the one by Giordano and others Giordano et al. (2015). Assume an ordering on all ranked models of a knowledge base , which is defined as follows: , if, for every , . Intuitively, ranked models lower down in the ordering correspond to descriptions of the world in which typicality of each situation (valuation) is maximised. It is easy to see that is a weak partial order. Giordano et al.. Giordano et al. (2015) showed that there is a unique -minimal element. The rational closure of is defined in terms of this minimum ranked model of .
Definition 2.2 (Rational Closure).
Let be a conditional knowledge base, and let be the minimum element of on ranked models of . The rational closure of is the defeasible consequence relation .
As an example, Figure 1 shows the minimum ranked model of w.r.t. . Hence we have that is in the rational closure of (but note it is not in the preferential closure of ).
Observe that there are two levels of typicality at work for rational closure, namely within ranked models of , where valuations lower down are viewed as more typical, but also between ranked models of , where ranked models lower down in the ordering are viewed as more typical. The most typical ranked model is the one in which valuations are as typical as allows them to be (the principle of presumption of typicality we alluded to above).
Rational closure is commonly viewed as the basic (although certainly not the only acceptable) form of non-monotonic entailment, on which other, more venturous forms of entailment can be and have been constructed Lehmann (1995); Kern-Isberner (2001); Casini et al. (2014); Booth et al. (2019); Casini et al. (2019).
3 Situated conditionals
We now turn to the heart of the paper, the introduction of a logic-based formalism for the specification of and reasoning with situated conditionals. For a more detailed motivation, let us consider a more technical version of the penguin-dodo example introduced in Section 1.
Example 3.1.
We know that birds usually fly (), and that penguins are birds () that usually do not fly (). Also, we know that dodos were birds () that usually did not fly (), and that dodos do not exist anymore. Using the standard ranked semantics (Definition 2.1), we have two ways of modelling the information above.
The first option is to formalise what an agent believes by referring to the valuations with rank in a ranked interpretation. That is, the agent believes is true iff holds. In such a case, means that the agent believes that dodos do not exist. The minimal model for this conditional knowledge base is shown in Figure 2 (left). The main limitation of this representation is that all exceptional entities have the same status as dodos, since they cannot be satisfied at rank . Hence we have , just as we have , and we are not able to distinguish between the status of the dodos (they do not exist anymore) and the status of the penguins (they do exist and are simply exceptional birds).
The second option is to represent what an agent believes in terms of all valuations with finite ranks. That is, an agent believes to hold iff holds. If dodos do not exist, we add the statement . The minimal model for this case is depicted in Figure 2 (right). Here we can distinguish between what is considered false (dodos exist) and what is exceptional (penguins), but we are unable to reason coherently about counterfactuals, since from we can conclude anything about dodos.
A situated conditional (SC for short) is a statement of the form , with , which is read as ‘given the situation , usually holds on condition that holds’. Formally, a situated conditional is a ternary relation on . We shall write as an abbreviation for . To provide a suitable semantics for SCs, we define a refined version of the ranked interpretations of Section 2 that we refer to as epistemic interpretations. A ranked interpretation can differentiate between plausible valuations (those in ) but not between implausible ones (those in ). In contrast, an epistemic interpretation can also tell implausible valuations apart. We thus distinguish between two classes of valuations: plausible valuations with a finite rank, and implausible valuations with an infinite rank. Within implausible valuations, we further distinguish between those that would be considered as possible, and those that would be impossible. This is formalised by assigning to each valuation a tuple of the form , where , or , where . The in is meant to indicate that has a finite rank, while the in is intended to denote that has an infinite rank, where finite ranks are viewed as more typical than infinite ranks. Implausible valuations that are considered possible have an infinite rank , where , while those considered impossible have the infinite rank , where is taken to be less typical than any of the other infinite ranks.
To capture this formally, let denote henceforth a set of ranks. We define the total ordering over Rk as follows: if and , or and , where for all .
Definition 3.1 (Epistemic Interpretation).
An epistemic interpretation is a total function from to Rk for which the following convexity property holds: (i) for every and every , if , then, for all s.t. , there is a s.t. , and (ii) for every and every , if , then, for all s.t. , there is a s.t. .
Observe that the version of convexity satisfied by epistemic interpretations is a straightforward extension of the convexity of ranked interpretations (Definition 2.1). Figure 3 depicts an epistemic interpretation in our running example.
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Casini et al.. Casini et al. (2020) have a similar definition of epistemic interpretations, but they do not allow for the rank .
We let , for some and , for some . Note that does not contain valuations with rank . We let , for all , , for all , and , for all .
Observe that epistemic interpretations are allowed to have no plausible valuations (), as well as no implausible valuations that are possible (). This means it is possible that for all , in which case , for all . Epistemic interpretations also allow for the case where all valuations are possible (that is, either plausible, or implausible but possible). This corresponds to the case where an epistemic interpretation does not have any valuation with rank .
Armed with the notion of epistemic interpretation, we can provide an intuitive semantics to situated conditionals.
Definition 3.2 (Satisfaction of Situated Conditionals).
Let be an epistemic interpretation. We say satisfies the situated conditional , denoted as and often abbreviated as , if
Intuitively, satisfaction of situated conditionals works as follows. If the situation is compatible with the plausible part of (the valuations in ), then holds if the most typical plausible models of are also models of . On the other hand, if the situation is not compatible with the plausible part of , i.e., all models of have an infinite rank, then holds if the most typical implausible (but possible) models of are also models of .
An immediate corollary of Definition 3.2 is that the rational conditionals defined in terms of ranked interpretations can be simulated with SCs by setting the situation to .
Definition 3.3 (Extracted Ranked Interpretation).
For an epistemic interpretation , we define the ranked interpretation extracted from as follows: for , , where , and for .
Corollary 3.1.
Let be an epistemic interpretation. Then iff .
Proof.
Assume . Then, by definition, we have if , and otherwise. If the former is the case, then, by the construction of , we have , and therefore . If, instead, the latter holds, then , from which it follows that , and therefore . For the other direction, assume . If , then, from the construction of , we have , from which we get . If , then, since , we must have , too. From the latter it follows that , and therefore . ∎
The principal advantage of situated conditionals and their associated enriched semantics in terms of epistemic interpretations is that they allow us to represent different degrees of epistemic involvement, with the finite ranks (the plausible valuations) representing the expectations of an agent. So being true in indicates that is expected. What an agent believes to be true is what is true in all the valuations with finite ranks. That is, the agent believes to be true iff . It is also possible to reason counterfactually. We can express that dodos would not fly, if they existed, in a coherent way. We can talk about dodos in a counterfactual situation or context, for example assuming that Mauritius had never been colonised (): the conditional is read as ‘In the situation of Mauritius not having been colonised, the dodo would not fly’. Moreover, we can reason coherently with a situated conditional, not even knowing whether its premises are plausible or counterfactual. To do so, it is sufficient to introduce statements of the form . If is plausible, this conditional is evaluated in the context of the finite ranks, exactly as if were being evaluated. On the other hand, if holds, will be evaluated referring to the infinite ranks. So, in the case of penguins and dodos, and express the information that penguins usually do not fly in the situation of penguins existing, and that dodos usually do not fly in the situation of dodos existing, regardless of whether the agent is aware of penguins or dodos existing or not. In contrast, a statement such as cannot be used to reason counterfactually about dodos, once we are aware that they do not exist (that is, ): given the latter, once we consider all the interpretations satisfying (that is, all the interpretations) our reasoning about dodos would be trivial, since we would be able to conclude everything about dodos, that is, we would be able to conclude for any proposition . Also, note that it is still possible to impose that something necessarily holds. The conditional holds only in epistemic interpretations in which all models of have as their rank. The following example illustrates these claims more concretely.
Example 3.2.
Consider the following rephrasing of the statements in Example 3.1. ‘Birds usually fly’ becomes . Defeasible information about penguins and dodos are modelled using and . Given that dodos don’t exist anymore, the statement leaves open the existence of dodos in the infinite rank, which allows for coherent reasoning under the assumption that dodos exist (the situation ). Moreover, information such as dodos and penguins necessarily being birds can be modelled by the conditionals and , relegating the valuations in to the rank . Figure 3 (below Definition 3.1) shows a model of these statements.
Next we consider the class of situated conditionals from the perspective of a list of situated rationality postulates in the KLM style. We start with the following ones:
Observe that they correspond exactly to the original KLM postulates, except that the notion of situation has been added.
Definition 3.4 (Basic Situated Conditional).
An SC is a basic situated conditional (BSC, for short) if it satisfies the situated rationality postulates.
An immediate corollary of this definition is that for a BSC with the situation fixed, is a rational conditional. We then get the following result.
theoremrestatableTheoremBSC Every epistemic interpretation generates a BSC, but the converse does not hold.
The reason why the converse of Theorem 3.4 does not hold is that the structure of a BSC is completely independent of the situation referred to in the situated KLM postulates. As a very simple instance of this problem, observe that BSCs are not even syntax-independent w.r.t. the situation. That is, we may have but , where . To put it another way, a BSC is simply a rational defeasible consequence relation with the situation playing no role whatsoever in determining the structure of the BSC. To remedy this, we require BSCs to satisfy the following additional postulates:
We shall refer to these as the situated AGM postulates for reasons to be outlined below.
Definition 3.5 (Full Situated Conditional).
A BSC is a full SC (FSC) if it satisfies the situated AGM postulates.
One way in which to interpret the addition of a situation to conditionals, from a technical perspective, is to think of it as similar to belief revision. That is, can be thought of as stating that if a revision with has taken place, then will hold on condition that holds. With this view of situated conditionals, the situated AGM postulates above are seen as versions of the AGM postulates for belief revision Alchourrón et al. (1985). The names of these postulates were chosen with the names of their AGM analogues in mind. The situated AGM postulates can be motivated intuitively as follows.
Together, Inc and Vac require that when the situation (or revision with) is compatible with what is currently plausible, then a conditional w.r.t. the situation (a ‘revison by’ ) is the same as a conditional where the situation is (where there is no ‘revision’ at all), but with added to the premise of the conditional. Ext ensures that situation is syntax-independent. Finally, SupExp and SubExp together require that if the situation is implausible (that is, the ‘revision’ with is incompatible with what is plausible), then a conditional w.r.t. the situation (a ‘revision by’ ) is the same as a conditional where the situation (or ‘revision’) is , but with added to the premise of the conditional.
It turns out that FSCs are characterised by epistemic interpretations, resulting in the following representation result.
theoremrestatableTheoremFSC Every epistemic interpretation generates an FSC. Every FSC can be generated by an epistemic interpretation.
The AGM-savvy reader may have noticed that the following two obvious analogues of the suite of situated AGM postulates are missing from our list above.
Succ requires situations to matter: a ‘revision’ by will always be successful. Cons states that we will obtain an inconsistency only when the situation is inconsistent.
It turns out that Succ holds for epistemic interpretations, but follows from the combination of the situated KLM and AGM postulates, while just one direction of Cons holds.
Corollary 3.2.
Every FSC satisfies Succ, but there are FSCs for which Cons does not hold. However, the right-to-left direction of Cons holds: If then .
Proof.
To prove that Succ holds, it suffices, by Theorem 3.4, to show that for all epistemic interpretations and all . To see that this holds, observe that .
To prove that Cons does not hold, it suffices, by Theorem 3.4, to show that there is an epistemic interpretation such that but . To construct such an , let (and so for all . It is easy to see that by picking any s.t. the result follows.
The fact that Cons does not hold can be explained by considering the epistemic interpretation where all valuations are taken to be impossible (that is, to have the rank ), in which case all statements of the form are true.
We conclude this section by considering the following two postulates.
Incons requires that all conditionals hold when the situation is inconsistent, while Cond requires that conditionals w.r.t. the situation be equivalent to the same conditional with added to the premise and with a tautologous situation (i.e., the situation is ), provided that is not inconsistent w.r.t. the tautologous situation.
Proposition 3.1.
Every FSC satisfies Incons and Cond.
Proof.
To prove that Incons holds, it suffices, by Theorem 3.4, to show that for all epistemic interpretations , and all . To see that this holds, observe that .
To prove that Cond holds, it suffices, by Theorem 3.4, to show that if , then iff for all epistemic interpretations , and all . So, suppose that . By Definition 3.2, this means that and also that . From this, by Definition 3.2, we need to show that iff for the result to hold, which follows immediately. ∎
4 Entailment
The previous section provides a framework for characterising the class of full situated conditionals in terms of epistemic interpretations. In this section, we move to an investigation of how we can reason within this framework. More precisely, the question of interest is the following: given a finite set of situated conditionals, or a situated conditional knowledge base (SCKB) , which situated conditionals can be said to be entailed from it? Lehmann and Magidor already pointed out that in a non-monotonic framework it is generally not appropriate to consider entailment relations that are Tarskian in nature. The reason for this is that such entailment relations are, by definition, monotonic. As a result, they tend to be too weak, inferentially speaking Lehmann and Magidor (1992). Rather, more suitable entailment relations can be defined by picking a single model of the knowledge base satisfying some desirable postulates. It is generally accepted that there is not a unique entailment relation for defeasible reasoning, with different forms of entailment being dependent on the kind of reasoning one wants to model Lehmann (1995); Casini et al. (2019). In the framework of preferential semantics, rational closure, recalled in Section 2, is generally recognised as a core form of entailment with other apt forms of entailment being refinements of rational closure.
We now present a version of rational closure, reformulated for our framework, that we refer to as minimal closure (MC). We adapt the notion of a minimal model Giordano et al. (2015), recalled in Section 2, for our framework, and show that for any SCKB the minimal model is unique.
The construction of the minimal model is obtained by creating a bridge between situated conditionals and epistemic interpretations on one hand and defeasible conditionals and ranked interpretations on the other. Some notions can naturally be extended from the latter framework to the former one. First of all, we can extend the notion of consistency. A set of defeasible conditionals is consistent iff it has a ranked model s.t. . This is the case since such a model does not satisfy the conditional , which captures absurdity in the conditional framework. This condition can easily be translated into our framework.
Definition 4.1 (SCKB Consistency).
An SCKB is consistent if it has an epistemic model s.t. .
In other words, an SCKB is consistent if it has an epistemic model that does not satisfy . is a notation for epistemic interpretations that mirrors the notation for ranked interpretations, that is, represents the set of worlds that have rank in .
Given Corollary 3.1, we can define the satisfaction of defeasible conditionals also for epistemic interpretations:
Note that an epistemic interpretation satisfies exactly the same defeasible conditionals of its extracted ranked interpretation (see Definition 3.3). That is, the ranks specified inside are totally irrelevant w.r.t. the satisfaction of the defeasible conditionals of the form . We can also intuitively define the converse operation of the extraction of a ranked interpretation from an epistemic interpretation: we can extract an epistemic interpretation from a given ranked interpretation.
Definition 4.2 (Extracted Epistemic Interpretation).
For a ranked interpretation , we define the epistemic interpretation extracted from as follows: for , , where , and , for .
It is easy to see that and are equivalent w.r.t. the satisfaction of defeasible conditionals.
The following corollary of Proposition 3.1, that is simply a semantic reformulation of the postulate Cond, will be central in connecting the satisfaction of situated conditionals to that of defeasible ones.
Corollary 4.1.
For every epistemic interpretation , if , then iff .
Proof.
Since it is just a semantic reformulation of the postulate Cond, it follows directly from the proof in Proposition 3.1 that Cond holds. ∎
Given Corollary 4.1, we define a simple transformation of situated conditional knowledge bases.
Definition 4.3.
Let be an SCKB; with we denote its conjunctive classical form, defined as follows: .
We can use the conjunctive classical form to define two models for an SCKB : the classical epistemic model and the minimal epistemic model. The former is the epistemic interpretation generated by the minimal ranked model of .
Definition 4.4 (Classical Epistemic Model).
Let be an SCKB, its conjunctive classical form, and the minimal ranked model of . The classical epistemic model of is the epistemic interpretation extracted from (see Definition 4.2).
Since is a ranked model of , so is . We need to check whether is also a model of .
Proposition 4.1.
Let be an SCKB, and let be defined as in Definition 4.4. Then, we have that is a model of .
Proof.
Let . Since is an epistemic model of and we have Corollary 4.1, if , then we conclude . Otherwise, we have , which implies , which in turn implies . ∎
Proposition 4.2.
Let be an SCKB. has an epistemic model iff has a ranked model.
Proof.
Proposition 4.1 shows that if has a ranked model, then has an epistemic model. For the opposite direction, assume that has an epistemic model . From , we define an epistemic model in the following way:
It is easy to check that is an epistemic model of . Moreover, thanks to Corollary 4.1, it is an epistemic model of : for every , if , then , by Corollary 4.1; if , then , and we can conclude .
Let be the ranked model corresponding to , that is,
Since for every pair of valuations in , is preferred to in iff is preferred to in , it is easy to see that if is an epistemic model of , then is a ranked model of . ∎
By linking the satisfaction of an SCKB to the satisfaction of its conjunctive form , we are able to define a simple method for checking the consistency of an SCKB, based on the materialisation of . The materialisation of a set of defeasible conditionals is the set of material implications corresponding to the conditionals in , defined in the following way:
Corollary 4.2.
An SCKB is consistent iff .
This corollary is an immediate consequence of Proposition 4.2 and the well-known property that a finite set of defeasible conditionals is consistent if and only if its materialisation is a consistent propositional knowledge base (Lehmann and Magidor, 1992, Lemma 5.21).
The results above show that a classical epistemic model serves as the basis for reducing SCKB consistency checking to simple propositional satisfiability checking. This is because it is a direct translation of a ranked interpretation into an equivalent epistemic interpretation. At the same time, since classical epistemic models are not sufficiently expressive to define appropriate forms of entailment, we now move to the definition of the minimal epistemic model, referring to the minimality order introduced for ranked interpretations in Section 2. We need to adapt, in an intuitive way, the notion of minimality defined for ranked interpretations to the present framework. In Section 3, we defined a total ordering over the tuples representing the ranks in epistemic interpretations. Let the ordering on all the epistemic models of an SCKB be defined as follows: , if, for every , , and there is a s.t. .
Definition 4.5 (Minimal Epistemic Model).
Let be a consistent SCKB, and be the set of its epistemic models. is a minimal epistemic model of if there is no s.t. .
We first define the construction of a model, given a consistent SCKB . Then we prove that it is actually the unique minimal epistemic model of w.r.t. the ordering .
Definition 4.6 (Construction of a Minimal Epistemic Model).
Let be a consistent SCKB, its conjunctive classical form, and be the minimal ranked model of . We pick out in a set
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