On Rational Entailment for Propositional Typicality Logic

Propositional Typicality Logic (PTL) is a recently proposed logic, obtained by enriching classical propositional logic with a typicality operator capturing the most typical (alias normal or conventional) situations in which a given sentence holds. The semantics of PTL is in terms of ranked models as studied in the well-known KLM approach to preferential reasoning and therefore KLM-style rational consequence relations can be embedded in PTL. In spite of the non-monotonic features introduced by the semantics adopted for the typicality operator, the obvious Tarskian definition of entailment for PTL remains monotonic and is therefore not appropriate in many contexts. Our first important result is an impossibility theorem showing that a set of proposed postulates that at first all seem appropriate for a notion of entailment with regard to typicality cannot be satisfied simultaneously. Closer inspection reveals that this result is best interpreted as an argument for advocating the development of more than one type of PTL entailment. In the spirit of this interpretation, we investigate three different (semantic) versions of entailment for PTL, each one based on the definition of rational closure as introduced by Lehmann and Magidor for KLM-style conditionals, and constructed using different notions of minimality.

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1 Introduction

Propositional Typicality Logic (PTL) [1, 2] is a recently proposed logic allowing for the representation of and reasoning with an explicit notion of typicality. It is obtained by enriching classical propositional logic with a typicality operator , the intuition of which is to refer to those most typical (or normal or conventional) situations in which a given sentence holds. PTL is characterised using a preferential semantics similar to that originally proposed by Shoham [25] and extensively developed by Kraus et al[20] and Lehmann and Magidor [22] in the propositional case, and by others [4, 7, 8, 18, 24, 14, 15] in more expressive languages.

In spite of the non-monotonic features introduced by the adoption of a preferential semantics for , the obvious definition of entailment for PTL, i.e., the one based on a Tarskian notion of logical consequence, remains monotonic. Of course, such a notion of entailment is inappropriate in non-monotonic contexts, in particular when reasoning about typicality, as is already clear from an enriched version of the classical Tweety example: If birds typically fly, and penguins are birds (and that is all we know), we would expect to be able to conclude that typical penguins are typical birds, and therefore that typical penguins fly. Learning that penguins typically do not fly should lead us to conclude that penguins are not typical birds, and to retract the conclusions about typical penguins being typical birds, and about typical penguins flying.

In this paper, we investigate three semantic versions of entailment for PTL, constructed using three different forms of minimality. All these are based on the notion of rational closure as defined by Lehmann and Magidor [22] for KLM-style conditionals in a propositional setting. We show that they can be viewed as distinct extensions of rational closure, equivalent with respect to the conditional language originally proposed by Kraus et al., but different in the PTL framework.

We shall study the aforementioned forms of entailment in an abstract formal setting, obtained by proposing a set of postulates that, at first glance, seem appropriate for any notion of entailment with regard to typicality. Our first important result is a negative one, though. It is an impossibility result proving that the set of postulates cannot all be satisfied simultaneously. A more detailed analysis of the result shows that, instead of being viewed as negative, this result should rather be interpreted as an indication that PTL allows for different types of entailment, corresponding to different subsets of the full set of postulates we provide. In line with this argument, we define three types of entailment for PTL corresponding to distinct subsets of the postulates, referred to as LM-entailment, PT-entailment, and PT’-entailment, a modification of the latter. Our argument for more than one type of entailment for the same logic is in line with the proposal put forward by Lehmann in the context of entailment for conditional knowledge bases, where he proposes both prototypical reasoning and presumptive reasoning as acceptable forms of entailment [21]. The details of the distinct forms of entailment need not concern us here. Rather, what is important is the acknowledgement of the existence of more than one form of entailment for the same representational formalism.

The remainder of the present paper is structured as follows. Section 2 provides the background and notation for the rest of the work. In Section 3 we discuss the complexities surrounding a notion of entailment for PTL. In Section 4 we put forward our postulates and show the impossibility result. In Section 5 we define LM-entailment while Section 6 is devoted to the definition of PT-entailment, and Section 7 to the definition of PT’-entailment. Section 8 addresses the implications of the impossibility result, making the case for three forms of PTL entailment. Section 9 concludes and discusses future work.

2 Logical preliminaries

Let  be a finite set of propositional atoms with at least two elements.222This (reasonable) assumption is needed for technical reasons. We use as meta-variables for atoms. Propositional sentences (and, in later sections, sentences of the richer language we shall introduce in Section 2.3 below) are denoted by , and are recursively defined in the usual way: . All the other Boolean connectives (, , , …) are defined in terms of  and  in the standard way. With  we denote the set of all propositional sentences.

We denote by the set of all propositional valuations , i.e., . Whenever it eases the presentation, we shall represent valuations as sets of literals (i.e., atoms or negated atoms), with each literal indicating the truth-value of the respective atom. Thus, for the logic generated from , the valuation in which  is true and  is false will be represented as . Satisfaction of a sentence by is defined in the usual truth-functional way and is denoted by .

2.1 KLM-style rational conditionals

In the conditional logic investigated by Kraus et al[20], often referred to as the KLM approach, one is interested in (defeasible) conditionals of the form , read as “typically, if , then ” (or, depending on the example at hand, as “s are typically s” and variants thereof). For instance, if , where , and stand for, respectively, “being a bird”, “being able to fly”, and “being a penguin”, the following are examples of defeasible conditionals: (birds typically fly), (penguins that are birds typically do not fly).

Kraus et al. put forward the following list of properties that the conditional  ought to satisfy in order to be considered as appropriate in a non-monotonic setting (these properties have been discussed at length in the non-monotonic reasoning community and we shall not do so here):

A conditional satisfying such properties is called a preferential conditional. We can require  to satisfy other properties as well, one of which is rational monotonicity:

A preferential conditional also satisfying (RM) is called a rational conditional.

The semantics of KLM-style rational conditionals is given by structures called ranked interpretations [22]:

Definition 2.1 (Ranked interpretation)

A ranked interpretation  is a function from to satisfying the following convexity property: for every , if , then, for every such that , there is a for which .

Observe that generates a modular order on as follows: if and only if (where for every ). If there is no ambiguity, we will omit the subscript and refer to the modular order as .333 Recall that, given a set , is modular if and only if there is a ranking function s.t. for every , if and only if . Note also that modular orders can be obtained from total preorders by imposing anti-symmetry.

In a ranked interpretation  the intuition is that valuations lower down in the ordering are deemed more normal (or typical) than those higher up, with those with an infinite rank (a rank of ) being regarded as so atypical as to be impossible.

The possible valuations in are defined as follows: . Given , we let . Given , we say  satisfies (is a ranked model of) the conditional  (denoted ) if all the -minimal -valuations also satisfy , i.e., if . We say  is a ranked model of a set of conditionals  if for every .

Sometimes it is convenient to represent a ranked interpretation as a partition of where, for , and where is some for which . That is, for each , is the set of all valuations of rank . We refer to such a ranked interpretation as an -rank interpretation.

Observe that the partition above has a finite number of cells, but includes the possibility for some of the s to be empty. This is necessary for two reasons. First, the cell  (the set of all impossible valuations) may be empty. Second, as we shall see below, this representation will often be used to compare ranked interpretations. In cases where such ranked interpretations do not have the same number of non-empty cells, this representation allows us to represent them as having the same (finite) number of cells, say and , where is the smallest integer such that .

Figure 1 depicts an example of a ranked interpretation for satisfying both and . (In our graphical representations of the ranked interpretations we frequently omit the rank .)

,  
,  ,  
Figure 1: A ranked interpretation for .

For a better understanding of the reasons behind the aforementioned properties and the semantic constructions, the reader is referred to the work of Kraus et al[20, 22].

2.2 Rational closure

Given a set of conditionals , reasoning in the KLM framework amounts to the derivation of new conditionals from . Towards this end, Lehmann and Magidor [22] proposed what they refer to as rational closure. Here we focus on the semantic version of rational closure they present.

Their idea was to define a preference relation  over the set of possible ranked interpretations and then to base entailment on choosing only the most preferred, i.e., minimal w.r.t. , ranked models of .

The relation can be described as follows. Consider any pair of ranked interpretations and . Then,

This is not exactly the semantic representation defined by Lehmann and Magidor, but this representation can easily be derived from other work on rational closure, such as that of Booth and Paris [3] and Giordano et al[19]. The idea is that those ranked interpretations should be preferred in which as many valuations as possible are judged to be as plausible as the background knowledge  allows. Observe also that one of the consequences of this ordering is that, all other things being equal, a ranked interpretation in which a valuation is deemed to be possible will be preferred over one in which the same valuation is seen as impossible.

Clearly forms a partial order over ranked interpretations. Lehmann and Magidor showed that for every set of conditionals , there exists a unique -minimum element among all the ranked models of . We will refer to this element as the LM-minimum. Then the rational closure of  is the set . Rational closure is commonly viewed as the basic (although certainly not the only acceptable) form of entailment over propositional conditional knowledge bases, on which other, more venturous, forms of entailment can be constructed. It is therefore an appropriate choice on which to base our investigations into versions of entailment for PTL.

2.3 Propositional Typicality Logic

PTL [1] is a logical formalism explicitly allowing for the representation of and reasoning about a notion of typicality. Syntactically, it extends classical propositional logic with a typicality operator , the intuition of which is to capture the most typical (alias normal or conventional) situations or worlds. Here we shall briefly present the main results about PTL relevant for our purposes.

The language of PTL, denoted by , is recursively defined by:

As before, denotes an atom and all the other Boolean connectives are defined in terms of and .

Let , where , and are as before and  represents “being an ostrich”. The following are examples of -sentences: (being a typical bird), (ostriches are not typical birds), (being a penguin or an ostrich is equivalent to being a bird and being a typical non-flying creature).

Intuitively, a sentence of the form is understood to refer to the typical situations in which  holds. Note that  can itself be a -sentence. The semantics of PTL is also in terms of ranked interpretations (see Definition 2.1). Satisfaction is defined inductively in the classical way, adding the following condition: if and there is no  such that and . That is, given , . In the ranked interpretation  of Figure 1, we have , and .

We say that is satisfiable in a ranked interpretation  if , otherwise  is unsatisfiable in . We say that  is a ranked model of  (denoted ) if .

A PTL knowledge base is a finite set of sentences . We define .

A useful property of the typicality operator  is that it allows us to express KLM-style conditionals. That is, for every ranked interpretation and every , if and only if . The converse does not hold since it can be shown that there are -sentences that cannot be expressed as a set of KLM-style -statements on  [2].

The representation result below, extending Theorem 3.12 of Lehmann and Magidor [22] to , shows that the formalisation of the KLM rational conditional  inside PTL is appropriate.

Observation 1 (Booth et al. [2], Corollary 22)

Let be a ranked interpretation and let and . Then is a rational conditional. Conversely, for every rational conditional , there exists a ranked interpretation such that, for every , if and only if .

For more details on PTL and the aforementioned properties, the reader is referred to the work by Booth et al[2].

3 The entailment problem for PTL

The purpose of this section is to provide a more formal motivation for the remainder of the paper. From the perspective of knowledge representation and reasoning (KR&R), a central issue is that of what it means for a PTL sentence to follow from a (finite) PTL knowledge base . An obvious approach to the matter is to embrace the notion of entailment advocated by Tarski [26] and largely adopted in the logic-based KR&R community.

Definition 3.1 (Ranked entailment and consequence)

Let and . We say ranked-entails  (noted ) if . Its associated ranked consequence operator is defined by setting, for , .

As we shall see below, this version of entailment is not appropriate in the context of PTL for a number of reasons. For one, consider the following definition of a conditional induced from a set of PTL sentences.

Definition 3.2 (Induced conditional relation)

Let . We define and .

It is worth investigating whether  is rational, i.e., whether it satisfies all the KLM properties for rationality from Section 2.1. The following proposition, which mimics a similar result by Lehmann and Magidor in the propositional case, shows that this is not the case:

Observation 2 (Booth et al. [2], Proposition 25)

is a preferential conditional, but is not necessarily a rational conditional.

Hence, ranked consequence as defined above delivers an induced defeasible conditional that is preferential but that need not be rational. This forms an argument against ranked entailment being an appropriate notion of entailment for PTL.

One of the principles to give serious consideration when investigating PTL entailment is the presumption of typicality [21, p. 63]. Informally, this means that one should assume that every situation is assumed to be as typical as possible. Sections 4 and 6 contain a formalisation of this principle. For now, we illustrate it with an example.

Example 3.1

Let (penguins are birds, and typical birds fly). Given just this information about birds and penguins, it is reasonable to expect both (typical penguins are typical birds) and therefore (typical penguins fly) to follow from . It is easy to see that with ranked entailment these requirements are not met, as ranked entailment is not ampliative, i.e., it does not allow for venturing beyond what is sanctioned by the knowledge base. ∎

Besides requiring PTL entailment to be ampliative, we also want it to be defeasible, that is, the conclusions derived under the presumption of typicality in an ampliative way can be retracted in case of new conflicting information. This is illustrated by the following example.

Example 3.2

Assume and (somehow) could follow from  in Example 3.1, but then we are informed that typical penguins do not fly. That is, let . While we want (penguins are not typical birds) to follow from , we do not want to follow from , which is not possible with ranked entailment. ∎

4 Towards a notion of entailment for PTL

We have seen that ranked entailment has some serious drawbacks in a non-monotonic context. Therefore, the question as to what logical consequence in PTL should mean remains mostly unanswered so far. In this section, we first specify and discuss a list of postulates formalising the requirements motivated in the last section and that, at first glance, seem reasonable for an appropriate notion of entailment in PTL. In the subsequent section, we consider specific alternatives to ranked entailment and check them against our postulates.

We start by introducing some notation. With , we denote any entailment relation on the language of PTL. Given an entailment relation , its associated consequence operator is defined in the usual way by setting, for each , .

Following the tradition in the non-monotonic reasoning literature, the obvious starting point is to consider some of the basic properties of classical consequence operators.

P1

(Inclusion)

P2

(Idempotence)

Idempotence specifies that a consequence operator behaves as a ‘once-off’ operation, in the same spirit as that of a closure operator. It implies also its finitary version, the Cumulativity Property: If , then . There is agreement in the literature that both Inclusion and Cumulativity are desirable properties to have.

Ranked entailment, as defined in Section 3, satisfies Properties P1 and P2. Nevertheless, , the associated consequence relation of ranked entailment, also satisfies the classical property of Monotonicity: If , then . As seen in Example 3.1, this is a property that we do not want to satisfy (certainly not in general).

So, we require  to be a non-monotonic consequence operator. This amounts to requiring  to satisfy the following two postulates:

P3

For every , (Ampliativeness)

P4

For some , but (Defeasibility)

Ampliativeness, a property generalising supra-classicality [23] (where the basic underlying entailment relation is classical), says that should be at least as venturous as its underlying ranked entailment. Defeasibility specifies that  should be flexible enough to disallow previously derived conclusions in the light of new (possibly conflicting) information. In Example 3.1, assuming is the case, then should no longer be concluded if  is added to . Note that Defeasibility actually implies a strict version of Ampliativeness which says should in some cases be more venturous than its underlying ranked entailment. (Since, if for all , then is just ranked entailment, which is monotonic.)

P2 and P3 together imply that the closure operation gives as output a theory that is closed under .

Lemma 4.1

If satisfies P2 and P3, then, for every ,

Proof.

Since clearly satisfies inclusion, is immediate. By P3 we have , that, by P2, implies . ∎

Similarly to KLM in the propositional case, we would ideally like the defeasible conditional induced by  (see Definition 3.2) to satisfy all the rationality properties:

P5

For every , is a rational conditional relation on  (Conditional Rationality)

As observed above, P5 requires the defeasible conditional induced by  to be rational—that is, to satisfy all the rationality properties. But from Theorem 3.12 of Lehmann and Magidor [22] it follows that every rational defeasible conditional can be obtained from a single ranked interpretation. So, from this it follows that requiring the defeasible conditional induced by  to be rational amounts to requiring that the defeasible conditional be generated by a single ranked interpretation. That is, by courtesy of this result, P5 can also be rephrased as follows:

P5’

For every , there is a ranked interpretation  s.t., for every , if and only if . ( Single Model)

The next postulate we consider, which is easily shown to be a strengthening of P5, simply applies this same requirement, not just to defeasible statements, but to all statements expressible in PTL:

P6

For every , there is a ranked interpretation  s.t., for all , if and only if (Single Model)

An important special case of a PTL knowledge base is when the individual elements of correspond to KLM-style conditionals.

Definition 4.1 ((Propositional) conditional knowledge base)

A PTL knowledge base will be called a (propositional) conditional knowledge base if each element of is of the form , for .

The next postulate says that if  is a propositional conditional knowledge base, then the result should coincide with Lehmann and Magidor’s definition of rational closure:

P7

If is a conditional knowledge base, then (Extends Rational Closure)

Clearly, P7 implies P4.

The following property was shown by Lehmann and Magidor to be satisfied by the rational closure for conditional knowledge bases.

P8

Let . Then if and only if (Strict Entailment)

P8 states that  should coincide with ranked entailment for those sentences not involving typicality. The motivation for Strict Entailment is twofold. First, it is a proposal for ranked entailment to be the lower bound for entailment w.r.t. classical sentences (those not involving typicality), a proposal that is not controversial. But secondly, it also requires entailment of classical sentences to correspond to exactly those sanctioned by ranked entailment. This can be viewed as adhering to the principle of minimal change. Being Tarskian, ranked entailment is monotonic, and the argument is therefore that, while non-monotonicity may be applicable for sentences involving typicality, it should not be applicable to classical statements.

We are also interested in a couple of progressively weaker versions of Strict Entailment (and the reasons will become clear later on). The first restricts it to hold only when is a conditional knowledge base.

P9

Let be a conditional knowledge base and . Then if and only if (Conditional Strict Entailment)

Note that P7 also implies P9. The latter implies that entailment for PTL coincides with classical propositional entailment in the case of propositional knowledge bases, as formalised by the next property.

P9’

Let and . Then if and only if entails in classical propositional logic. (Classical Entailment)

Since for every , entails in classical propositional logic if and only if , and any is equivalent , P9’ is indeed a weakening of P9 (provided that P8 also holds).

Finally, we consider another property shown by Lehmann and Magidor to be satisfied by the rational closure for conditional knowledge bases.

P10

Let . Then if and only if (Typical Entailment)

The motivation for P10 is similar to that for P8 above. Consequences of the form are those for which  holds in the most typical situations. So, on the one hand, P10 is a proposal for ranked entailment to provide a lower bound for those statements holding in the most typical situations. But as with P8 above, it also provides an upper bound, thereby requiring that of those statements holding in typical situation exactly those sanctioned by ranked entailment ought to be regarded as being entailed by the knowledge base. The argument for this is that ranked entailment is monotonic and, applying the principle of minimal change, it is only when dealing with atypical situations that ranked entailment is not always sufficient.

Although these postulates all seem reasonable on their own, it turns out that they cannot all be satisfied simultaneously. In fact, this impossibility result already holds for a strict subset of the postulates.

Theorem 4.1

There is no PTL consequence operator satisfying all of P1, P2, P3, P5, P8 and P10.

Proof.

About (P5), requiring to satisfy (RM) is equivalent to requiring that, for every knowledge base and whatever formulas , if and , then we have .

Assume satisfies the given properties, and let . By Strict Entailment (P8), (because of e.g. the 2-rank model of ). By Typical Entailment (P10), (because of e.g. the 1-rank model of ). By Inclusion (P1) , and then by (RM) we must conclude that , that is, ; since must satisfy LLE, the latter implies , that is, .

Since by Inclusion (P1) , we have . Since and is monotonic, we have . Then, by Lemma 4.1, that assumes P2 and P3, we have that .

Since , we have that , that is, , that is, by Lemma 4.1, , against (P8). ∎

While, at first glance, this seems to be a negative result, our contention is that it should be interpreted as an indication that a logic as expressive as PTL admits more than one form of entailment. We elaborate directly on this point in Section 8, and indirectly in Sections 5 and 6, where we define and discuss two instances of entailment for PTL.

5 LM-entailment

We now come to our first construction of an entailment relation in PTL. The idea is to try to lift the rational closure construction from conditional knowledge bases to arbitrary knowledge bases in . We first observe that there is nothing to stop us from using the preference relation (see Section 2.2) to compare ranked interpretations of any PTL knowledge base . The question then is, does there always exist a unique LM-minimum element of the ranked models of , as there does in the restricted conditional case? And if so, how can we construct it? We now answer these questions.

We assume as input a PTL knowledge base , where each sentence is in normal form:

Definition 5.1 (Normal form)

is in normal form if it is of the form , where and the , and are all purely propositional sentences.

Theorem 5.1

The normal form is complete for , i.e., for every sentence there is a (finite) set of sentences , each one in normal form, such that .

Proof.

From the results by Booth et al[1, Section 4], it follows that we need only consider sentences with non-nested instances of the typicality operator. So we let be such a sentence. We let the set of typicality atoms be the propositional atoms occurring in together with every sentence of the form where is a propositional sentence (we refer to the latter as pure typicality atoms). And we define the set of typicality literals in the obvious way: the set of typicality atoms and their negations. The set of pure typicality literals consists of the pure typicality atoms and their negations.

Now we define typicality conjunctive normal form as a conjunctive normal form defined on typicality atoms. It follows immediately that can be rewritten as a sentence, say , in typicality conjunctive normal form. Let  be the set of conjuncts occurring in . We show below how to rewrite each conjunct in  into a sentence in normal form. The resulting set  of sentences in normal form is the set referred to above.

By construction, each sentence is a disjunction of typicality literals. We separate them into three disjoint sets, the set of propositional literals, the set of positive pure typicality literals (with cardinality of, say , where ) and the set of negative pure typicality literals (with cardinality of, say , where ). Let be the disjunction of propositional literals, denote the positive pure typicality literals by , and the negative pure typicality literals by . It follows immediately that can be rewritten as the sentence . ∎

For any ranked interpretation , and , let be the ranked interpretation such that for every , and for every . That is, is the ranked interpretation obtained from by turning all valuations not in into impossible valuations. Similarly, let be the ranked interpretation such that for every , and for every . That is, is the ranked interpretation obtained from by increasing the rank of all valuations not in by 1.

We now construct a sequence of ranked interpretations as follows:

Step 1

Set for all , , and ;

Step 2

(separate the valuations which satisfy w.r.t. the current ranked interpretation from those that do not);

Step 3

If , then return (if there is no change in the new then set the rank of those valuations that do not satisfy w.r.t. to and return the interpretation that remains);

Step 4

Otherwise (otherwise create a new ranked interpretation by increasing the rank of every valuation not in by 1);

Step 5

and (separate the valuations which satisfy w.r.t. the current ranked interpretation from those that do not, and increment );

Step 6

Go to Step 3.

Algorithm 1 below gives a compact description of the above steps.

Input:
Output:
1 ;
2 Let be the universe of valuations for the vocabulary ;
3 for every ;
4 ;
5 ;
6 ;
7 while  do
8        ;
9        ;
10        ;
11       
12 ;
return
Algorithm 1 LM-minimal
Example 5.1

Let us assume, for the sake of the example, that we are only talking about birds. Let (the most typical things are neither penguins nor robins, typical penguins are typical non-flying birds, and typical robins are typical flying birds, penguins are not robins). The procedure initialises with all valuations being assigned the rank of . The only valuations that satisfy all three sentences w.r.t.  are those satisfying both and . Thus and so we obtain by changing the rank of all valuations not in to . Note that and , so we can see that none of the valuations in is able to satisfy either or w.r.t. . As a consequence, and so the procedure terminates here with as the ranked interpretation in which all valuations in ( and ) have rank and all other valuations have rank . See Figure 2 for the ranked interpretations generated by this example. ∎

  , , , , , , ,

  , , , , , ,

, , , , , ,

with the valuations of rank  omitted:  ,

Figure 2: The ranked interpretations generated in Example 5.1.

We now need to show that: (i) the algorithm always terminates; (ii) it returns a ranked model of , and (iii) for any other ranked model  of , we have . We know the following about (i) and (ii):

Lemma 5.1

The following hold for each :

  1. , i.e., ;

  2. For all , if , then ;

  3. is a ranked interpretation.

Proof.

See A.1. ∎

From Item 1 in Lemma 5.1 above, we know the algorithm terminates, since it generates a sequence of ranked interpretations (by Item 3) in which the set of valuations satisfying increases monotonically from one ranked interpretation to the next. Since each of these is finite, and since there is a finite number of valuations, the stopping criterion in Step 3 of the algorithm is guaranteed to occur eventually.

To show that the algorithm returns a ranked model of it suffices to show the following.

Lemma 5.2

For every and every , is a ranked model of .

Proof.

See A.2. ∎

So, at each stage of the algorithm, the current ranked interpretation, when those valuations not satisfying are excluded, forms a ranked model of . Since the output takes precisely this form we have the following result.

Proposition 5.1

.

Proof.

Follows from Lemma 5.2 and the construction of . ∎

Next we want to show that for any other ranked model  of , we have .

Lemma 5.3

Let and let be any other ranked model of . Let . If for all , then .

Proof.

See A.3. ∎

From this lemma we can state:

Proposition 5.2

Consider any and let be a ranked model of . Then .

We are now in a position to define our first form of entailment for PTL.

Definition 5.2 (LM-entailment)

Let and . We say LM-entails , denoted , if . Its corresponding consequence operator is defined as .

The next result outlines which properties from the previous section are satisfied by .

Theorem 5.2

satisfies P1–P7, P9, and P10, but not P8.

Proof.

For P1, Proposition 5.1 guarantees that is a model of . About P2, by Proposition 5.2, is the LM-minimum model of . If , must also be the LM-minimum model of . For P3, note that is a ranked model of (Lemma 5.1, Item 3, plus Proposition 5.1), and so if , then . P4 is an immediate consequence of the satisfaction of P7.444For this conclusion we need the requirement (specified in Section 2) that contains at least two elements. P5 is an immediate consequence of the satisfaction of P6. The latter holds by definition of . For P7, see Section 2.2. P9 is an immediate consequence of the satisfaction of P7.

Now consider P10. From right to left, it is an immediate consequence of P3. From left to right, assume there is a formula that is in , but not in . It means that there is a ranked model of that has in its lower layer a propositional valuation s.t. ; but, given that the model  defining is the LM-minimum model of , then also the lower layer of must contain the valuation , against the hypothesis.

Failure of P8 can be seen in Example 5.1. There we have (there is no penguin) because holds in both valuations occurring in . But , because there does exist a ranked model of for which , for instance the model appearing in Example 6.1 below. Thus LM-entailment forces us to infer from . ∎

In summary then, LM-entailment satisfies all our postulates, except for Strict Entailment (P8). Lest this be seen as a negative result, bear in mind that LM-entailment satisfies Conditional Strict Entailment (P9), the weakened version of Strict Entailment, and therefore also Classical Entailment.

In the next section we turn to a form of entailment satisfying Strict Entailment, but at the price of having to forego Conditional Rationality, and therefore the Single Model postulate as well.

6 PT-entailment

In this section we consider another option for entailment based on a version of minimality, and derived from the characterisation of rational closure by Giordano et al[17, 19]. The general idea is to respect the principle of presumption of typicality (see Section 3), We shall refer to this form of entailment as Presumption of Typicality entailment, shortened to PT-entailment. Such a principle indicates the way in which the property (RM) should be satisfied. If we have in our knowledge base , then, in order to satisfy (RM), we have to add either or . The presumption of typicality requires that, whenever possible, we prefer the latter (that corresponds to a constrained application of monotonicity) over the former. Semantically, given the ranked models of a knowledge base , this corresponds to considering only those models in which every valuation is taken as typical as possible, that is, it is ‘pushed downward’ in the model as much as possible, modulo the satisfaction of .

In order to identify the interpretations that are necessary for the definition of a notion of entailment, we introduce a preference relation on the set of ranked interpretations that follows directly from the presumption of typicality.

Definition 6.1 (Relation )

For two ranked interpretations and , if and only if for every , . if and only if and not .

It is easy to check that is a pre-order. Consistent with the principle of presumption of typicality, as a guideline in the choice of the relevant interpretations, the relation can be used to identify the relevant interpretations for the definition of a notion of entailment: we choose the models of  in which the valuations are presumed to be as typical as possible, that is, the relevant models are those that are in . Then, entails if and only if holds in all the (preferred) models in .

If we consider knowledge bases composed only of classical non-monotonic conditionals , it corresponds exactly to LM-minimality as defined in the previous section. Nevertheless, given the extra expressive power of PTL, we obtain the surprising result that the two semantic constructions are not equivalent anymore. Moreover, in the present context, this notion of minimality can give back a number of minimal models, as the following example shows.

Example 6.1

Consider the knowledge base from Example 5.1. Then, one can see that