A General Framework for Modelling Conditional Reasoning – Preliminary Report

We introduce and investigate here a formalisation for conditionals that allows the definition of a broad class of reasoning systems. This framework covers the most popular kinds of conditional reasoning in logic-based KR: the semantics we propose is appropriate for a structural analysis of those conditionals that do not satisfy closure properties associated to classical logics.

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1 Introduction

Conditionals are generally considered the backbone of human (and AI) reasoning: the “if-then” connection between two propositions is the stepping stone of arguments and a lot of the research effort in formal logic has focused on this kind of connection. A conditional connection satisfies different properties according to the kind of arguments it is used for. The classical material implication is appropriate for modelling the “if-then” connection as it is used in Mathematics, but the equivalence between the material implication and is not appropriate for many other contexts. Different kinds of reasoning use different kinds of conditionals, modelling, among others, presumptive reasoning (e.g. “Birds typically fly”), normative reasoning (e.g. “if you have had alcohol, you should not drive”), casual reasoning (e.g. “if you throw a stone against that window, then you will break it”), probabilistic reasoning

(e.g. “if you go out in this weather, you will probably get a cold”),

fuzzy reasoning (e.g. “if the temperature is hot, then the fan speed is high”), or counterfactual reasoning (e.g. “if I were you, I wouldn’t do that”).

In different contexts we associate to the “if-then” expressions distinct modalities, each of them validating different argumentation patterns. A common way of formalising different reasoning patterns that are or are not endorsed in a specific reasoning context is through structural properties. That is, formal constraints specifying that a set of conditionals is closed under certain reasoning patterns. This kind of analysis was used already in classical logic, as the class of Tarskian logical consequence relations have been characterised in terms of three main properties:

Reflexivity:

Monotonicity:

    (Mon)

Cut:

    (Cut)  .

Referring to structural properties in analysing conditional logics has become a standard in some areas [5, 13, 12]. However, let us note that while some properties may appear obvious in everyday reasoning, these may become in fact undesirable depending on the reasoning context in which we apply them. For example, a property like

Right Conjunction:

    (And)

dictates that if an agent believes “if then ” and “if then ”, then it should also believe that “if then and ” ( stands for conditional implication). For instance, if an agent believes that presumably birds fly and that presumably winter days are cold, it is reasonable to require for a rational agent to abide to the (And) property, and, thus, to believe the conjunction of the two, i.e. presumably birds fly and winter days are cold.

While the (And) property is required in presumptive reasoning, it is not considered appropriate for other kinds of reasoning, as, for example, in a probabilistic context [6] or in deontic reasoning. In the latter case, in some kind of normative reasoning involving incompatible preferences, (And) is not a desirable reasoning pattern: an agent could believe (“On Saturday night I would like to go to a party”) and (“On Saturday night I would like to stay home watching TV”), but not (“On Saturday night I would like to go to a party and to stay home watching TV”).

Another property that is usually satisfied in most of the reasoning contexts is

Right Weakening:

    (RW)  .

(RW) simply states that if an agent believes “if then ”, then it believes also “if then ” for any (classical) consequence of . For example, it is reasonable to impose that believing that presumably birds fly implies also believing that presumably birds move, since flying implies moving.

However, there are contexts in which (RW) gives back counter-intuitive results, as in some forms of deontic and causal reasoning [1], as illustrated by the following examples:

  • “if you are involved in a car accident, you should remain on the spot” is an acceptable norm, but “if you are involved in a car accident, you should remain on the spot or paint yourself in blue” is not as acceptable;

  • “if you turn the wheel of a moving car, the car will move in a circle” is meaningful, while “If you turn the wheel of a moving car, the car will move” is not really that meaningful;

  • “if you throw a stone against the window, it will break” is meaningful, but “If you throw a stone against the window, it will break or Ann will drink tea” is not.

(RW) is a property that is strongly connected to the traditional semantics that is used to formalise conditional reasoning, i.e. possible-worlds semantics. In fact, most formalisations of conditional reasoning have been built using a possible-worlds semantics by referring more or less directly to classical modal operators. Using such an approach it has been possible to define logical systems modelling various kinds of non-classical reasoning.

On the other hand, relying on possible worlds means relying on closed logical theories, and such an approach enforces some properties (e.g. logical omniscience) that may be in conflict with some modelling goals. Some works have already considered ways of combining a possible world approach with some constrained forms of (RW) [1, 17]. Let us anticipated that, in contrast to those approaches, we will consider here a kind of intentional semantics instead.

One limit of the possible-worlds approach to the formalisation of conditionals “if condition holds, then effect holds with a given modality” is that it accounts for the modality that is associated with the truth of  given the truth of . However, it does not account for whether the truth of  given the truth of  has any relevance for the kind of reasoning we are considering. The centrality of the notion of relevance in conditional reasoning has already been pointed out in [3]. However what ‘relevance’ means in the context of conditional reasoning remains still vague nowadays.

As we are going to show in the next section, our formalisation focuses on choice functions that model what the agent considers as relevant effects and relevant conditions. Our work is somewhat inspired by [18] that also suggested the use of choice functions in modelling the semantics of conditionals.

The paper is organised as follows. In the next section we introduce some background concepts we will rely on in our formalisation of conditionals. In Section 3 we illustrate our formalisation of conditionals, while Section 4 describes how we may accommodate various structural properties within our approach. Section 5 discusses how to formalise entailment relations in our framework and shows possible future developments. Eventually, Section 6 summarises our contribution.

2 Preliminaries

We use a conditional language containing conditionals of the form . We do not consider here the possibility of nesting the conditionals or combining them via propositional operators.

Let be a finitely generated propositional language, with logical connectives and and propositional symbol having usual meaning. Capital letters will be used to refer to propositions, while will refer to sets of propositions. With we denote the classical propositional consequence relation.

Our language will be , the conditional language built on top of : namely,

On the semantics side we will use a relation among propositional formulae, where iff , so that generates the classical lattice semantics over propositional formulas, with and represented by the join and meet operations, respectively. The relations and are defined as usual from . Note that, using as a representation of , represents and , while is a representative of . Of course, is reflexive and transitive.

With we denote the minimal elements in w.r.t. , i.e.  , while and (we will write for ).

We are going to use a well-known order among sets of formulae, based on : the Smyth order over power sets (see, e.g. [21, Section 3] for a short introduction).111Orders of this type are often used in the context of so-called power domains [8, 7, 16, 20, 22]. Specifically,

We also write iff and .

A choice function is a set-valued function , mapping a formula to a set of formulae. We say that is Smyth-monotone, or simply S-monotone, iff for every , if then . Furthermore, is a fixed-point of iff (see, e.g. [21]).

Eventually, we say that is -closed, where , iff for all , if and then .222Note that for order matters as is not symmetric. On the other hand, we will say that is -closed, where , iff for all , if and then .

3 Semantics

We build our semantics on top of two choice functions, and , representing what an agent considers as relevant connections. Specifically, a conditional interpretation is a pair

s.t.  and . represents the relevant effects of a proposition, and the possible conditions for a proposition to hold.

Definition 1 (Satisfaction).

Let be a conditional interpretation. satisfies a conditional , denoted , iff the following conditions hold:

  1. there is s.t. and ; and

  2. .

is satisfiable (has a model) if there is a conditional interpretation such that . A set of conditionals is satisfiable (has a model) iff each conditional in it is so.

Fig. 1 gives a graphical representation of the satisfaction relation: iff there is a “triangle” . We indicate with that there is a triangle passing through some .

The meaning of the above definition has an epistemic flavour: an agent accepts a conditional connection between and if is a logical consequence of some relevant effect of (), and is recognised as a relevant condition for ().

Figure 1: Graphical representation of .

Given an interpretation , with we indicate the set of conditionals satisfied by , i.e. .

Let us note that our class of interpretations is quite generic and, in particular, can represent any set of conditionals. In fact, given a set of conditionals , we may define a model characterising it, that is, satisfying exactly the conditionals in ). To do so, given , we construct a conditional interpretation w.r.t. 

in the following way:

  1. define the following sets: and .

  2. for every , we set

characterises , as the following proposition proves.

Proposition 1.

Given a set of conditionals , is its characteristic model, that is, a conditional is in iff .

Proof.

From left to right. Assume is in . Then, by definition of , we have that and there is a s.t. (it could be itself). Hence .

From right to left. Assume . Then , and that, by construction of , can be only if . ∎

Please note that, as we have proved Proposition 1 for any arbitrary set of conditionals , the following immediate corollary tells us that the class of conditional interpretations do not impose any form of closure under any structural property.

Corollary 1.

The class of conditional interpretations can represent any set of conditionals.

Corollary 2.

Any set of conditionals is satisfiable.

4 Structural Properties

In the following, we are going to show that by constraining the functions and , it is possible to enforce the closure of the set of conditionals under structural properties that are considered as appropriate for modelling various kinds of reasoning. We start by analysing some classical reasoning patterns.

At first, as and range over formulae and not over possible worlds, i.e. logically closed theories, Definition 1 does not imply any form of closure under logical equivalence. Such a behaviour may be desirable in some epistemic contexts in which we would like to avoid some form of a priori logical omniscience [4]. That is, the well known reasoning patterns of Left Logical Equivalence (LLE) and Right Logical Equivalence (RLE) do not hold in general in our framework. However, if these are desired, it is quite straightforward to enforce (LLE) and (RLE) in our setting. Specifically, for

Left Logical Equivalence:
(LLE)

it suffices to impose the following semantic constraints on a conditional interpretation :

(LLE)

for all :

  1. if , then ;

  2. is -closed.

Similarly, for

Right Logical Equivalence:
(RLE)

the semantic constraint to be imposed on a conditional interpretation is:

(RLE)

for all :

  1. if , then .

The conditions (LLE) and (RLE) characterise the classes of the conditional interpretations satisfying, respectively, (LLE) and (RLE). In fact, it can be shown that333Since the proof is straightforward we omit it.

Proposition 2.

A set of conditionals is closed under (LLE) (resp. (RLE)) iff it can be characterised by a conditional model that satisfies (LLE) (resp. (RLE)).

Another basic property is Reflexivity, that simply states that for every proposition it holds ‘If , then ’. Despite appearing as an obviously valid conditional, there are some contexts in which it is not a desirable property. Consider for example a deontic system expressing recommendations, in which is read as “if holds, then would be preferrable”. This kind of conditionals can result quite counter-intuitive if it embeds reflexivity (see e.g. [12]): while “if there is an act of violence, then you should call the police” appears to be a reasonable conditional, to be forced to conclude “if there is an act of violence, then there should be an act of violence” is counter-intuitive. Reflexivity does not hold in our framework, though if we would like to have this pattern, it suffices to impose a simple constraint on conditional interpretations. For

Reflexivity:
(Ref)

the semantic constraint to be imposed on a conditional interpretation is:

(Ref)

for all :

  1. is a fixed-point of both and .

Proposition 3.

A set of conditionals is closed under (Ref) iff it can be characterised by a conditional model that satisfies (Ref).

Proof.

From right to left. Assume that is characterised by some conditional model that satisfies (Ref), that is, . We have to show that is closed under (Ref), and it is immediate to see that (Ref) implies for every .

From left to right. Let be a set of conditionals closed under (Ref). We need to prove that there is a conditional interpretation characterising it and satisfying (Ref). We can define such an by slightly modifying the characteristic model . Specifically, it suffices to consider , where and , for every . Clearly, satisfies (Ref). The proof that iff is analogous to the proof of Proposition 1, considering also that is closed under (Ref). ∎

As next, we consider more elaborate structural properties. We start with considering the Cut reasoning pattern, one of the main structural properties in classical logic (cf. Section 1). So, for

Cut:
(Cut)

the semantic constraints to be imposed on a conditional interpretation are:

(Cut)

for all :

  1. If , then ;

  2. If and , then .

Then, we can show that

Proposition 4.

A set of conditionals is closed under (Cut) iff it can be characterised by a conditional model that satisfies (Cut).

Proof.

From right to left. Assume that is characterised by some conditional model that satisfies (Cut), that is . We need to prove that is closed under (Cut). Suppose and . Then there is some s.t. . Since there is some s.t. , that is, . Regarding , we have and , hence . and imply . Therefore, is closed under (Cut).

From left to right. Let be a set of conditionals closed under (Cut). We need to prove that there is a conditional interpretation characterising it and satisfying (Cut). Let us consider the characteristic model .We need to prove that it satisfies the two conditions of (Cut). So, assume and . implies , that by Proposition 1 implies . By construction of , implies that . From and (Cut) we have that , and, by Proposition 1, . Therefore, there is a s.t. and . That is, holds. Regarding the second condition on , let and . By construction of , and are in , and by (Cut) . Therefore, by construction of , , which concludes the proof. ∎

As next, we address monotonicity (cf. Section 1), also a main property of classical logic. It states that strengthening the antecedent of a conditional from a logical point of view, we still preserve the effects. For example, the conditional in a monotonic system imposes to conclude that any kind of horse is a mammal, e.g. . That is, (Mon) makes our conditionals strict, in the sense that they do not admit exceptions. So, for

Monotonicity:
(Mon)

the semantic constraints to be imposed on a conditional interpretation are:

(Mon)
  1. is S-Monotone;

  2. is -closed.

Proposition 5.

A set of conditionals is closed under (Mon) iff it can be characterised by a conditional model that satisfies (Mon).

Proof.

From right to left. Assume that is characterised by some conditional model that satisfies (Mon), that is, . We need to prove that is closed under (Mon). Assume and , i.e. . implies that there is some s.t. . By S-Monotonicity and, thus, there is some s.t. , that implies . As is -closed and , implies . Hence . Therefore, is closed under (Mon).

From left to right. Let be a set of conditionals closed under (Mon). We need to prove that there is a conditional interpretation characterising it and satisfying (Mon). Let us consider the characteristic model of as by Proposition 1. We need to prove that it satisfies the two conditions of (Mon). So, let , and let . By construction of , implies that , and by (Mon) . By construction of , either , or there is a s.t. . Hence is S-Monotone. Also the -closure of is an immediate consequence of the closure under (Mon) of and the definition of in , , which concludes the proof. ∎

As by Section 1, (And) is a property that appears desirable in many contexts, but may have some exceptions. For

Right Conjunction:
(And)

the semantic constraints to be imposed on a conditional interpretation that characterise (And) are:

(And)

for all :

  1. if , then ;

  2. .

Proposition 6.

A set of conditionals is closed under (And) iff it can be characterised by a conditional model that satisfies (And).

Proof.

From right to left. Assume that is characterised by some conditional model that satisfies (And), that is, . We need to prove that is closed under (And). Assume and . Then there is some s.t.  and some s.t. . contains some s.t. and some s.t. . By the first condition of (And) we have , and as a consequence we have and eventually . Regarding , we have and , hence . , and together imply . Therefore, is closed under (And).

From left to right. Let be a set of conditionals closed under (And). We need to prove that there is a conditional interpretation characterising it and satisfying (And). Let us consider the characteristic model of as by Proposition 1. We need to prove that it satisfies the two conditions of (And). So, assume there are three propositions s.t.  and . From the construction of we have that implies that , and that for any , if , then . Hence we have and , but not , against the closure of under (And). Regarding the second condition, for all , iff . Let . Then , , and, by (And), , that implies , which concludes the proof. ∎

Reasoning by cases is another well-known characteristics of classical reasoning, which is formalised by the Left Disjunction reasoning pattern. To deal with it, for

Left Disjunction:
(Or)

the semantic constraints to be imposed on a conditional interpretation that characterise (Or) are:

(Or)

for all :

  1. ;

  2. is -closed.

Proposition 7.

A set of conditionals is closed under (Or) iff it can be characterised by a conditional model that satisfies (Or).

Proof.

From right to left. Assume that is characterised by a conditional model that satisfies (Or), that is, . We need to prove that is closed under (Or). Assume and . Then there are s.t. , and s.t. . Then there must be some s.t. , , and ( itself satisfies the constraint), and the minimal among them w.r.t. are in by condition 1. of (Or). Hence in there is some s.t. . But, and imply also that and, thus, as is -closed, . Therefore, we can conclude . Therefore, is closed under (Or).

From left to right. Let be a set of conditionals closed under (Or). We need to prove that there is a conditional interpretation characterising it and satisfying (Or). So, let us consider the characteristic model as by Proposition 1. At first, we show that satisfies the second condition of (Or). In fact, by construction of , for all , if then . Therefore, as is closed under (Or), must be -closed. On the other hand, if does not satisfy the first condition of (Or), we transform into a model by extending only. Specifically, it is sufficient that for every disjunction we add the set to . Now, it is easily verified that indeed satisfies exactly the same set of conditionals as , i.e. . In fact, in we have an extension of , while stays the same. Therefore, as by construction of , iff , the same holds for and, thus, the set of satisfied conditionals by remains the same as for , i.e. , which concludes the proof. ∎

As mentioned in Section 1, Right Weakening is a property that is generally desirable in many context with some exceptions. To support the reasoning pattern of

Right Weakening:
(RW)

the semantic constraint to be imposed on a conditional interpretation that characterise (RW) is:

(RW)

for all :

  1. if then .

Proposition 8.

A set of conditionals is closed under (RW) iff it can be characterised by a conditional model that satisfies (RW).

Proof.

From right to left. Assume that is characterised by a conditional model that satisfies (RW), that is, . We need to prove that is closed under (RW). So, assume and , i.e. . Then there is some s.t. , and consequently . Since , , by condition 1. we have . Hence . is closed under (RW).

From left to right. Let be a set of conditionals closed under (RW). We need to prove that there is a conditional interpretation characterising it and satisfying (RW). So, consider the characteristic model , assume , i.e.  , and let . By construction of , implies , that, by (RW), implies . By construction of , , as desired. ∎

So far, we have taken under consideration most of the properties characterising classical entailment. However, we still miss two important consistency properties: namely, ex falso quodlibet and consistency preservation. The former is a classical property strongly connected with classical implication and entailment, and stating that we can conclude anything from a false premise. This property, for example, is not fully desirable in counterfactual reasoning, where we would like to be able to reason coherently about false situation, but that are at least conceivable. Nevertheless, to support the reasoning pattern of

Ex Falso Quodlibet:
(EFQ)

the semantic constraints to be imposed on a conditional interpretation that characterise (EFQ) are:

(EFQ)

for all : if , then

  1. ;

  2. , for all .

Proposition 9.

A set of conditionals is closed under (EFQ) iff it can be characterised by a conditional model that satisfies (EFQ).

Proof.

From right to left. Assume that is characterised by a conditional model that satisfies (EFQ), that is, . We need to prove that is closed under (EFQ). Assume . We need to prove that holds for all . implies , hence, by (EFQ), we have , and, thus, . Therefore, and, thus, is closed under (EFQ).

From left to right. Let be a set of conditionals closed under (EFQ). We need to prove that there is a conditional interpretation characterising it and satisfying (EFQ). So, consider the characteristic model and let . By (EFQ), follows, and, since , holds. Furthermore, by (EFQ), holds, for all . Therefore, by construction of , holds, for any and, thus, satisfies (EFQ), which concludes the proof. ∎

Please note that (EFQ) is an immediate consequence of (RLE), (And) and (RW). However, we may have contexts that do not satisfy some of these three properties, but still satisfies (EFQ). If this is the case, the semantic constraint (EFQ) has to be considered.

Consistency preservation tells us that we cannot conclude absurdity from a classically consistent premise. To support the reasoning pattern of

Consistency Preservation:
(Con)

the semantic constraint to be imposed on a conditional interpretation that characterise (Con) is:

(Con)

for all ,

  1. if , for some , then .

Please note that only if we assume (RLE) we can express (Con) in the classical (equivalent) forms

where the reading of the latter is: “if is not a tautology then the conditional cannot be concluded”.

Proposition 10.

A set of conditionals is closed under (Con) iff it can be characterised by a conditional model that satisfies (Con).

Proof.

From right to left. Assume that is characterised by a conditional model that satisfies (Con), that is . We need to prove that is closed under (Con). So, assume and , i.e. . We need to prove that holds. implies that there is s.t. , hence . Therefore, by (Con) we have , that is, .

From left to right. Let be a set of conditionals satisfying (Con). We need to prove that there is a conditional interpretation characterising it and satisfying (Con). We prove that the characteristic model is such an interpretation, by proving that for any , if then there is no s.t. . Let and ; hence and . By (Con), . By construction of , implies that is not in , since otherwise we would have .

A stronger property that connects conditional reasoning to classical entailment is supraclassicality, that is, the conditional systems extends classical reasoning. To support the reasoning pattern of

Supraclassicality:
(Sup)

the semantic constraints to be imposed on a conditional interpretation that characterise (Sup) are:

(Con)

for all ,

  1. is a fixed-point of ;

  2. .

Proposition 11.

A set of conditionals is closed under (Sup) iff it can be characterised by a conditional model that satisfies (Sup).

Proof.

From right to left. Assume that is characterised by a conditional model and, thus, , that satisfies (Sup). We need to prove that is closed under (Sup). So, assume , i.e. . Then , hence , , and , hence .

From left to right. Let be a set of conditionals satisfying (Sup). We need to prove that there is a conditional interpretation characterising it and satisfying (Sup). Consider the characteristic model : it clearly satisfies the second condition, the one over . It is possible it does not satisfy the condition over , in case contains some conditional with . To cover such a case it is sufficient to modify into a model in the same way as done in the proof of Proposition 3. is a characteristic model of satisfying both the conditions in (Sup). ∎

Please note that (Sup) is a consequence of (Ref) and (RW) together, but it is not equivalent to the combination of those two properties; and if we change the second condition in (Sup) into , we model the classical propositional entailment (proof omitted).

A main portion of the research in conditional reasoning has focused on forms of defeasible reasoning. Defeasible reasoning is characterised by a degree of uncertainty connected some of the drawn conclusions that may be revised when faced with more complete and specific information. Presumptive reasoning, that is, reasoning based on expectations, represents the most popular context in which it is necessary to constraint (Mon). The basic form of constrainted monotonicity is Cautious Monotonicity. To support the reasoning pattern of

Cautious Monotonicity:
(CM)

the semantic constraints to be imposed on a conditional interpretation that characterise (CM) are:

(CM)

for all ,

  1. if , then ;

  2. if then .

Proposition 12.

A set of conditionals is closed under (CM) iff it can be characterised by a conditional model that satisfies (CM).

Proof.

From right to left. Assume that can be characterised by a conditional model and, thus, , that satisfies (CM). We need to prove that is closed under (CM). So, assume and . Therefore, there is some s.t. , and . As a consequence, there is some s.t. , that is, . Regarding , we have and , hence . Therefore, holds.

From left to right. Let be a set of conditionals closed under (CM). We need to prove that there is a conditional interpretation characterising it and satisfying (CM). Consider the characteristic model as by Proposition 1. We need to prove that it satisfies the two conditions of (CM). Let and . implies , which by Proposition 1 implies . By construction of , implies that . From and (CM) we have that , and, by Proposition 1, . That is, there is a s.t. and . Therefore, holds. Regarding the second condition on , let . By construction of , and are in , and by (CM) , that is, by construction of , , which concludes the prove. ∎

Beyond being a desirable property from the point of view of many reasoning contexts, such as presumptive and prototypical reasoning [9], (CM) if formally important because combining it with (Cut) we obtain Cumulativity:

Cumulativity:
(Cumul)

(Cumul) is formally important because entailment relations satisfying (Cumul) satisfy also Idempotence, a classical closure property.

The semantic constraints to be imposed on a conditional interpretation that characterise (Cumul) are obtained by combining (Cut) and (CM): that is,

(Cumul)

for all ,

  1. If then ;

  2. If then ( iff ).

Proceeding in this way we can introduce many other structural properties / reasoning patterns as formal constraints specified over the the functions and . For example, consider (AntiRW), a form of constrained (RW) [1]:

Anti Right Weakening:
(AntiRW)

Or, equivalently,

(AntiRW*)

(AntiRW), that is implied by (RW), states that we can weaken the conclusions, but, once we block the right weakening process, we cannot recover it anymore. It is a property that, for example, appears appropriate for some causal or deontic forms of reasoning (see [1] for details).

We can enforce (AntiRW) in our framework via the following semantic constraints:

(AntiRW)

for all ,

  1. if , and then implies .

Proposition 13.

A set of conditionals is closed under (AntiRW) iff it can be characterised by a conditional model that satisfies (AntiRW).

Proof.

From right to left. Assume that can be characterised by a conditional model and, thus, , that satisfies (AntiRW). We prove that is closed under (AntiRW*) (that is equivalent to (AntiRW)). So, assume and , with . Then there is some s.t. . Also, and , that, by condition 1. of (AntiRW), imply . The latter and imply , as desired.

From left to right. Let be a set of conditionals closed under (AntiRW*). We need to prove that there is a conditional interpretation characterising it and satisfying (AntiRW). Let us consider the characteristic model , and we prove that it satisfies the condition (AntiRW). So, let , , and . By the construction of we have and . Since and is closed under (AntiRW*), , that implies . Hence condition 1. is satisfied, which completes the prove. ∎

Finally, let be the set of structural properties presented in this section. We have taken under consideration each of them, and we have given a semantic counterpart in our framework. Each semantic property is a sufficient condition for obtaining a characterising model, but not a necessary condition. Specifically, given any set of conditionals closed under some structural property (X), we have proved that there must be a characterising model satisfying (X), not that every model characterising must satisfy (X).

In the following, we clarify whether all these semantic properties are compatible among them. That is, given a set of conditionals closed under some of the structural properties in , we are going to answer to the problem whether there is a characterising model closed under all the correspondent semantic properties.

Proposition 14.

Let be a set of structural properties in , and be the set of the correspondent semantic properties. If a set of conditionals is closed under the properties in , then there is a conditional interpretation characterising and satisfying all the properties in .

Proof.

(Sketch) Let . If the proof is straightforward: as seen in the proof of the propositions in this section, given a set satisfying any property in , the characteristic model satisfies the correspondent semantic property. So, if we are dealing only with properties in , the characteristic model of is the model we are looking for. It remains to take under consideration the combinations between properties in and .

For (Ref) in Proposition 3 we have extended in the model into a function s.t. for every proposition , and that the new model satisfies the same set of conditionals . It is easy to check that the satisfaction of any property in and of their semantic counterparts is preserved in this extension of , with the only exception of (LLE), that requires a further extension of : namely, for every , . It is easy to check that, given any set of conditionals closed under (LLE) and (Ref), this further extension of w.r.t.  does not affect neither the set of conditionals satisfied by the model (that is, it is still the characteristic model of the initial set ), nor the satisfaction of the other semantic properties.

For (Sup) we introduce the same extension to , and the same argument applies.

For (Or) in Proposition 7, we define a model that extends by adding to , for every disjunction . Again, this change of does not affect any of the other semantic properties, apart from (LLE), that requires an extra change as for (Ref) and (Sup): we need to extend imposing to any s.t. for some disjunction . As for (Ref) and (Sup), this extra change does not affect the set of the satisfied conditionals and the satisfaction of the other semantic properties, which completes the prove. ∎

5 Entailment and Future Work

Most of the results in this paper are representational ones showing how conditional interpretations are appropriate for modelling different forms of closure. The next step is the definition of an actual reasoning systems in this framework: we start from a finite set of conditionals , and we would like to derive new conditionals according to reasoning patterns satisfied, or, more generally, according to some predefined functions and . In this preliminary report, we present only intuition behind our approach that aims at modelling conditionals entailed by predefined functions and .

To do so, we consider the following example for illustrative purposes, showing how one may derive new conditionals, for instance, under (Ref) and (Cut).

Example 1.

Let (we use only the initials of the propositional letters in what follows). The conditionals in represent the information an agent is aware of. That is, if then the agent is aware that is a relevant effect of and is a relevant condition for . Formally, this translate into a model where, for every ,

Hence in the present case we have , and for any other formula ; , and for any other formula . This model satisfies only the conditionals in , and in order to impose the closure under (Ref) and (Cut), we impose the satisfaction of (Ref) and (Cut) by extending and into, respectively, and : in order to satisfy (Ref) we add to and for every formula , while to satisfy (Cut) we need to add to (for condition 1.) and to (for condition 2.). Hence, we end up with the model with , and for any other formula ; , and for any other formula . To determine which conditionals are satisfied by , we have to look for ‘triangles’ (see Fig. 1) that occur under , and . In this case, one may verify that indeed satisfies also , i.e., (“a feline is a mammal”) and all the reflexive conditionals.

Therefore, the main idea to formalise reasoning is, given a knowledge base , to build a model characterising and then to modify its and according to the reasoning patterns we would like to implement. The first step is to define closure operations over and