We extend the expressivity of classical conditional reasoning by introducing situation as a new parameter. The enriched conditional logic generalises the defeasible conditional setting in the style of Kraus, Lehmann, and Magidor, and allows for a refined semantics that is able to distinguish, for example, between expectations and counterfactuals. We introduce the language for the enriched logic and define an appropriate semantic framework for it. We analyse which properties generally associated with conditional reasoning are still satisfied by the new semantic framework, provide a suitable representation result, and define an entailment relation based on Lehmann and Magidor's generally-accepted notion of Rational Closure.
3. Situated conditionals Back to our problem, let us present an extended version of the (admittedly over-used) penguin example.
Example 1. Suppose we know that birds usually fly ( b |∼ f), that penguins are birds (p → b) that usually do not fly (p |∼ ¬
f). Also, we know that dodos were birds (d → b) that usually did not fly ( d |∼ ¬ exist anymore. Using the standard ranked semantics (Defiholds. In such a case, ⊤ |∼ ¬ pretation. That is, the agent believes is true iff ⊤ |∼
d means that the agent believes that dodos do not exist. A model for this conditional knowledge base is shown in Figure 1 (left). The main limitation of this representation is that all exceptional entities d |∼ ⊥ ifed at rank ⊤ |∼ ¬ 0. Hence we have ⊤ |∼ ¬
p, just as we have d, 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 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 ¬ do not exist, we add the statement d |∼ ⊥ |∼ ⊥ holds. If dodos
. A model for this case is depicted in Figure 1 (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.
The first option is to formalise what an agent believes by referring to valuations with rank 0 in a ranked inter- ations: plausible valuations with a finite rank , and implaunition 1) we have two ways of modelling this information. temic interpretations, a refined version of the ranked inhave the same status as dodos, since they cannot be satis- ∈ N ∪ {∞}. The in ⟨, ⟩ is intended to indicate that sible or counterfactual. In the case of penguins and dodos, for example, it allows us to state that penguins usually do not fly assuming to be in a situation in which penguins existing, and that dodos usually do not fly, assuming dodos exist, while being unaware of whether or not penguins and dodos actually exist. At the same time, it remains possible to make statements about what necessarily holds, regardless of any plausible or counterfactual premise.
A situated conditional (SC) is a statement with , , , which is read as ‘given the situation ,
|∼ , ∈ ℒ terpretations. We distinguish between two classes of valusible 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 ∈ N, or ⟨∞, ⟩ where has a finite rank , while the ∞ in ⟨∞, ⟩ is intended to indicated 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 ∈ N, while those considered impossible have the infinite rank ⟨∞, ∞⟩, where ⟨∞, ∞⟩ is taken to be less typical than any of the other infinite ranks.
Formally, let R =def {⟨, ⟩ | ∈ N} ∪ {⟨∞, ⟩ | ∈ N ∪ {∞}}. We define the total ordering
⪯ follows: ⟨1, 1⟩ ⪯ ⟨ 2, 2⟩ if and only if 1 = 2 and 1 ≤ 2, or 1 = and 2 = ∞, where <
∞ for all ∈ N}. We need to extend the notion of convexity of ranked interpretations to epistemic interpretations: let e
over R as be a function from to R. e is said to be convex (w.r.t.⪯ ) if and only the following holds: i) If e() = ⟨, ⟩, then, for all s.t. 0 ≤
< , there is a ∈ s.t. e( ) = ⟨, ⟩; and ii) if e() = ⟨∞, ⟩ for ∈ N, then, for all s.t. 0 ≤ < , there is a ∈ s.t. e( ) = ⟨∞, ⟩.
Definition 2.
An epistemic interpretation E is a total noted R ⊩ |∼ ) if minJ R ⊩ ¬
|∼ ⊥ abbreviate ¬ |∼ ⊥
iff R ⊆ as .
KR ⊆
J K to as a ranked model of |∼ . It is easily verified that
J
K , with R referred been colonised, the dodo would not fly’. Moreover, it is possible to reason coherently with situated conditionals . Hence we frequently without needing to know whether their premises are plauinterpretations as follows: |∼ is satisfied in
R (de- counterfactual conditionals such as ‘Had Mauritius not f), and that dodos do not holds on condition that holds’.
To provide a suitable semantics for SCs we define epis∞ 2 0 ∖ (J0K ∪ J1K ∪ J2K) pdbf, pdbf, pdbf 1 pdbf, pdbf, pdbf, pdbf ∞ ∖ (J0K ∪ J1K ∪ J2K) 2 1
pdbf pdbf, pdbf, pdbf, pdbf, pdbf
0 pdbf, pdbf, pdbf
We introduce a logic of situated conditionals to overcome this problem. The central insight is that adding an explicit notion of context to standard conditionals allows for a refined semantics of this enriched language in which the problems described in Example 1 can be dealt with adequately. It also allows us to reason coherently with ple 1 satisfying ⊤ |∼ ¬ the KB expanded with d |∼ ⊥ .
d. Right: a ranked interpretation of ⟨∞, 1⟩ ⟨∞, 0⟩ ⟨, 2⟩ ⟨, 1⟩ pdbf, pdbf pdbf, pdbf
pdbf pdbf, pdbf ⟨, 0⟩ pdbf, pdbf, pdbf
Intuitively, this definition evaluates |∼ as follows. defined for ranked models [
If the situation is compatible with the plausible part of For a detailed explanation of the properties characterisE (the valuations in E ) then |∼ holds if the most ing FSC’s, the proof of the representation theorem, and a typical plausible models of
∧ are also models of .
On the other hand if the situation is not compatible with the plausible part of E (that is, all models of have an infinite rank) then |∼ holds if the most typical implausible (but possible) models of ∧ of . SCs and epistemic interpretations allow to model
are also models more correctly the conditionals in Example 1.
Example 2. Consider the following rephrasing of the
statements in Example 1. ‘Birds usually fly’ becomes
presentation of the minimal closure, we refer the reader
to the technical report [
f and d |∼ d ¬f. Given that dodos don’t exist anymore, the statement d |∼ ⊤ ⊥ leaves open the existence of dodos in the infinite rank, which is based on a semantic construction that has proven useful in the area of belief change and that is more general and also more fine-grained than the standard preferenwhich allows for coherent reasoning under the assump- tial semantics; (iii) an investigation of the properties that and d ∧ ¬b |∼ d∧¬b ⊥ Jp ∧ ¬b
K ∪ J a model of these statements.
K tion that dodos exist (the context d). Moreover, information such as dodos and penguins necessarily being birds can be modelled by the conditionals p ∧ ¬b |∼ p∧¬b ⊥
, relegating the valuations in d ∧ ¬b to the rank ⟨∞, ∞⟩. Figure 2 shows
We have identified relevant situated rationality postu- soning. lates, that represent desirable properties for SCs: (Ref) |∼ (And) (RW) (Inc) (Ext) |∼ ,
|∼ |∼ ∧ |∼ ,
|= → |∼ |∼ ∧ |∼ ⊤
≡ |∼ iff |∼ (LLE) |= ↔ , (Or) (RM) (SupExp)
|∼ |∼ , |∼ , ∨ |∼ ∧ |∼
|∼ |∼ ̸ |∼ ¬ |∼ |∼ ∧ ∧ |∼ (Vac) ⊤ ̸ |∼ ⊤ ¬, ∧ |∼ ⊤
(SubExp) |∼ ⊤ ⊥, ∧ |∼ |∼ ∧ situated conditionals satisfy and of their appropriateness for knowledge representation and reasoning, in particular when reasoning about information that is incompatible with background knowledge, and (iv) the definition of a form of entailment for contextual conditional knowledge bases based on the widely-accepted notion of rational closure, which is reducible to classical propositional rea
Next steps are the extension of this approach to other
logics. Description Logics, for which rational closure has
already been reformulated [
We also plan to investigate refinements of RC such as
lexicographic closure [
A conference version of this work was presented at
AAAI-21 [
The work of Giovanni Casini was partially supported by TAILOR (Foundations of Trustworthy AI – Integrating Reasoning, Learning and Optimization), a project funded by EU Horizon 2020 research and innovation programme under GA No 952215.
This work was supported in part by the ANR Chaire IA BE4musIA: BElief change FOR better MUlti-Source Information Analysis (ANR-20-CHIA-0028).