Vol-3197/short4
Jump to navigation
Jump to search
Paper
| Paper | |
|---|---|
 edit
| |
| description | scientific paper published in CEUR-WS Volume 3197 |
| id | Vol-3197/short4 |
| wikidataid | Q117341871→Q117341871 |
| title | Situated Conditionals - A Brief Introduction |
| pdfUrl | https://ceur-ws.org/Vol-3197/short4.pdf |
| dblpUrl | https://dblp.org/rec/conf/nmr/CasiniMV22 |
| volume | Vol-3197→Vol-3197 |
| session | → |
Situated Conditionals - A Brief Introduction
Situated Conditionals - A Brief Introduction
(Extended Abstract)
Giovanni Casini1,2,3 , Thomas Meyer2,3,1 and Ivan Varzinczak4,5,3,1
1
ISTI - CNR, Pisa, Italy
2
University of Cape Town, South Africa
3
CAIR, South Africa
4
CRIL, Univ. Artois & CNRS, France
5
Stellenbosch University, South Africa
Abstract
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.
Keywords
Conditional reasoning, non-monotonic reasoning, counterfactual reasoning, defeasible reasoning, belief revision
1. Introduction is ℒ = {𝛼, 𝛽, . . .}. The set of all valuations (worlds)
is denoted 𝒰 = {𝑢, 𝑣, . . .}. Whenever it eases presenta-
Conditionals are at the heart of human everyday reason- tion, we represent valuations as sequences of atoms (e.g.,
ing and play an important role in the logical formalisa- p) and barred atoms (e.g., p), with the usual understand-
tion of reasoning. Two very common interpretations, that ing. E.g., the valuation bfp conveys the idea that b is
are also strongly interconnected, are conditionals repre- true, f is false, and p is true. 𝑣 satisfies 𝛼 is indicated by
senting expectations (‘If it is a bird, then presumably 𝑣 ⊩ 𝛼, while J𝛼K = def
{𝑣 ∈ 𝒰 | 𝑣 ⊩ 𝛼} and for 𝑋 ⊆ ℒ,
it flies’), and conditionals representing counterfactuals J𝑋K = def
⋂︀
𝛼∈𝑋 J𝛼K. 𝑋 |= 𝛼 denotes classical propositional
(‘If Napoleon had won at Waterloo, all Europe would be entailment. Given a set of valuations 𝑉 , fml(𝑉 ) indicates
speaking French’). The first example above assumes that a formula characterising the set 𝑉 .
the premises of conditionals are consistent with what is A defeasible conditional |∼ is a binary relation on ℒ. A
believed, while the second example assumes that those suitable semantics for rational conditionals is provided by
premises are inconsistent with an agent’s beliefs. This ranked interpretations.
poses a formal problem for the classical semantics of
conditional reasoning, that we are going to explain in Ex- Definition 1. A ranked interpretation R is a function
ample 1, but let us introduce some formal preliminaries from 𝒰 to N ∪ {∞}, satisfying the following convexity
first. A conference version of this work has been pre- property: for every 𝑖 ∈ N, if R(𝑢) = 𝑖, then, for every 𝑗
sented at AAAI-21 [1], while an extended technical report 0 ≤ 𝑗 < 𝑖, there is a 𝑢′ ∈ 𝒰 for which R(𝑢′ ) = 𝑗.
is available online [2].
Figure 1 gives an example of two ranked interpretations.
For a given ranked interpretation R and valuation 𝑣, we
2. Formal background denote with R(𝑣) the rank of 𝑣. The number R(𝑣) in-
dicates the degree of atypicality of 𝑣. So the valuations
We assume a finite set of propositional atoms 𝒫 = judged most typical are those with rank 0, while those with
{𝑝, 𝑞, . . .}, while the set of all propositional sentences an infinite rank are judged so atypical as to be implausible.
We can therefore partition the set 𝒰 w.r.t. R into the set
NMR 2022: 20th International Workshop on Non-Monotonic Reason- of plausible valuations 𝒰R = {𝑢 ∈ 𝒰 | R(𝑢) ∈ N}, and
𝑓 def
ing, August 07–09, 2022, Haifa, Israel ∞ def 𝑓
implausible valuations 𝒰R = 𝒰 ∖ 𝒰R .
$ giovanni.casini@isti.cnr.it (G. Casini); tmeyer@cair.org.za
Let R be a ranked interpretation and let 𝛼 ∈ ℒ. Then
(T. Meyer); ivarzinczak@icloud.com (I. Varzinczak)
J𝛼K𝑓R =𝒰 R ∩J𝛼K, and minJ𝛼KR ={𝑢 ∈ J𝛼KR | R(𝑢) ≤
def 𝑓 𝑓 def 𝑓
� 0000-0002-4267-4447 (G. Casini); 0000-0003-2204-6969
(T. Meyer); 0000-0002-0025-9632 (I. Varzinczak) R(𝑣) for all 𝑣 ∈ J𝛼KR }. A defeasible conditional 𝛼 |∼ 𝛽
𝑓
© 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0). can be given an intuitive semantics in terms of ranked
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073
CEUR Workshop Proceedings (CEUR-WS.org)
151
�interpretations as follows: 𝛼 |∼ 𝛽 is satisfied in R (de- counterfactual conditionals such as ‘Had Mauritius not
noted R ⊩ 𝛼 |∼ 𝛽) if minJ𝛼K𝑓R ⊆ J𝛽K, with R referred been colonised, the dodo would not fly’. Moreover, it is
to as a ranked model of 𝛼 |∼ 𝛽. It is easily verified that possible to reason coherently with situated conditionals
R ⊩ ¬𝛼 |∼ ⊥ iff 𝒰R 𝑓
⊆ J𝛼K. Hence we frequently without needing to know whether their premises are plau-
abbreviate ¬𝛼 |∼ ⊥ as 𝛼. 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
3. Situated conditionals existing, and that dodos usually do not fly, assuming do-
dos exist, while being unaware of whether or not penguins
Back to our problem, let us present an extended version and dodos actually exist. At the same time, it remains
of the (admittedly over-used) penguin example. possible to make statements about what necessarily holds,
Example 1. Suppose we know that birds usually fly (b |∼ regardless of any plausible or counterfactual premise.
f), that penguins are birds (p → b) that usually do not fly A situated conditional (SC) is a statement 𝛼 |∼𝛾 𝛽,
(p |∼ ¬f). Also, we know that dodos were birds (d → b) with 𝛼, 𝛽, 𝛾 ∈ ℒ, which is read as ‘given the situation 𝛾,
that usually did not fly (d |∼ ¬f), and that dodos do not 𝛽 holds on condition that 𝛼 holds’.
exist anymore. Using the standard ranked semantics (Defi- To provide a suitable semantics for SCs we define epis-
nition 1) we have two ways of modelling this information. temic interpretations, a refined version of the ranked in-
The first option is to formalise what an agent believes terpretations. We distinguish between two classes of valu-
by referring to valuations with rank 0 in a ranked inter- ations: plausible valuations with a finite rank, and implau-
pretation. That is, the agent believes 𝛼 is true iff ⊤ |∼ 𝛼 sible valuations with an infinite rank. Within implausible
holds. In such a case, ⊤ |∼ ¬d means that the agent be- valuations we further distinguish between those that would
lieves that dodos do not exist. A model for this conditional be considered as possible, and those that would be impos-
knowledge base is shown in Figure 1 (left). The main limi- sible. This is formalised by assigning to each valuation 𝑢
tation of this representation is that all exceptional entities a tuple of the form ⟨𝑓, 𝑖⟩ where 𝑖 ∈ N, or ⟨∞, 𝑖⟩ where
have the same status as dodos, since they cannot be satis- 𝑖 ∈ N ∪ {∞}. The 𝑓 in ⟨𝑓, 𝑖⟩ is intended to indicate that
fied at rank 0. Hence we have ⊤ |∼ ¬p, just as we have 𝑢 has a finite rank, while the ∞ in ⟨∞, 𝑖⟩ is intended to
⊤ |∼ ¬d, and we are not able to distinguish between the indicated that 𝑢 has an infinite rank, where finite ranks are
status of the dodos (they do not exist anymore) and the viewed as more typical than infinite ranks. Implausible
status of the penguins (they are simply exceptional birds). valuations that are considered possible have an infinite
The second option is to represent what an agent believes rank ⟨∞, 𝑖⟩ where 𝑖 ∈ N, while those considered impossi-
in terms of all valuations with finite ranks. That is, an ble have the infinite rank ⟨∞, ∞⟩, where ⟨∞, ∞⟩ is taken
agent believes 𝛼 to hold iff ¬𝛼 |∼ ⊥ holds. If dodos to be less typical than any of the other infinite ranks.
def
do not exist, we add the statement d |∼ ⊥. A model for Formally, let R = {⟨𝑓, 𝑖⟩ | 𝑖 ∈ N} ∪ {⟨∞, 𝑖⟩ | 𝑖 ∈
this case is depicted in Figure 1 (right). Here we can N ∪ {∞}}. We define the total ordering ⪯ over R as
distinguish between what is considered false (dodos exist) follows: ⟨𝑥1 , 𝑦1 ⟩ ⪯ ⟨𝑥2 , 𝑦2 ⟩ if and only if 𝑥1 = 𝑥2 and
and what is exceptional (penguins), but we are unable 𝑦1 ≤ 𝑦2 , or 𝑥1 = 𝑓 and 𝑥2 = ∞, where 𝑖 < ∞ for all
to reason coherently about counterfactuals, since from 𝑖 ∈ N}. We need to extend the notion of convexity of
d |∼ ⊥ we can conclude anything about dodos. ranked interpretations to epistemic interpretations: let e
be a function from 𝒰 to R. e is said to be convex (w.r.t.⪯)
if and only the following holds: i) If e(𝑢) = ⟨𝑓, 𝑖⟩, then,
∞ 𝒰 ∖ (J0K ∪ J1K ∪ J2K) ∞ 𝒰 ∖ (J0K ∪ J1K ∪ J2K) for all 𝑗 s.t. 0 ≤ 𝑗 < 𝑖, there is a 𝑢𝑗 ∈ 𝒰 s.t. e(𝑢𝑗 ) =
2 pdbf , pdbf , pdbf 2 pdbf ⟨𝑓, 𝑗⟩; and ii) if e(𝑢) = ⟨∞, 𝑖⟩ for 𝑖 ∈ N, then, for all 𝑗
1 pdbf , pdbf , pdbf , pdbf 1 pdbf , pdbf , s.t. 0 ≤ 𝑗 < 𝑖, there is a 𝑢𝑗 ∈ 𝒰 s.t. e(𝑢𝑗 ) = ⟨∞, 𝑗⟩.
0 pdbf , pdbf , pdbf 0 pdbf , pdbf , pdbf
Definition 2. An epistemic interpretation E is a total
Figure 1: Left: a ranked interpretation of the KB in Exam- function from 𝒰 to R that is convex.
ple 1 satisfying ⊤ |∼ ¬d. Right: a ranked interpretation of
the KB expanded with d |∼ ⊥. We let 𝒰E𝑓 =
def
{𝑢 ∈ 𝒰 | E(𝑢) = ⟨𝑓, 𝑖⟩ for some 𝑖 ∈ N}
∞ def
and 𝒰E = {𝑢 ∈ 𝒰 | E(𝑢) = ⟨∞, 𝑖⟩ for some 𝑖 ∈ N}.
def
We let minJ𝛼KE = {𝑢 ∈ J𝛼K | E(𝑢) ⪯ E(𝑣) for all
We introduce a logic of situated conditionals to over-
𝑣 ∈ J𝛼K}, minJ𝛼K𝑓E = def
{𝑢 ∈ J𝛼K ∩ 𝒰E𝑓 | E(𝑢) ⪯ E(𝑣) for
come this problem. The central insight is that adding an 𝑓
all 𝑣 ∈ J𝛼K ∩ 𝒰E }, and minJ𝛼K∞ def
E = {𝑢 ∈ J𝛼K ∩ 𝒰E |
∞
explicit notion of context to standard conditionals allows
E(𝑢) ⪯ E(𝑣) for all 𝑣 ∈ J𝛼K∩𝒰E∞ }. We can now provide
for a refined semantics of this enriched language in which
a semantic definition of situated conditionals in terms of
the problems described in Example 1 can be dealt with
epistemic interpretations.
adequately. It also allows us to reason coherently with
152
� ⟨∞, ∞⟩ Jp ∧ ¬bK ∪ Jd ∧ ¬bK
These properties are inspired by both the KLM char-
⟨∞, 1⟩ pdbf , pdbf
acterisation of conditional reasoning [3, 4] and the AGM
⟨∞, 0⟩ pdbf , pdbf approach to belief revision [5]. A situated conditional re-
⟨𝑓, 2⟩ pdbf lation that is closed under all these properties is a Full
⟨𝑓, 1⟩ pdbf , pdbf Situated Conditional (FSC). A representation theorem
⟨𝑓, 0⟩ pdbf , pdbf , pdbf connects the class of FSC’s to the class of epistemic inter-
pretations.
Figure 2: Model of the statements in Example 2. Theorem 1. Every epistemic interpretation generates an
FSC. Every FSC can be generated by an epistemic inter-
pretation.
Definition 3. E ⊩ 𝛼 |∼𝛾 𝛽 (abbreviated as 𝛼 |∼E𝛾 𝛽) if
Beyond investigating the properties characterising the
minJ𝛼 ∧ 𝛾K𝑓E ⊆ J𝛽K if J𝛾K ∩ 𝒰E𝑓 ̸= ∅; class of epistemic interpretations, we have also modeled a
{︂
∞ first form of non-monotonic entailment relation, minimal
minJ𝛼 ∧ 𝛾KE ⊆ J𝛽K otherwise.
closure, that is based on the classical rational closure
Intuitively, this definition evaluates 𝛼 |∼𝛾 𝛽 as follows. defined for ranked models [4].
If the situation 𝛾 is compatible with the plausible part of For a detailed explanation of the properties characteris-
E (the valuations in 𝒰E𝑓 ) then 𝛼 |∼𝛾 𝛽 holds if the most ing FSC’s, the proof of the representation theorem, and a
typical plausible models of 𝛼 ∧ 𝛾 are also models of 𝛽. presentation of the minimal closure, we refer the reader
On the other hand if the situation 𝛾 is not compatible to the technical report [2].
with the plausible part of E (that is, all models of 𝛾 have
an infinite rank) then 𝛼 |∼𝛾 𝛽 holds if the most typical
implausible (but possible) models of 𝛼∧𝛾 are also models 4. Concluding remarks
of 𝛽. SCs and epistemic interpretations allow to model
more correctly the conditionals in Example 1. The main contributions of this work can be summarised as
follows: (i) the motivation for and the provision of a sim-
Example 2. Consider the following rephrasing of the ple situation-based form of conditional which is general
statements in Example 1. ‘Birds usually fly’ becomes enough to be used in several application domains (e.g.,
b |∼⊤ f. Defeasible information about penguins and do- planning [2, Example 5.1]); (ii) an intuitive semantics
dos are modelled using p |∼p ¬f and d |∼d ¬f. Given which is based on a semantic construction that has proven
that dodos don’t exist anymore, the statement d |∼⊤ ⊥ useful in the area of belief change and that is more general
leaves open the existence of dodos in the infinite rank, and also more fine-grained than the standard preferen-
which allows for coherent reasoning under the assump- tial semantics; (iii) an investigation of the properties that
tion that dodos exist (the context d). Moreover, informa- situated conditionals satisfy and of their appropriateness
tion such as dodos and penguins necessarily being birds for knowledge representation and reasoning, in particular
can be modelled by the conditionals p ∧ ¬b |∼p∧¬b ⊥ when reasoning about information that is incompatible
and d ∧ ¬b |∼d∧¬b ⊥, relegating the valuations in with background knowledge, and (iv) the definition of a
Jp ∧ ¬bK ∪ Jd ∧ ¬bK to the rank ⟨∞, ∞⟩. Figure 2 shows form of entailment for contextual conditional knowledge
a model of these statements. bases based on the widely-accepted notion of rational
closure, which is reducible to classical propositional rea-
We have identified relevant situated rationality postu- soning.
lates, that represent desirable properties for SCs: Next steps are the extension of this approach to other
logics. Description Logics, for which rational closure has
|= 𝛼 ↔ 𝛽, 𝛼 |∼𝛾 𝛿
(Ref) 𝛼 |∼𝛾 𝛼 (LLE) already been reformulated [6, 7, 8], are the first candidates.
𝛽 |∼𝛾 𝛿
𝛼 |∼𝛾 𝛽, 𝛼 |∼𝛾 𝛿 𝛼 |∼𝛾 𝛿, 𝛽 |∼𝛾 𝛿 We also plan to investigate refinements of RC such as
(And) (Or)
𝛼 |∼𝛾 𝛽 ∧ 𝛿 𝛼 ∨ 𝛽 |∼𝛾 𝛿 lexicographic closure [9] and their variants [10, 11, 12].
𝛼 |∼𝛾 𝛽, |= 𝛽 → 𝛿 𝛼 |∼𝛾 𝛽, 𝛼 ̸|∼𝛾 ¬𝛿 A conference version of this work was presented at
(RW) (RM)
𝛼 |∼𝛾 𝛿 𝛼 ∧ 𝛿 |∼𝛾 𝛽 AAAI-21 [1], and, while an extended version of the paper
𝛼 |∼𝛾 𝛽 ⊤ ̸|∼⊤ ¬𝛾, 𝛼 ∧ 𝛾 |∼⊤ 𝛽 is under review at the moment, a technical report can be
(Inc) (Vac) found online [2].
𝛼 ∧ 𝛾 |∼⊤ 𝛽 𝛼 |∼𝛾 𝛽
𝛾≡𝛿 𝛼 |∼𝛾∧𝛿 𝛽
(Ext) (SupExp)
𝛼 |∼𝛾 𝛽 iff 𝛼 |∼𝛿 𝛽 𝛼 ∧ 𝛾 |∼𝛿 𝛽
𝛿 |∼⊤ ⊥, 𝛼 ∧ 𝛾 |∼𝛿 𝛽
(SubExp)
𝛼 |∼𝛾∧𝛿 𝛽
153
�Acknowledgments Proceedings of the 14th European Conference on
Logics in Artificial Intelligence (JELIA), number
The work of Giovanni Casini was partially supported by 8761 in LNCS, Springer, 2014, pp. 92–106.
TAILOR (Foundations of Trustworthy AI – Integrating [11] G. Casini, T. Meyer, I. Varzinczak, Taking defeasi-
Reasoning, Learning and Optimization), a project funded ble entailment beyond rational closure, in: F. Cal-
by EU Horizon 2020 research and innovation programme imeri, N. Leone, M. Manna (Eds.), Proceedings of
under GA No 952215. the 16th European Conference on Logics in Artifi-
This work was supported in part by the ANR Chaire cial Intelligence (JELIA), number 11468 in LNCS,
IA BE4musIA: BElief change FOR better MUlti-Source Springer, 2019, pp. 182–197.
Information Analysis (ANR-20-CHIA-0028). [12] G. Casini, U. Straccia, Defeasible inheritance-based
description logics, JAIR 48 (2013) 415–473.
References
[1] G. Casini, T. A. Meyer, I. Varzinczak, Contex-
tual conditional reasoning, in: Thirty-Fifth AAAI
Conference on Artificial Intelligence, AAAI 2021,
Thirty-Third Conference on Innovative Applications
of Artificial Intelligence, IAAI 2021, The Eleventh
Symposium on Educational Advances in Artifi-
cial Intelligence, EAAI 2021, Virtual Event, Febru-
ary 2-9, 2021, AAAI Press, 2021, pp. 6254–6261.
URL: https://ojs.aaai.org/index.php/AAAI/article/
view/16777.
[2] G. Casini, T. A. Meyer, I. Varzinczak, Situated
conditional reasoning, CoRR abs/2109.01552
(2021). URL: https://arxiv.org/abs/2109.01552.
arXiv:2109.01552.
[3] S. Kraus, D. Lehmann, M. Magidor, Nonmono-
tonic reasoning, preferential models and cumulative
logics, Artificial Intelligence 44 (1990) 167–207.
[4] D. Lehmann, M. Magidor, What does a conditional
knowledge base entail?, Art. Int. 55 (1992) 1–60.
[5] C. Alchourrón, P. Gärdenfors, D. Makinson, On the
logic of theory change: Partial meet contraction and
revision functions, Journal of Symbolic Logic 50
(1985) 510–530.
[6] P. Bonatti, Rational closure for all description logics,
Artificial Intelligence 274 (2019) 197–223.
[7] G. Casini, U. Straccia, Rational closure for defeasi-
ble description logics, in: T. Janhunen, I. Niemelä
(Eds.), Proceedings of the 12th European Confer-
ence on Logics in Artificial Intelligence (JELIA),
number 6341 in LNCS, Springer-Verlag, 2010, pp.
77–90.
[8] L. Giordano, V. Gliozzi, N. Olivetti, G. Pozzato,
Semantic characterization of rational closure: From
propositional logic to description logics, Art. Int.
226 (2015) 1–33.
[9] D. Lehmann, Another perspective on default rea-
soning, Annals of Math. and Art. Int. 15 (1995)
61–82.
[10] G. Casini, T. Meyer, K. Moodley, R. Nortjé, Rel-
evant closure: A new form of defeasible reasoning
for description logics, in: E. Fermé, J. Leite (Eds.),
154
�