Lexicographic inference [1] is a well-behaved and popular approach to reasoning with non-monotonic conditionals. In recent work we have shown that lexicographic inference satisfies syntax splitting, which means we can restrict our attention to parts of the belief base that share atoms with a given query. In this paper, we introduce the concept of conditional syntax splitting, inspired by the notion of conditional independence as known from probability theory. We show that lexicographic inference satisfies conditional syntax splitting, and connect conditional syntax splitting to several known properties from the literature on non-monotonic reasoning, including the drowning effect.
Lexicographic inference [
An intuitively related problem that was somewhat surprisingly
shown to be independent of syntax splitting in [
The drowning problem seems to be related to syntax splitting. Intuitively, {(|)} is unrelated to the rest of the belief base, in the sense that having beaks has nothing to do with flying or having wings, as long as we know we are talking about birds. However, (unconditional) syntax splitting does not allow to capture this kind of independence, since the atom prohibits the belief base from being split into information about flying and wings on the one hand, and information about beaks on the other hand. It is exactly this kind of conditional independencies between conditionals that we seek to formally capture and study in this paper. In more detail, the contributions of the paper are the following: 1. we introduce and study the notion of conditional splitting of a belief base, a property of conditional belief bases, and generalize the concept of syntax splitting, a property of inductive inference operators, to conditional syntax splitting, thus bringing the idea of conditional independence into the realm of inductive inference operators;
Outline of this Paper: We first state all the necessary preliminaries in Section 2 on propositional logic (Section 2.1), reasoning with non-monotonic conditionals (Section 2.2), inductive inference (Section 2.3), System Z (Section 2.4) and lexicographic inference (Section 2.5). In Section 3 we define and study the concept of conditional syntax splitting. In Section 4, we show that lexicographic inference satisfies conditional syntax splitting. In Sections 5 and 6, we show how properties of inductive inference operators previously only discussed informally, namely the drowning effect (Section 5) and the properties introduced by Lehmann 1995 (Section 6) can be seen as special cases of conditional syntax splitting. Finally, we discuss related work in Section 7 and conclude in Section 8.
4. we show how Lehmann’s so-called desirable closure prop- both antecedent and conclusion ((|)() = 1); it falsifies, or
erties [
Given a total preorder (in short, TPO) ⪯ on possible worlds,
representing relative plausibility, ⪯ iff ⪯ ′ for some
∈ min⪯ (Mod()) and some ′ ∈ min⪯ (Mod()). This
allows for expressing the validity of defeasible inferences via
stating that |∼ ⪯ iff (∧) ≺ (∧¬) [
In the following, we briefly recall some general preliminaries on propositional logic, and technical details on inductive inference.
1Θ⪯ |Θ 2Θ iff 1Θ ⪯ 2Θ.
(2)
Note that on the right hand side of the iff condition above 1Θ, 2Θ
are considered as propositions in the superlanguage ℒ(Ω) , hence
1Θ ⪯ 2Θ is well defined [
Similarly, any inference relation |∼ on ℒ(Σ) induces a
marginalized inference relation |∼ |Θ on ℒ(Θ) by setting
For a set At of atoms let ℒ(At) be the corresponding
propositional language constructed using the usual connectives ∧ (and), |∼ |Θ iff |∼ (3)
∨ (or), ¬ (negation), → (material implication) and ↔
(material equivalence). A (classical) interpretation (also called pos- for any , ∈ ℒ(Θ) .
sible world) for a propositional language ℒ(At) is a function An obvious implementation of total preorders are ordinal
: At → {⊤, ⊥}. Let Ω( At) denote the set of all interpretations conditional functions (OCFs), (also called ranking functions)
for At. We simply write Ω if the set of atoms is implicitly given. : Ω → N ∪ {∞} with − 1(0) ̸= ∅. [
Given a language ℒ, conditionals are objects of the form (|)
where , ∈ ℒ. The set of all conditionals based on a language
ℒ is defined as: (ℒ|ℒ) = {(|) | , ∈ ℒ}. We follow
the approach of [
Examples of inductive inference operators include system P
[
As already indicated in the previous subsection, inference relations can be obtained on the basis of TPOs respectively OCFs: ((|))() = ⎧ 1 : |= ∧ ⎨
0 : |= ∧ ¬ ⎩ : |= ¬ where stands for unknown or indeterminate. In other words, a possible world verifies a conditional (|) iff it satisfies conditional belief base ∆ a total preorder ⪯ Δ on Ω s.t. |∼ ⪯ Δ for every (|) ∈ ∆ (i.e. s.t. DI is ensured). A model-based (1) inductive inference operator for OCFs (on Ω ) is a mapping C that assigns to each conditional belief base ∆ an OCF Δ on Ω s.t. ∆ is accepted by Δ(i.e. s.t. DI is ensured).
Examples of inductive inference operators for OCFs System
Z ([
To define the property of syntax splitting [
Definition 3 (Independence (Ind), [
As an example of a (non-)inference, observe that e.g. ⊤ |∼ Δ ¬
We recall lexicographic inference as introduced by [
Thus, Ind requires that inferences from one sub-language are independent from formulas over the other sublanguage, if the belief base splits over the respective sublanguages. In other words, information on the basis of one sublanguage does not influences inferences made in the other sublanguage. Rel, on the other hand, restricts the scope of inferences, by requiring that inferences in a sublanguage can be made on the basis of the conditionals in a conditional belief base formulated on the basis of that sublanguage.
SynSplit combines these two properties.
We present system defined in [
For ∈ ∆ we define: Δ( ) = iff ∈ ∆ and (∆ 0, . . . , ∆ ) is the Z-partioning of ∆ . Finally, the ranking function Δ is defined via: Δ () = max{( ) | () = 0, ∈ ∆ } + 1, with max ∅ = − 1. The resulting inductive inference operator Δ is denoted by .
In the literature, system has also been called rational closure
[
We now illustrate OCFs in general and System in particular with the well-known “Tweety the penguin”-example. avoid clutter.
In [
Example 4 (Example 3 ctd.). For the Tweety belief base ∆ as in Example 3 we obtain the following lex()-vectors: lex() (0,1) (0,0) lex() (1,0) (1,0) lex() (0,2) (0,0) lex() (0,1) (0,0)
(0, 0) ≺ lex (1, 0) ≺ lex (0, 1) ≺ lex (0, 2).
Observe that e.g. ⊤ |∼ lΔex¬ (since lex(⊤ ∧ ¬) = (0, 0) ≺ lex lex(⊤ ∧ ) = (1, 0)) and ∧ |∼ lΔex.
We now introduce a conditional version of syntax splitting. A first central idea is the syntactical notion of conditional splitting, a property of belief bases.
Definition 6. We say a conditional belief base ∆ can be split into subbases ∆ 1,∆ 2 conditional on a sub-alphabet Σ 3, if ∆ ⊂ (ℒ(Σ ∪ Σ 3) | ℒ(Σ ∪ Σ 3)) for = 1, 2 s.t. Σ 1, Σ 2 and Σ 3 are pairwise disjoint and Σ = Σ 1 ∪ Σ 2 ∪ Σ 3, writing: ∆ = ∆ 1 ⋃︁ ∆ 2 | Σ 3
Intuitively, a conditional belief base can be split into Σ 1 and Σ 2 conditional on Σ 3, if every conditional is built up from atoms in Σ 1 ∪ Σ 3 or atoms in Σ 2 ∪ Σ 3.
The above notion of conditional syntax splitting, however, is Proposition 2. Let a conditional belief base ∆ = too strong, in the sense that it does not warrant satisfaction of ∆ 1 ⋃︀Σ1,Σ2 ∆ 2 | Σ 3 be given. If there is a ∈ ℒ(Σ 3) s.t. conditional variants of relevance and independence (we will define for every conditional in (|) ∈ ∆ = ∆ 1 ⋃︀Σ1,Σ2 ∆ 2 | Σ 3: them in formal detail below) for lexicographic inference. The underlying problem is that toleration might not be respected by 1. ∈ ℒ(Σ 1) ∪ ℒ(Σ 2), or ≡ . conditional belief bases that conditionally split: 3. ⋀︀(|)∈Δ → ̸⊢ ⋁︀{¬ | ( |′) ∈ ∆ , ′ ≡
} for = 1, 2.1 Then ∆ = ∆ 1 ⋃︀sΣ1,Σ2 ∆ 2 | Σ 3.
Example 5. Let ∆ =
{(|), (¬|), (| ∧ )}. Then ∆ = {(|), (¬|)}
{(| ∧ )} | {, } ⋃︁ {},{} However, this notion of purely syntactical conditional independence is not reflected on the level of tolerance (and therefore Proof. Consider some 3 ∈ Ω(Σ 3). If 3 ̸|= we are done. entailment). Indeed, {(| ∧ )} (trivially) tolerates itself, i.e. Suppose therefore 3 |= . With the third condition, there is an {(|∧)}(| ∧ ) = 0, yet ∆ does not tolerate (| ∧ ), i.e. 2′3 ∈ Mod(⋀︀(|)∈Δ2 → ) s.t. 2′3 |= for every Δ(| ∧ ) = 1. ( |′) ∈ ∆ 2 s.t. ′ ≡ . Notice that, in view of the first two
This means that for system Z and lexicographic entaiment, conditions, for any (|) ∈ ∆ , either 2 |= → or ≡ conditional relevance (now only introduced informally) is vio- or ≡ . Furthermore, in thec, aaslesowhe2re3|=≡ ,. A2lto′3ge|=ther,. lated for this belief base. In more detail, even though ∆ = Since ∈ ℒ(Σ 2) or ≡ {(|), (¬|)} ⋃︀{},{}{(| ∧ )} | {, }, we have e.g. 23 |= ⋀︀(|)∈Δ2 → . Thus, ∆ = ∆ 1 ⋃︀sΣ1,Σ2 ∆ 2 | ⊤ ̸ |∼ l{e(x|∧)}¬( ∧ ) whereas ⊤ |∼ lΔex¬( ∧ ) (and likewise Σ 3. for system ).
What happens here is that (|) and (¬|) act as “constraints” on and being true together, which on its turn is needed for (| ∧ ) to be tolerated. In other words, pure syntactic conditional splitting is not reflected on the semantic level (in contradistinction to unconditional splitting). We can exclude such cases by using the following weaker notion of safe conditional syntax splitting: Definition 7. A conditional belief base ∆ = ∆ 1 ⋃︀Σ1,Σ2 ∆ 2 | Σ 3 can be safely split into subbases ∆ 1, ∆ 2 conditional on a sub-alphabet Σ 3, writing: ∆ = ∆
s 1 ⋃︁ ∆ 2 | Σ 3,
Σ1,Σ2 if for every 3 ∈ Ω(Σ ∪Σ 3), there is a ∈ Ω(Σ ⋁︀( |)∈Δ ∧ ¬ (for , = 1, 2 and ̸= ).
) s.t. 3 ̸|=
A simpler case of this condition is a belief base where all antecedents derive from the common alphabet Σ 3 and all consequents derive from either Σ 1 or Σ 2.
Notice that e.g. the conditional belief base from Example 2 has the form described in Proposition 2: Example 6. Consider again ∆ from Example 2, and let Σ 1 = {, }, Σ 2 = {} and Σ 3 = {}. Observe that: ∆ = {( |), (|), (¬ |)} ⋃︁ {(|)} | Σ 3.
Σ1,Σ2 Furthermore, the first two items in Proposition 2 are satisfied as every conditional is either completely on the basis of the alphabet {, } or has as an antecedent or a consequent . Finally, the last condition is satisfied as { → } ̸|= ¬ and { → , → , → ¬ } ̸|= ∨ ¬. We thus see that ∆ = {( |), (|), (¬ |)} ⋃︀sΣ1,Σ2 {(|)} | Σ 3.
The notion of safe splitting is explained as follows: ∆ can be The bicycle example is also of this form: safely split into ∆ 1 and ∆ 2 conditional on Σ 3 if it can be split in ∆ 1 and ∆ 2 conditional on Σ 3, and additionally, for every world Example 7. Consider again ∆ from Example 1. We see that: 3 in the subsignature Σ ∪ Σ 3, we can find a world in s the subsignature Σ (, = 1, 2 and ̸= ) s.t. no conditional {(|), (|)} ⋃︁ {( |)} | {}. ∈ ∆ is falsified by 3 (or, equivalently, by 3). We {,},{} will show some more syntactical formulated conditions that ensure Remark 1. Note that a weaker prerequisite such as taking only safe splitting below. the first two conditions in Proposition 2 does not work: in more de
We argue here that safe splitting faithfully captures indepen- tail, requiring that there is a ∈ ℒ(Σ 3) s.t. for every conditional sdiegnncaetusreo fΣ t3w. oIncdoeendd,itsiaofneaslpbliettliinegf rbeaqsueisrecsotnhadtit(i1o)naalll oconnaditsiuobn-- in (|) ∈ ∆ = ∆ 1 ⋃︀Σ1,Σ2 ∆ 2 | Σ 3: als are built up from the sub-signatures Σ 1 ∪ Σ 3 or Σ 2 ∪ Σ 3 (i.e. 1. ∈ ℒ(Σ 1) ∪ ℒ(Σ 2), ∆ 1 ⋃︀Σ1,Σ2 ∆ 2 | Σ 3), and (2) that any information on Σ ∪ Σ 3 is 2. ∈ ℒ(Σ 1) ∪ ℒ(Σ 2), or ≡ . compatible with ∆ , i.e. no world 3 causes a conditional in ∆ to be violated. In other words, toleration with respect to ∆ is In other words, these two conditions say that conditionals are independent of ∆ . either fully from the language based on either Σ 1 or Σ 2, or their
We now delineate some more syntactic conditions that ensure antecedent is fully based on Σ 3, and there is only a single formula safe syntax splitting. These conditions are typically easier to allowed to occur as such. However, this notion is not consistent check, and might reasonably be expected to hold for certain with toleration. Consider ∆ = {(|⊤), (¬|), (|)}. Then ∆(n=oaft uc∆roanld1iltai⋃no︀gnΣua1la,sΣgie2n∆ ∆sc2)enu|asΣ irni3og,se.(l2e)Fmaoelrnlteasxnoatfemcthepedleec,nomtisf maitnodnhoscluodbns-saetlhqpauhteabn(1etst) ∆ = {(|⊤), (¬|)} {⋃},︁{}{(|)} | {} Σ 3 are equivalent, and (3) all material versions of the conditional and {(|)} tolerates itself (trivially), yet {(|⊤), (¬|), (|)} sub-base ∆ are consistent with the set of consequents of the does not tolerate (|). It is perhaps not surprising that such a conditionals whose antecedent uses atoms in the common sub- purely syntactic condition is elusive. alphabet Σ 3, then ∆ can be safely split into ∆ 1 and ∆ 2 condi- 1Or, equivalently, { → | (|) ∈ Δ}∪{ | ( |′) ∈ Δ, ′ ≡ tional on Σ 3. } ̸⊢ ⊥.
Safe conditional splitting of a conditional belief base is consistent with toleration, in the sense that ∆ = ∆ 1 ⋃︀sΣ1,Σ2 ∆ 2 | Σ 3 implies that toleration of a conditional (|) by ∆ is equivalent to toleration of (|) by the conditional sub-base ∆ in which it occurs. This gives further evidence to the fact that safe conditional splitting adequately captures the notion of independence of sub-bases: toleration of a conditional is independent of a (conditionally) unrelated sub-base.
CInd CRel CInd CRel ⪯ Δ iff ⪯ Δ ⪯ Δ iff ⪯ Δ Δ() ⩽ Δ() iff Δ() ⩽ Δ() Δ() ⩽ Δ() iff Δ () ⩽ Δ ()
We now connect CInd to the notion of conditional
independence of TPOs as known from belief revision. For this, we need
the following notion taken from [
We now move to the formulation of conditional syntax splitting, Proposition 4. An inductive inference operator for TPOs C : a property of inductive inference relations that expresses that the ∆ ↦→⪯ Δ on ℒ satisfies (CInd) iff for any ∆ = ∆ 1 ⋃︀sΣ1,Σ2 ∆ 2 | independencies between sub-bases of conditionals, as encoded in Σ 3, it holds that Σ 1 =⪯| Σ 2|Σ 3. safe splitting, are respected by an inductive inference relation.
Conditional independence (CInd) and safe conditional relevance (CRel) are defined analogous to ( Ind) and (Rel), but now 1), assuming that a conditional belief base can be safely split and taking into account we have full information on the “conditional pivot” Σ 3: Proof. For the ⇒-direction, suppose that C satisfies (CInd) and ∆ = ∆ 1 ⋃︀sΣ1,Σ2 ∆ 2 | Σ 3. Consider some 11, 21 ∈ Ω(Σ 2 ∈ Ω(Σ 2) and 3 ∈ Ω(Σ 3) and suppose that 113 ≺ 113. Thus, 113 ∨ 213 |∼ ⪯ ¬11. Thus, with (CInd), 1123 ∨ 2123 |∼ ⪯ ¬11 and thus 1123 ≺ 2123. The other direction of equation (4) is analogous. iDf effinoritaionny 8∆=. ∆ An 1in⋃d︀usΣc1t,iΣv2e ∆inf2er|eΣ nc3e, aonpderfaotorranCy sat,isfies∈( ℒC(IΣnd)), ∆ =F o∆r th1e⋃⇐︀sΣ-1d, Σir2ec∆ti2on|, Σsu3ppaonsde Σ1 |∼ =Δ⪯| Σ 2f|oΣr 3soamnde su,ppos∈e ∈ ℒ(Σ ) (for , ∈ {1, 2}, ̸= ) and a complete conjunction ℒ(Σ 1) and some complete conjunction ∈ ℒ(Σ 3). Notice ∈ ℒ(Σ 3), that since is a complete conjunction, there is a unique world |∼ Δ iff |∼ Δ 3 ∈ Ω(Σ 3) s.t. 3 |= . Then ≺ . Consider Thus, an inductive inference operator satisfies conditional in- now some arbitrary ∈ ℒ(Σ 2). Notice that ⪯ . ddietpioennadleonnceΣ if3,, fwohreanneyv∆er wtheathsaavfeelayllstphleitsneincteos s∆ar1y ainnfdo∆rm2actioonn- Tmaikne⪯ soMmoed(21223),∈m31in12⪯3M∈odM(od().F)o(raasny ∈11ℒ12(Σ31∈), about Σ 3, inferences from one sub-language are independent from and thus 11123 ≺ 31123 (since ≺ ). With formulas over the other sub-language. independence, 113 ≺ 213. Again with independence, 11223 ≺ 21223. Since 22 |= , 11223 ∈ Mod() Definition 9. An inductive inference operator C satisfies ( CRel) and thus there is some 31323 min⪯ Mod() with if for any ∆ = ∆ 1 ⋃︀sΣ1,Σ2 ∆ 2 | Σ 3, and for any , ∈ ℒ(Σ ) 31323 ≺ 21223. Since 212∈23 ∈ min⪯ Mod(), (for ∈ {1, 2}) and a complete conjunction ∈ ℒ(Σ 3), we have established that |∼ ⪯ .
|∼ Δ iff |∼ Δ
Thus, CRel restricts the scope of inference by requiring that inferences in the sub-language Σ 1 ∪ Σ 3 can be made on the basis of the conditionals on the basis of that sub-language.
Syntax splitting (CSynSPlit) combines the two properties (CInd) and (CRel):
Proposition 4 establishes a correspondence between the property CInd of inductive inference operators, and the notion of conditional independence for TPOs, as already known from belief revision.
Proposition 5. An inductive inference operator for TPOs C : ∆ ↦→⪯ Δ on ℒ satisfies (CRel) iff for any ∆ = ∆ 1 ⋃︀sΣ1,Σ2 ∆ 2 | Σ 3, it holds that ⪯ Δ = ⪯ Δ|Σ .
Definition 10. An inductive inference operator C satisfies conditional syntax splitting (CSynSPlit) if it satisfies ( CInd) and (CInd).
Proof. For the ⇒-direction, suppose that C satisfies (CRel) and consider some ∆ = ∆ 1 ⋃︀sΣ1,Σ2 ∆ 2 | Σ 3. Suppose 1113 ≺ Δ 2123. Then 1113 ∨ 2123 |∼ Δ ¬2123. With We Wfirset naonwalypsreoctheeedprwopitehrttihees sotfudCyoIfncdonandditiConRaleslyfnotraxTPspOlsit.tiWnge. (CRel), 1113 ∨ 2123 |∼ Δ¬2123 and thus 1113 ≺ Δ 2123, iffrostr nToPtiOcse rtehsaptecCtivIneldy afonrdOCCRFesl ifsoerqinudivuaclteinvtetiontfheerefnocleloowpienrgattowros i.e.For11th13e⪯ ⇐Δ-|Σdi1rec21tio23n., Tsuhpepootsheerthdaitre∆ c=tio∆ n is1an⋃︀asΣlo1g,Σo2u s∆. 2 | Σ 3 properties (for any ∆ = ∆ 1 ⋃︀sΣ1,Σ2 ∆ 2 | Σ 3 and for , ∈ and ⪯ Δ = ⪯ Δ|Σ . Suppose now |∼ Δ. Then ≺ Δ ℒ(Σ ), complete conjunction ∈ ℒ(Σ 3), ∈ ℒ(Σ ), , = and thus ≺ Δ , which implies |∼ Δ . 1, 2 and ̸= ):
We now analyze conditional syntax splitting for inductive infer- Lemma 9. Let a conditional belief base ∆ 1 ⋃︀sΣ1,Σ2 ∆ 2 | Σ 3 ence operators for OCFs. Thanks to the close relationship between with its corresponding Z-partition (∆ 0, . . . , ∆ ) be given. Then rankings and probabilities, there is a straightforward adaptation for every 0 ⩽ ⩽ 2: of conditional independence for OCFs [19, Chapter 7]. (, ∆ ) = (13, ∆ 1 ) + (23, ∆ 2 ) − (3, ∆ 1 ) = |{(|) ∈ ∆ 1 | 13 |= ∧ ¬}| +|{(|) ∈ ∆ 2 | 23 |= ∧ ¬}| −|{ (|) ∈ ∆ 1 ∩ ∆ 2 | 123 |= ∧ ¬}|.
Proposition 6. An inductive inference operator for OCFs C : ∆ ↦→ Δ satisfies CInd iff for any ∆ = ∆ 1 ⋃︀sΣ1,Σ2 ∆ 2 | Σ 3 we have Σ 1 =| Σ 2|Σ 3.
Proof. We first recall the following Lemma from [
We now show that: (†) for any ∈ ℒ(Σ ) and 3 ∈ ℒ(Σ 3), (3) = min{ (13) | 23 |= }. Wlog let = 1 and 1=22.O3b|=serve}th.atCboyndsiedfineitriosno,me (132)=3 m∈inM{od(1,21,3(23)3|) Lℒe(Σm1m)aan1d0.W3∈heΩ(rΣe ∆ =3)∆ , 1∈⋃m︀sΣi1n,⪯Σ2lΔex∆ (M2 o| dΣ(3,3∧∈{)1),i2ff},1 ∈∈ s.t. (123) = (3). With Lemma 7, (123) = (13) + (23) − (3). Suppose now towards a contra- min⪯ lΔex1 (Mod1,3(3 ∧ )) and 2 ∈ min⪯ lΔex2 Mod2,3(3). diction that (13) > (3), i.e. there is some ⋆1⋆23 s.t. ⋆13 |= 3 and (⋆13)+ (⋆23) < (13)+ (23). Proof. For the ⇒-direction, suppose ∈ min⪯ lΔex(Mod(3)). Suppose first that (⋆23) > (23). Then (⋆13) < Suppose now towards a contradiction that either (a) 23 ̸∈ (13) and thus (⋆123) < (123), contradiction. min⪯ lΔex2 (Mod2,3(3)) or (b) 1 ̸∈ min⪯ lΔex1 (Mod1,3(3 ∧ )). Suppose therefore that (⋆23) ⩽ (23). Then we can derive that (⋆123) < (123), contradiction. ad. (b) Suppose that 23 ∈ min⪯ lΔex2 (Mod2,3(3)) (the case
From the †, it follows immediately that (3) < (3) where also 23 ̸∈ min⪯ lΔex2 (Mod2,3(3)) is similar). iff (3) < (3) for any , ∈ ℒ(Σ ), ∈ ℒ(Σ ) and 3 ∈ ℒ(Σ 3) (for , = 1, 2 and ̸= ). 1 ̸∈ min⪯ lΔex1 (Mod1,3(3 ∧ )) implies that there is Proposition 8. An inductive inference operator for OCFs C : ∆ ↦→ Δ satisfies CRel iff for any ∆ = ∆ 1 ⋃︀sΣ1,Σ2 ∆ 2 | Σ 3, it holds that Δ = Δ |Σ1∪Σ3.
In this section, we show that for any conditional belief base that safely splits conditionally, conditional syntax splitting is satisfied. We first need to show some intermediate results.
Fact 1. Where ∆ = ∆ 1 ⋃︀sΣ1,Σ2 ∆ 2 | Σ 3, ∈ {1, 2} and (|) ∈ ∆ , Δ((|)) = Δ((|))
some ′13 ∈ Mod1,3(3∧) s.t. ′13 ≺ lΔex1 13, i.e. there is some ⩾ 0 s.t. for every > , (′13, ∆ 1) = (13, ∆ 1) and (′13, ∆ 1 ) < (13, ∆ 1 ) (with Lemma 9) this implies:
(′123, ∆ ) = (′13, ∆ 1) + (23, ∆ 2) − (3, ∆ 1) = (123, ∆ ) = (13, ∆ 1) + (23, ∆ 2) − (3, ∆ 1) and
(′123, ∆ ) = (′13, ∆ 1 ) + (23, ∆ 2 ) − (3, ∆ 1 ) < (123, ∆ ) = (13, ∆ 1 ) + (23, ∆ 2 ) − (3, ∆ 1 ) which (since ′123 ∈ Mod(3)) contradicts ∈ min⪯ lΔex(3). ad. (a) Similar.
The ⇐-direction is similar. 2Notice that it follows from Fact 1 that, given the Z-partition (Δ1,...,Δ) of Δ and Σ ⊆ Σ, Δ = Δ ∩ (ℒ|ℒ) for any 0 ⩽ ⩽ .
Proposition 11. lex satisfies CInd. basis of examples such as the Tweety-example, but no generic Proof. Suppose that ∆ = ∆ 1 ⋃︀sΣ1,Σ2 ∆ 2 | Σ 3. We show that forImn atlhdisespcarpipetri,ownehahsabveeednegvievleonp.ed the necessary tools to talk Σ 1 and Σ 2 are independent w.r.t. ⪯ lΔex conditional on Σ 3, i.e. for about the drowning effect in a formally precise manner. Indeed, any ∈ {le1x, 2}, for any 1 , 2 ∈ Ω , 3 ∈ Ω 3, ∈ {1, 2}, ̸= , the first crucial notion is that of unrelatedness of propositions. 1 3 ⪯ Δ 2 3 iff 1 3 ⪯ lΔex 2 3 for all ∈ Ω . This notion is formally captured by safe splitting into subbases With Proposition 4 this is sufficient to show the proposition. For (Definition 7) : given a belief base ∆ , a proposition is unrelated simplicity, we let = 1 and = 2, the other case follows by to a proposition iff ∆ can be safely split into subbases ∆ 1, ∆ 2 symmetry. conditional on a sub-alphabet Σ 3, i.e. ∆ = ∆ 1 ⋃︀sΣ1,Σ2 ∆ 2 | Σ 3,
For the ⇒-direction, suppose that 113 ⪯ lΔex 213. We show and ∈ ℒ(Σ 2) and ∈ ℒ(Σ 1 ∪ Σ 3). This means that the that 1123 ⪯ lΔex 2123 for all 2 ∈ Ω 2. We first make the abstract situation of the drowning problem can be precisely defollowing observation that follows in view of Lemma 10. For any scribed by conditional syntax splitting. We see that the drowning ⋆2 ∈ min⪯ lΔex2 (Mod2,3(3)) and = 1, 2: effect is nothing else than a violation of the postulate of conditional independence (CInd): if we know that a typical property 13 ≈ lΔex 1⋆23. of -individuals ( |∼ Δ) is unrelated to an exceptional subclass of , then we can also derive that if something is Thus, for any 2 ∈ min⪯ lΔex2 (Mod2,3(3)): is typically ( |∼ Δ). We illustrate this with the Tweety-example: 1123 ⪯ lΔex 2123.
This means (with Lemma 9) that there is some ⩾ 0 s.t. (113, ∆ 1 ) = (213, ∆ 1 ) for every > and (113, ∆ 1 ) ⩽ (213, ∆ 1 ). Thus, for any 2 ∈ Ω 2, it holds that (for any > ): Example 8 (Example 2 ctd.). We already established in Example 6 that ∆ = {( |), (|), (¬ |)} ⋃︀{,},{}{(|)} | {}. It is now not hard to see that any inductive inference operator C that satisfies ( DI) and (CInd) avoids the drowning effect. In more detail, we have: (113, ∆ 1 ) + (23, ∆ 2 ) − (3, ∆ 1 ) (213, ∆ 1 ) + (23, ∆ 2 ) − (23, ∆ 1 ) and: = ⩽ (113, ∆ 1 ) + (23, ∆ 2 ) − (3, ∆ 1 ) (213, ∆ 1 ) + (23, ∆ 2 ) − (3, ∆ 1 ).
With Lemma 9 we have that (123, ∆ ) = (13, ∆ 1) + (23, ∆ 2) − (3, ∆ 3) for = 1, 2 and ⩽ and thus
lex 1223 (for any 2 ∈ Ω 2). 1123 ⪯ Δ
The ⇐-direction is similar.
Proposition 12. lex satisfies Rel. 13 |= . ∈ ℒ1, this implies that if 13 ∈ Proof. This follows immediately from Lemma 10. Indeed, we have that (for , ∈ ℒ1 and a complete conjunction ∈ ℒ3): Lem|m∼alΔex10, iff ∈formailnl⪯ lΔex (∈m)inif⪯flΔex(1), 13 |= . With ∈ min⪯ lΔex1 (). Since
min⪯ lΔex1 () then
From Proposition 11 and 12, we can immediately obtain the main result of this section: Theorem 1. lex satisfies SynSplit.
As mentioned in the introduction, the drowning effect, illustrated
by Example 2, is intuitively related to syntax splitting. In more
detail, the drowning effect is constituted by the fact that according
to some inductive inference operators (e.g. system ), exceptional
subclasses (e.g. penguins) do not inherit any properties of the
superclass (e.g. birds), even if these properties are unrelated to the
reason for the subclass being exceptional (e.g. having beaks). To
the best of our knowledge, discussion of the drowning effect in
the literature has been restricted to informal discussions on the
|∼ Δ
∧ |∼ Δ
by DI
by CInd and (5)
For any inductive inference operator that additionally satisfies
Cut (i.e. from |∼ and ∧ |∼ derive |∼ ), a postulate
that holds for any inductive inference operator based on TPOs
[
Proposition 13. Any inductive inference operator that satisifes ( CInd) does not show the drowning problem (for ∆ = {( |), (|), (¬ |)} ⋃︀{,},{}{(|)} | {}).
We note that the drowning effect has not been defined in the literature in a general sense. Above, we provided, to the best of our knowledge, the first general definition of the drowning problem. As we defined the drowning effect as a violation of the postulate of conditional independence, it is trivial that any inductive inference operator that satisfies ( Cind) does not show the drowning problem interpreted in this general, formal sense.
Lehmann 1995 remarks that in addition to the properties encoded
by rational closure, other properties for inductive inference
operators might be desirable. In particular, he lists four properties:
presumption of typicality, presumption of independence, priority
of typicality and respect for specificity . All of these properties
are only explained informally and illustrated using examples in
[
We now describe the four properties from [
This section thus shows that all four properties proposed in [
The main contributions of this paper are the following: (1) we define the concept of conditional syntax splitting for inductive inference operators, thus bringing a notion of conditional independence between sub-signatures to the realm of inductive inference operators; (2) we show that lexicographic inference satisfies conditional syntax splitting; (3) we show how the drowning effect can be seen as a violation of conditional syntax splitting, and (4) we show how Lehman’s desirable properties can be derived from (conditional) syntax splitting.
There are several main avenues for further work. Firstly, it
will be interesting to investigate whether other inductive
inference operators that satisfy (non-conditional) syntax splitting, such
as c-representations ([
Priority of typicality Priority of typicality gives, in situations where the presumption of typicality and the presumption of independence clash, priority to the former. It is illustrated using the conditional belief base ∆ 3 = {(|), (¬| ∧ )}. The presumption of typicality justifies us in deriving ∧ ∧ |∼ Δ3 ¬ (since no reason can be found for accepting ∧ |∼ Δ3 ¬), whereas we can derive both ∧ ∧ |∼¬ and ∧ ∧ |∼ Δ3 with the presumption of independence. Priority of typicality demands that priority is given to ∧ ∧ |∼ Δ3 ¬.
Fact 4. Let the inductive inference operator satisfying Ind and ∆ 3 = {(|), (¬| ∧ )} be given. Then ∧ ∧ |∼ Δ3 ¬. Proof. Let Σ 1 = {, , } and Σ 2 = {}. Then clearly ∆ 3 = ∆ 3 ⋃︀Σ1,Σ2 ∅ and thus with (Ind) we have: ∧ |∼ Δ3 ¬ iff ∧ ∧ |∼ Δ3 ¬. Since (¬| ∧ ) ∈ ∆ 3, with (DI), we have ∧ |∼ Δ3 ¬.
Lehmann 1995 uses a second example to illustrate priority of
typicality:
Example 9 (Example 5 in [
The work of Jesse Heyninck was partially supported by Fonds Wetenschappelijk Onderzoek – Vlaanderen (project G0B2221N).