Ranking functions and total preorders on worlds are two common models for epistemic states that can represent conditional beliefs. To further explore the connection between these frameworks, we consider mappings among models of both frameworks. Especially interesting are mappings that preserve desirable properties like syntax splittings, or are compatible with operations like marginalization and conditionalization. In this paper, we introduce postulates for such mappings. We evaluate the postulates for mappings within and across the two frameworks, establishing dependencies as well as incompatibilities among the postulates. The results will be useful for transferring methods developed for ranking functions to the total-preorder framework and the other way round.
In the field of knowledge representation, there is a long tradition of employing
conditionals as fundamental objects. A conditional formalizes a defeasible rule
“If A then usually B” for logical formulas A, B and is often denoted as (B|A). As
conditional logic is more expressive than propositional logic, it requires a richer
semantics as well. There are di↵erent approaches to semantics for conditional
logic, e.g., [
In this paper, we focus on these two kinds of models for conditionals, ranking
functions (or ordinal conditional functions, OCFs) and total preorders on worlds
(TPOs). Both models have their own advantages. TPOs are used in
characterisation theorems for AGM revisions [
We formalize functions on these models within and across the two
frameworks as epistemic state mappings and propose postulates that govern epistemic
state mappings. The postulates require the epistemic state mappings to
preserve certain properties of the models like the entailed inference relation and
syntax splittings. Syntax splitting is a concept describing that beliefs about
different parts of the signature are uncorrelated [
In summary, the main contributions of this paper are: – Introduction of epistemic state mappings for TPOs and OCFs – Coverage of marginalization and conditionalization for the iterated case via the introduction of restricted TPOs and restricted OCFs – Formalization of desirable properties of epistemic state mappings in terms of general postulates – Establishment of relationships among the postulates and of realizability results for the postulates and for subsets thereof.
This paper is a shorter version of a text submitted to NMR2021 [
The following text is structured as follows. After giving some background on conditional logic, ranking functions and total preorders in Section 2, we introduce marginalization, conditionalization, and syntax splitting in Section 3. We proceed to introduce the concept of epistemic state mappings and postulates for such mappings in Section 4. Then, we analyse the relationship among the postulates for epistemic state mappings from TPOs to TPOs in Section 5.1 and among the postulates for epistemic state mappings from OCFs to OCFs in Section 5.2. In Section 5.3, we consider epistemic state mappings from OCFs to TPOs, and in Section 5.4 we consider epistemic state mappings from TPOs to OCFs. In Section 6, we conclude and point out future work. 2
Background: Conditional Logic, Ranking Functions, and Total Preorders A (propositional) signature is a finite set ⌃ of identifiers. For a signature ⌃ , we denote the propositional language over ⌃ by L⌃ . Usually, we denote elements of the signatures with lowercase letters a, b, c, . . . and formulas with uppercase letters A, B, C, . . .. We may denote a conjunction A ^ B by AB and a negation ¬A by A for brevity of notation. The set of interpretations over a signature ⌃ is denoted as ⌦ ⌃ . Interpretations are also called worlds and ⌦ ⌃ is called the universe. An interpretation ! 2 ⌦ ⌃ is a model of a formula A 2 L⌃ if A holds in !. This is denoted as ! |= A. The set of models of a formula (over a signature ⌃ ) is denoted as Mod ⌃ (A) = {! 2 ⌦ ⌃ | ! |= A}. A formula A entails a formula B if Mod ⌃ (A) ✓ Mod ⌃ (B). Two formulas A and B are equivalent, denoted as A ⌘ B, if they have the same models, i.e., Mod ⌃ (A) = Mod ⌃ (B).
A conditional (B|A) connects two formulas A, B and represents the rule “If A then usually B”. For a conditional (B|A) the formula A is called the antecedent and the formula B the consequent of the conditional. The conditional language over a signature ⌃ is denoted as (L|L)⌃ = {(B|A) | A, B 2 L⌃ }. The set (L|L)⌃ is a flat conditional language as it does not allow nesting conditionals.
We use a three-valued semantics of conditionals in this paper [
There exist di↵erent kinds of models for epistemic states that can handle conditionals. Two approaches to this are ranking functions and total preorders on possible worlds.
A ranking function [
A total preorder (TPO) is a total, reflexive, and transitive binary relation. The meaning of a total preorder on ⌦ ⌃ as model for an epistemic state is that !1 is at least as plausible as !2 if !1 !2 for !1, !2 2 ⌦ ⌃ . The strict version of a TPO is the relation defined by !1 !2 if !1 !2 and !2 6 !1. TPOs on worlds are extended to consistent formulas by A B if min(Mod ⌃ (A), ) min(Mod ⌃ (B), ). A TPO models a conditional (B|A), denoted as |= (B|A), if AB AB, i.e., if the verification of the conditional is strictly more plausible than its falsification. 3
Marginalization, Conditionalization, Syntax Splitting We want to consider transformations among models of epistemic states represented by ranking functions or total preorders. To establish a notion for the domain of such transformations, we define the sets MTPO (⌃ ) and MOCF (⌃ ) containing all models over a certain signature ⌃ :
MTPO (⌃ ) = { ✓ ⌦ ⌃ ⇥ ⌦ ⌃ |
total preorder over ⌦ ⌃ } MOCF (⌃ ) = { : ⌦ ⌃ 7! N0 [ {1} | ranking function} 3.1
Conditionalization and Marginalization on TPOs and OCFs Two operations on epistemic states that we will use in this paper are conditionalization and marginalization. Conditionalization restricts the set of worlds that are considered in an epistemic state. After the conditionalization with a formula A the resulting state only considers the elements of Mod ⌃ (A) as possible worlds.
A notion of conditionalization for TPOs where the models of A are shifted to
the uppermost layer has been introduced in [
Definition 1 (restricted OCF/TPO). A restricted ranking function over a set M ✓ ⌦ ⌃ is a function : M ! N0 [ {1} such that 1(0) 6= ; . Restricted ranking functions are extended to formulas by (A) = min!2 Mod(A)\ M (!) with min; (. . .) = 1 .
a restricted total preorder. Restricted total preorders on worlds are extended to formulas by A B if min(Mod ⌃ (A) \ M, ) min(Mod ⌃ (B) \ M, ).
The intuition of restricted OCFs and TPOs is the same as for usual OCFs and TPOs: Worlds with lower rank or position in the ordering are more plausible. For a signature ⌃ , a formula A 2 L⌃ and with MA = Mod ⌃ (A) we define MTPO (⌃, A ) = { ✓
MA ⇥
MA |
total preorder over MA}
MOCF (⌃, A ) = { : MA ! N0 [ {1} | ranking function}.
The restricted OCFs and TPOs properly include the original notation as
MI (⌃ ) = MI (⌃, >) for I 2 {TPO , OCF }. For state 2 MI (⌃, A ), we call
sig ( ) = ⌃ the signature of and dom( ) = Mod ⌃ (A) the domain of . Now
we can define conditionalization using restricted OCFs/TPOs. This definition of
conditionalization is based on the usual notion of conditionalization [
Definition 2 (conditionalization of restricted OCFs/TPOs). The conditionalization of ranking functions over Mod ⌃ (B) to the models of a formula A 2 L⌃ is a function MOCF (⌃, B ) ! M OCF (⌃, A ^ B), 7! |A such that |A(!) = (!) (A) for ! 2 Mod ⌃ (A ^ B).
formula A 2 L⌃ is a function MTPO (⌃, B ) ! M TPO (⌃, A ^ B), 7! | A such that !1 |A !2 i↵ !1 !2 for !1, !2 2 Mod ⌃ (A ^ B).
Marginalization on the other hand restricts the epistemic state to a
subsignature of the original signature. We extended the notion of marginalization
in [
Definition 3 (marginalization). The marginalization of OCFs from a
signature ⌃ to a sub-signature ⌃ 0 ✓ ⌃ is a function MOCF (⌃ ) ! M OCF (⌃ 0), 7!
|⌃ 0 such that |⌃ 0 (!) = (!) for ! 2 ⌦ ⌃ 0 [
is a function MTPO (⌃ ) ! M TPO (⌃ 0), 7! |⌃ 0 such that !1 |⌃ !2 i↵
!1 !2 for !1, !2 2 ⌦ ⌃ 0 [
Note that a world ! of a sub-signature ⌃ 0 ✓ ⌃ is considered as a formula when evaluated in the context of ⌃ , i.e., the rank of ! 2 ⌦ ⌃ 0 with respect to an OCF over ⌃ is the rank of the world !0 2 ⌦ ⌃ with the lowest rank that coincides with ! on the variables in ⌃ 0.
The marginalizations of OCFs and TPOs presented above are special cases
of general forgetful functors Mod (%) from ⌃ -models to ⌃ 0-models given in [
For MI (⌃, >) = MI (⌃ ), the marginalization/conditionalization of the restricted OCFs/TPOs coincides with the marginalization/conditionalization of OCFs/TPOs. 3.2
Syntax splitting was first introduced as property of belief sets in [
The notion of syntax splitting was extended to other representations of
epistemic states such as TPOs and OCFs in [
i = 1, . . . , n – and, for i = 1, . . . , n and !1, !2 2 dom( ), An and Ai 2 ⌃ i for !6=i = !6=i 1 2 implies !1 !2 i↵ !1i |⌃ i !2i .
A syntax splitting {⌃ 1, . . . ⌃ n} is denoted by ⌃ = ⌃ 1 [ ˙· · · [ ˙ ⌃ n.
Definition 5 (syntax splitting for restricted OCFs (adapted from [
splitting for if – there are formulas A1, . . . , An such that A ⌘ A1 ^ · · · ^ An and Ai 2 ⌃ i for i = 1, . . . , n – and there are ranking functions i 2 MOCF (⌃ i, Ai) for i = 1, . . . , n such that (!) = 1(!1) + · · · + n(!n) for ! 2 dom( ).
This is denoted as = 1
· · · n.
The definitions of syntax splitting for restricted TPOs/OCFs yield the original definition of syntax splitting for TPOs/OCFs by considering MI (⌃, >). 4
Postulates for Mappings on Epistemic States To formalize the transformations among models of epistemic states we introduce epistemic state mappings.
Definition 6. Let I1, I2 2 {TPO , OCF }. An epistemic state mapping from I1 to I2, denoted as ⇠ : I1 I2, is a function family ⇠ = (⇠ ⌃,A ) for signatures ⌃ and formulas A 2 L⌃ with ⇠ ⌃,A : MI1 (⌃, A ) ! M I2 (⌃, A ) such that A ⌘ B implies ⇠ ⌃,A = ⇠ ⌃,B .
Example 1. The family of functions ⇠ reverse that reverses every TPO defined by ⇠ ⌃,rAeverse ( ) = 0 with !1 0 !2 i↵ !2 !1 for a signature ⌃ , A 2 L⌃ , 2 MTPO (⌃, A ), and !1, !2 2 Mod ⌃ (A) is an epistemic state mapping from TPOs to TPOs.
Every epistemic state mapping represents a way to transform epistemic states of kind I1 to epistemic states of kind I2. Desirable properties of epistemic state mappings (⇠ ⌃,A ) can be stated in the form of postulates. Some of these postulates use the fact that both OCFs and TPOs induce a TPO on their domain. For a TPO , the induced ordering l is the order itself. For an OCF , the induced ordering l is given by !1 l !2 i↵ (!1) 6 (!2) for !1, !2 2 dom( ).
(IE) |= (C|D) i↵ ⇠ ⌃,A ( ) |= (C|D). (wIE)) |= (C|D) implies ⇠ ⌃,A ( ) |= (C|D). (wIE() ⇠ ⌃,A ( ) |= (C|D) implies |= (C|D).
Let !1, !2 in dom( ). (Ord) !1 l !2 i↵ (wOrd)) !1 l !2 (wOrd() !1 l⇠ ⌃,A ( ) !2
!1 l⇠ ⌃,A ( ) !2. implies !1 l⇠ ⌃,A ( ) !2.
implies !1 l !2. (SynSplit) If sig ( ) = ⌃ 1 [ ˙· · · [ ˙ ⌃ n is a syntax splitting for , then ⌃ 1 [ ˙ · · · [ ˙ ⌃ n is a syntax splitting for ⇠ ⌃,A ( ). (SynSplitb) If ⌃ = ⌃ 1 [ ˙ ⌃ 2 is a syntax splitting for , then ⌃ 1 [ ˙ ⌃ 2 is a syntax splitting for ⇠ ⌃,A ( ).
Let ⌃ 0 ✓ ⌃ with ⌃ 0 6= ; and A0 2 L⌃ 0 such that Mod ⌃ 0 (A0) = {!0 | ! 2
(Marg) ⇠ ⌃ 0,A0 ( |⌃ 0 ) = ⇠ ⌃,A ( )|⌃ 0 Let F 2 L⌃ with Mod ⌃ (F ) \ dom( ) 6= ; . (Cond) ⇠ ⌃,A ^ F ( |F ) = ⇠ ⌃,A ( )|F
The postulate (IE) requires inferential equivalence and states that the epistemic state mapping may not change the set of conditionals accepted by an epistemic state. The epistemic state and its mapping induce the same inference relation with respect to conditionals. This is a quite strong postulate, the postulates (wIE)) and (wIE() are weaker versions of (IE). Postulate (wIE)) states that an epistemic state mapping may not remove conditionals from the set of inferred conditionals. Postulate (wIE() states that after an epistemic state mapping, we may not accept additional conditionals.
The postulate (Ord) expresses the postulate (IE) in terms of the induced total preorders of the epistemic states. Analogously (wOrd)) and (wOrd() represent (wIE)) and (wIE(), respectively.
(SynSplit) states that an epistemic state mapping should preserve syntax splittings of the epistemic state. (SynSplitb) is a special case of (SynSplit) for syntax splittings in two sub-signatures.
The postulate (Marg) ensures the compatibility of an epistemic state mapping with marginalization. It states that changing the order in which marginalization and the epistemic state mapping are applied does not matter. This postulate is illustrated in Figure 1a. Similarly, the postulate (Cond) ensures the compatibility of an epistemic state mapping with conditionalization. (Cond) is illustrated in Figure 1b.
It is easy to see that (IE) is equivalent to the conjunction of (wIE)) and (wIE() and that (Ord) is equivalent to the conjunction of (wOrd)) and (wOrd(). Other relationships among the postulates, such as the following, are less obvious.
Proof. Let ⇠ : I1 I2 be a epistemic state mapping with I1, I2 2 {TPO , OCF }. Ad (2): “( ” Let (⇠ ⌃,A ) satisfy (wOrd)). Let 2 MI1 (⌃, A ) and = ⇠ ⌃,A ( ). If |= (D|C), then min(Mod ⌃ (CD), l ) l min(Mod ⌃ (CD), l ). MI (⌃, A )
⇠ MI (⌃, A ) ·|⌃ 0 ·|⌃ 0
MI (⌃ 0, A)
⇠ MI (⌃ 0, A)
⇠ MI (⌃, A ) MI (⌃, A ) ·|F ·|F
MI (⌃, AF )
⇠ MI (⌃, AF ) Illustration of (Marg).
Illustration of (Cond).
In this case, (wOrd)) implies min(Mod ⌃ (CD), l ) l min(Mod ⌃ (CD), l ). This is equivalent to |= (C|D). Therefore, (⇠ ⌃,A ) satisfies (wIE)).
“) ” Let (⇠ ⌃,A ) satisfy (wIE)). Let 2 MI1 (⌃, A ) and = ⇠ ⌃,A ( ). Let !1, !2 2 ⌦ with !1 l !2. Then, |= (!1|!1 _ !2). (wIE)) implies that |= (!1|!1 _ !2). Therefore, !1 l !2. We see that (⇠ ⌃,A ) satisfies (wOrd)). Ad (3): “( ” Let (⇠ ⌃,A ) satisfy (wOrd(). Let 2 MI1 (⌃, A ) and = ⇠ ⌃,A ( ). If |= (D|C), then min(Mod ⌃ (CD), l ) l min(Mod ⌃ (CD), l ). In this case, (wOrd() implies min(Mod ⌃ (CD), l ) l min(Mod ⌃ (CD), l ). This is equivalent to |= (D|C). Therefore, (⇠ ⌃,A ) satisfies (wIE().
“) ” Let (⇠ ⌃,A ) satisfy (wIE(). Let 2 MI1 (⌃, A ) and = ⇠ ⌃,A ( ). Let !1, !2 2 ⌦ with !1 l !2. Then, |= (!1|!1 _ !2). (wIE() implies that |= (!1|!1 _ !2). Therefore, !1 l !2. We see that (⇠ ⌃,A ) satisfies (wOrd(). Ad (1): This follows from (2) and (3) as (IE) is the conjunction of (wIE)) and (wIE() and (Ord) is the conjunction of (wOrd)) and (wOrd().
In the next sections, we will investigate the introduced postulates further for specific combinations of I1 and I2. 5
Epistemic State Mappings of Di↵erent Types In this section we investigate epistemic state mappings from TPOs to TPOs (subsection 5.1), OCFs to OCFs (sub-section 5.2), OCFs to TPOs (sub-section 5.3), and TPOs to OCFs (sub-section 5.4) in more detail. 5.1
Let us first consider epistemic state mappings from TPOs to TPOs. If we want (IE) or the equivalent (Ord) to hold, we do not have much choice.
From Proposition 2 it follows that (IE) or (Ord) imply (SynSplit), (Cond), and (Marg) for epistemic state mappings from TPOs to TPOs as the identity fulfils these postulates. 5.2
Mapping Ranking Functions to Ranking Functions Now consider the case where we map OCFs to OCFs. For such epistemic state mappings all postulates are compatible, in the sense that all postulates can be satisfied simultaneously by some epistemic state mapping.
some a 2 N+ fulfils (Ord), (SynSplit), (Cond), and (Marg).
OCF , 7! a · for
The epistemic state mappings in Proposition 3 are the only epistemic state mappings fulfilling all postulates.
and (Cond) have the form 7! a · for some a 2 N+.
Proof. Let ⇠ : OCF OCF be an epistemic state mapping. First, we consider OCFs with two worlds in their domain. Let !1, !2 be worlds and 1 : {!1, !2} ! N0, {!1 7! 0, !2 7! 1}. Let 01 = ⇠ ( 1). Let a = 01(!2) 01(!1). Because of (Ord), we have a > 0.
Let !3, !4 be worlds with {!1, !2} \ { !3, !4} = ; . We show that for 2 : {!3, !4} ! N0, {!3 7! 0, !4 7! b} with b 2 N+ and 02 = ⇠ ( 2) it holds that 02(!4) 02(!3) = a · b by induction over b.
Base Case: Let b = 1. Consider the OCF 3 : {!1, !2, !3, !4} ! N0, {!1, !3 7! 0, !2, !4 7! 1}. Let 03 = ⇠ ( 3). (Ord) requires that 03(!1) = 03(!3) and 03(!2) = 03(!4). (Cond) requires that 03(!2) 03(!1) = a. Therefore, 03(!4) 03(!3) = a = a · b. With (Cond) follows that 02(!4) 02(!3) = a · b. Induction Step: Let b = n+1. Let !5 2/ {!1, !2, !3, !4} be an additional world. Consider the OCF 4 : {!3, !4, !5} ! N0, {!3 7! 0, !5 7! n, !4 7! n + 1}. Let 04 = ⇠ ( 4). The induction hypothesis in combination with (Cond) requires that 04(!5) 04(!3) = a · n and 04(!4) 04(!5) = a. Hence, 04(!4) 04(!3) = a · (n + 1). With (Cond) follows that 02(!4) 02(!3) = a · b.
The statement proven by induction is extended to situations with {!1, !2} \ {!3, !4} 6= ; by comparing OCFs over {!1, !2} and {!3, !4} with OCFs over {!5, !6} for additional worlds !5, !6 as in the base case of the induction.
Now, consider any restricted ranking function . Let ! 2 dom( ) and !0 2 1(0) and 0 = ⇠ ( 5). Because of (Ord) and because the minimal rank of an OCF is 0, we have that 0(!0) = 0. The result above and (Cond) yields 0(!) = 0(!) 0(!0) = a · ( (!) (!0)) = a · (!). 5.3
Mapping Ranking Functions to Total Preorders In this sub-section, we want to investigate epistemic state mappings from OCFs to TPOs on worlds.
The mapping ⌧ ⇤ is surjective: For a given TPO it is easy to construct an OCF such that = ⌧ ⇤ ( ). But ⌧ ⇤ is not injective as there are more ranking functions than TPOs for any given (non-empty) signature. Furthermore, ⌧ ⇤ satisfies the other introduced postulates for epistemic state mappings.
Hence, (Ord) implies (SynSplit), (Marg), and (Cond) for epistemic state mappings from OCFs to TPOs. 5.4
Mapping Total Preorders to Ranking Functions Finally, we consider epistemic state mappings that map a TPO to an OCF. Since the functions in ⌧ ⇤ are not bijective, we cannot simply reverse them. On the contrary, there is more than one epistemic state mapping ⇢ : TPO OCF that fulfils (Ord). That is not surprising as an OCF contains more information than a TPO over the same domain. The additional information is the absolute distance between worlds. The functions in ⇢ need to fill in this missing information. Example 2. Let ⇢ : TPO OCF be an epistemic state mapping defined as follows. For 2 MTPO (⌃, A ) let L0 = min(dom( ), ) and Lk = min(dom( ) \ (L0 [ · · · [ Lk 1), ). Every set Lk corresponds to the k-th layer of . The sets Li and Lj are disjunct for i 6= j. We define ⇠ ( ) = with (!) = k such that ! 2 Lk for every ! 2 dom( ). For example, the TPO ab ab, ab ab over ⌃ = {a, b} is mapped to : {ab 7! 0, ab 7! 1, ab 7! 1, ab 7! 2} by ⇢ .
This epistemic state mapping ⇢ fulfils (Ord).
To limit the possible outcomes of the transformation, we consider additional postulates such as (SynSplit). Unfortunately, there is no epistemic state mapping ⇢ that fulfils both (Ord) and (SynSplit).
This incompatibility persists if we consider the weaker (SynSplitb) instead of (SynSplit) and (wIE)) instead of (IE).
fulfils both (wIE)) and (SynSplitb).
Proof. Let ⌃ = {a, b, c, d} be a signature and be the TPO over ⌃ displayed in Figure 3. This TPO has the syntax splitting {a, b} [ ˙{ c, d}. Assume there is an OCF with syntax splitting {a, b} [ ˙{ c, d} such that !1 !2 implies (!1) < (!2). Then there are OCFs 1 : ⌦ {a,b} ! N0 and 2 : ⌦ {c,d} ! N0 such that = 1 2. Let 1(ab) = 0, 1(ab) = i, 1(ab) = j, 1(ab) = k, 2(cd) = 0, 2(cd) = l, 2(cd) = m, and 2(cd) = n. As !1 !2 implies (!1) < (!2) for every !1, !2 2 ⌦ ⌃ we have that m + j = 1(ab) + 2(cd) = (abcd) < (abcd) = 1(ab) + 2(cd) = i + n. MOCF (⌃, A ) MTPO (⌃, A )
MTPO (⌃ 0, A)
MTPO (⌃, A ) ⌧ ⇤
⇢ MOCF (⌃ 0, A)
MOCF (⌃, A )
MOCF (⌃, AF ) ab cd ·|F ·|F
⌧ ⇤ MTPO (⌃, AF ) ab cd ab cd ab cd ab cd ab cd ab cd ab cd ab cd ab cd ab cd ab cd ab cd ab cd
ab cd ab cd
Analogously, we get j > n from (abcd) > (abcd) and m > i from (abcd) > (abcd). The combination of these inequations is a contradiction. The assumed ranking function cannot exist.
The combination of (wIE() and (SynSplit) is consistent, as we will see later.
Any epistemic state mapping ⇢ from TPOs to OCFs satisfying (Ord) is compatible with ⌧ ⇤ (Prop. 5) with respect to marginalization and conditionalization.
(Ord). For every TPO 2 MTPO (⌃, A ) and ⌃ 0 ✓ ⌃ and F 2 L⌃ it holds that ⌧ ⇤ (⇢ ( )|⌃ 0 ) = |⌃ 0 and ⌧ ⇤ (⇢ ( )|F ) = |F .
It would be useful if a transformation from a TPO to an OCF preserved marginalization and conditionalization in the way ⌧ ⇤ does for the other direction. But Postulate (Cond) is unfulfillable in combination with (Ord). (Cond) is even incompatible with the weaker Postulate (wIE)).
Proof. Let ⌃ = {a, b} be a signature and in Figure 6a. We have 1|a = 2|a and 1, 2 be the TPO over ⌃ displayed
1|b = 2|b. Let 1 = ⇢ ( 1) and a¯b ab a¯b a¯¯b a¯b ab a¯b
a¯¯b 1 2 1
2 a¯b ab a¯¯b a¯b a¯b ab a¯¯b a¯b TPOs 1 and 2 on ⌃ = {a, b} showing TPOs 1 and 2 on ⌃ = {a, b} showing that (Cond) is incompatible with (wIE) ) that (Marg) is incompatible with (wIE) ) for mappings from TPOs to OCFs in the for mappings from TPOs to OCFs in the proof of Proposition 10. proof of Proposition 11. 2 = ⇢ ( 2). If (wIE)) and (Cond) were true it would imply 1(ab) = 1|b(ab) = 2|b(ab) = 2(ab) and 1(ab) = 1|a(ab) = 2|a(ab) = 2(ab). This contradicts (wIE)) as (wIE)) requires 1(ab) > 1(ab) and 2(ab) < 2(ab).
(Marg) is also unfulfillable in combination with (Ord) or (wIE)) in general.
Proof. Let ⌃ = {a, b} be a signature and 1, 2 be the total preorders over ⌃ displayed in Figure 6b. Let ⌃ 1 = {a} and ⌃ 2 = {b}. We have 1|⌃ 1 = 2|⌃ 1 and 1|⌃ 2 = 2|⌃ 2 . Let 1 = ⇢ ( 1) and 2 = ⇢ ( 2). If (wIE)) and (Marg) were true it would imply 1(ab) = 1|⌃ 1 (ab) = 2|⌃ 1 (ab) = 2(ab) and 1(ab) = 1|⌃ 2 (ab) = 2|⌃ 2 (ab) = 2(ab). This contradicts (wIE)) as (wIE)) requires 1(ab) > 1(ab) and 2(ab) < 2(ab).
The Propositions 8, 10, and 11 all showed that (wIE)) in combination with some of the other postulates cannot be fulfilled. (wIE() on the other hand can be fulfilled in combination with these other postulates. But the following triviality result shows that there is only one epistemic state mapping fulfilling the combination of (wIE() and (Cond).
(wIE() and (Cond) maps every TPO to the uniform ranking function uni. Proof. Let ⇢ be an epistemic state mapping fulfilling (wIE() and (Marg). Let ⌃ be a signature and !1, !2 2 ⌦ ⌃ with !1 6= !2. Choose a third world !3 2 ⌦ ⌃ with !3 2/ {!1, !2} and consider the TPOs !3 1 !2 1 !1 1 !4, . . . , !n and !3 2 !1 2 !2 2 !4, . . . , !n with {!4, . . . , !n} = ⌦ ⌃ \ {!1, !2, !3}. Let 1 = ⇢ ( 1) and 2 = ⇢ ( 2). The postulate (wIE() requires that 1(!2) 6 1(!1) and 2(!1) 6 2(!2). (⇤ ) Let A = !3 _ !1 and B = !3 _ !2. Conditionalization yields 0A= 1|A = 2|A and 0B= 1|B = 2|B. Postulate (Cond) requires 1|A = ⇢ ( 0A) = 2|A and 1|B = ⇢ ( 0B) = 2|B. This implies 1(!1) = 2(!1) and 1(!2) = 2(!2). With (⇤ ) it follows that 1(!2) 6 1(!1) = 2(!1) 6 2(!2) = 2(!2). Therefore, we can replace the 6 in this chain of (in-)equations by =. Let C = !1 _ !2. We can see that both 1|C = {!1 !2} and 2|C = {!2 !1} are mapped to the uniform ranking function uni due to (Cond).
Since we can choose any two worlds as !1, !2 in this argumentation, (Cond) requires that any TPO is mapped to the uniform ranking function uni. This mapping to uni fulfils all three postulates.
Note that these results only apply to epistemic state mappings defined for all TPOs. Mappings that are defined over a certain subset of TPOs might still fulfil combinations of postulates. 6
Conclusion In this paper, we introduced the notion of epistemic state mappings, i.e., mappings within and across the frameworks of OCFs and TPOs. We proposed postulates for these mappings that ensure the preservation of certain properties of the epistemic state across the mapping. The properties considered include the set of entailed conditionals and syntax splitting. Other postulates ensure compatibility with the operations marginalization and conditionalization, respectively. Furthermore, we investigated the relationships among the proposed postulates in general and for each combination of the considered framework. Some postulates are entailed by other postulates, e.g., (SynSplit) entails (SynSplitb), (IE) is equivalent to (Ord). We also showed that there are combinations of the postulates which cannot be satisfied simultaneously, e.g., there is no epistemic state mapping from TPOs to OCFs that fulfils both (wIE)) and (SynSplitb). In some cases we identified all possible epistemic state mapping satisfying a certain combination of postulates, e.g., the only mapping from TPOs to OCFs that fulfils both (wIE)) and (SynSplitb) is the trivial mapping of every TPO to uni.
Our current work includes extending the investigation of epistemic state mappings and their properties for establishing further relationships between OCFs and TPOs. Furthermore, we will consider epistemic state mappings among particular subclasses of TPOs and OCFs. We expect to find interesting and relevant subclasses such that epistemic state mappings over these subclasses fulfil combinations of postulates that are not fulfilled by epistemic state mappings over the full sets of TPOs and OCFs.