The logic behind desirable sets of things, and its filter representation

We identify the (filter representation of the) logic behind the recent theory of coherent sets of desirable (sets of) things, which generalise coherent sets of desirable (sets of) gambles as well as coherent choice functions, and show that this identification allows us to establish various representation results for such coherent models in terms of simpler ones.


INTRODUCTION
A very important aspect of the theory of imprecise probabilities [1,16,25,27] is that it allows for partial specification of probability models (such as probability measures) and, equally importantly, that it enables us to do conservative inference. To give an example, if we specify bounds on the probabilities of a number of events, then the theory's concerned with, amongst other things, inferring the implied bounds on the probabilities of other events.
Such conservative probabilistic inference can be represented quite intuitively and effectively by considering simple desirability statements [21,28]: if some uncertain rewardsalso called gambles-are considered desirable to a subject, what does that imply about the desirability (or otherwise) of other gambles? That a subject considers a given gamble to be desirable is then considered as a simple statement, very much like her asserting a proposition in a propositional logic context. Inferring from a collection of such desirability statements which other gambles are desirable, is then effectively a matter of deductive inference based on a number of so-called coherence rules, very much like logical inference is based on the conjunction and modus ponens rules. This observation has led to a theory of coherent sets of desirable gambles [4,6,13,14,17,18,28]: sets of gambles that are deductively closed under the inference based on the coherence rules.
There's, however, a further complication to be dealt with. Indeed, a desirability statement for a gamble is tantamount to a pairwise comparison, and more specifically a strict preference, between this gamble and the zero gamble-the status quo. It was recognised quite early on in the development of the theory by Isaac Levi [16] that certain aspects of conservative probabilistic inference demand looking further than merely pairwise preferences between gambles. This has led to the introduction of choice functions, a tool from social choice theory [24], into the field of imprecise probabilities [15,19,22,26]. In a number of recent papers [7,9,10,12], we showed that working with the resulting so-called coherent choice functions is mathematically equivalent to doing inferences with desirable sets of gambles, rather than with desirable gambles, where a set of gambles is judged to be desirable as soon as at least one of its elements is: coherent choice functions can be seen as a special case of coherent-deductively closed-sets of desirable sets of gambles.
In very recent work [8], Jasper De Bock has taken the idea of moving from desirable gambles to desirable sets of gambles a significant step further, by recognising that it can be applied to any context involving conservative inference based on a closure operator. In his abstract generalisation step, gambles are replaced by abstract objects, called things, and it's assumed that some abstract property of things, called their desirability, can be inferred from the desirability of other things through inference rules that are summarised by the action of some closure operator. This leads to a theory of coherent-deductively closed-sets of desirable sets of things.
At about the same time, Catrin Campbell-Moore [3] showed that statements about, and the inference behind, the desirability of gambles, and to some extent also the desirability of sets of gambles, 1 can be represented by filters of probability measures. This showed that the conservative inference mechanism behind these desirability models can also be interpreted as that of propositional logic involving statements about some 'ideal unknown' probability measure.
In the present paper, we combine ideas from both these recent developments, by showing how the conservative inference mechanism behind coherent-deductively closeddesirable sets of things is related to that of propositional logic involving statements about some 'ideal unknown' coherent set of desirable things. We do this by showing that the collection of all coherent sets of desirable sets of things is an intersection structure that's order isomorphic to the set of all proper filters on a specific distributive lattice of events, where the events are appropriately chosen collections of coherent sets of desirable things.
Besides identifying the nature of the inference mechanism behind coherent sets of desirable sets of things (and in particular, coherent sets of desirable sets of gambles and therefore also coherent choice functions), our results also allow us to prove powerful, interesting as well as useful representation theorems for such coherent sets of desirable sets of things in terms of simpler, so-called conjunctive, models.
We'll freely use basic concepts and results from order theory, and we'll assume the reader to be familiar with most of them, so especially in the context of proofs, we'll limit ourselves to pointers to the literature. For a nice introduction, we refer to Davey and Priestley's book [5].
Our argumentation's structured as follows. In Section 2, we summarise the basic ideas behind coherent sets of desirable sets of things, and identify the order-theoretic underpinnings of the inference mechanism behind them. We distinguish between the standard coherence notion, which connects the desirability of arbitrary sets of things, and the less restrictive finite coherence notion, which essentially only focusses on the desirability of finite sets of things. We show that the coherent sets of desirable things can be embedded into the (finitely) coherent sets of desirable sets of things, in the form of the so-called conjunctive models. In Section 3, we explain how each desirability statement for a given (finite) set of things can be identified with a so-called (finitary) event, i.e., the specific subset of the collection of all coherent sets of desirable things it's compatible with. In Section 4, we identify the order-theoretic nature of the collection of all finitary events as a bounded distributive lattice, and for the collection of all events as a completely distributive complete lattice. Section 5 is a very short primer on order-theoretic and set-theoretic filters, and the prime filter representation theorem.
After setting up all this background material, we show in Section 6 that the collection of all finitely coherent sets of desirable sets of things is order-isomorphic to the collection of all proper filters of finitary events, and then explain how this order isomorphism leads in a straightforward manner to a representation of finitely coherent sets of desirable sets of things as limits inferior of conjunctive ones, providing a simple alternative proof to, and significantly extending similar representation results by Catrin Campbell-Moore [3]. Similarly, we show in Section 7 that the collection of all coherent sets of desirable sets of things is order-isomorphic to the collection of all proper principal filters of events, and then explain how this order isomorphism can be used straightforwardly to a prove a representation of coherent sets of desirable sets of things as intersections of conjunctive ones, leading to a simple alternative proof for similar results by Jasper De Bock [8]. We pay extra attention 1 More specifically, the desirability of sets of gambles then needs to satisfy a so-called mixingness requirement; see also the discussion in Section 10. to the finitely coherent models in Section 8, where we use the prime filter representation theorem to show that under some extra conditions, a so-called finitary subclass of them can also be represented as intersections of conjunctive ones, which generalises our earlier results on desirable gamble sets [9,10,12], and also provides a simple alternative proof for a similar result in Ref. [8].
Sections 9 and 10 are devoted to a discussion of two important special cases, where desirable things take the more concrete form of asserted propositions and desirable options-generalisations of gambles-respectively.

SETS OF DESIRABLE (SETS OF) THINGS: AN OVERVIEW
Let us begin by giving a brief overview of Jasper De Bock's theory of desirable things [8] that's sufficient for the purposes of this paper. We use somewhat different notations, and our overview differs in a few technical details, but the ideas and the conclusions we draw from them are the same.
2.1. Desirable things. We begin by considering a set T of things t that may or may not have a certain property. Having this property makes a thing desirable.
You, our subject, may entertain ideas about which things are desirable, and You represent these ideas by providing a (not necessarily exhaustive) set of things that You find desirable. We'll call such a subset S ⊆ T a set of desirable things, or SDT for short (plural: SDTs): a set with the property that You think each of its elements desirable. We denote by P(T) the set of all subsets S of T, or in other words, the collection of all candidate sets of desirable things.
Such sets of desirable things can be ordered by set inclusion. We interpret S 1 ⊆ S 2 to mean that S 1 is less informative, or more conservative, than S 2 , simply because a subject with a set of desirable things S 1 finds fewer things desirable than a subject with set of desirable things S 2 .
Our basic assumption's that, mathematically speaking, there are a number of rules that underlie the notion of desirability for things, and that the net effect of these rules can be captured by a closure operator and a set of forbidden things.
We recall that a closure operator on a non-empty set G is a map Cl : P(G) → P(G) satisfying: C1. A ⊆ Cl(A) for all A ⊆ G; C2. if A ⊆ B then Cl(A) ⊆ Cl(B) for all A, B ⊆ G; C3. Cl(Cl(A)) = Cl(A) for all A ⊆ G. Again, P(G) denotes the set of all subsets of the set G. A closure operator Cl is called finitary 2 if it's enough to know the closure of finite sets, in the following sense: Cl(A) = {Cl(F) : F ∈ P(G) and F ⋐ A} for all A ∈ P(G), where we use the notation '⋐' to mean 'is a finite subset of', and agree to call the empty set / 0 finite. First of all, as already suggested above, we assume that there's some inference mechanism that allows us to infer the desirability of a thing from the desirability of other things. This inference mechanism is represented by a closure operator Cl D : P(T) → P(T), in the following sense: D1. if all things in S are desirable, then so are all things in Cl D (S). We collect all sets of things that are closed under the inference mechanism in the set D, so D := {S ∈ P(T) : Cl D (S) = S}.
Secondly, we assume that there's a set of so-called forbidden things T − , which are never desirable: 2 Davey and Priestley [5,Definition 7.12] use the term 'algebraic', but 'finitary' seems to be the more common name for this concept, so we'll stick to that. D2. no thing in T − is desirable, so if all things in S are desirable, then S ∩ T − = / 0. Of course, because we assume that all things in a set of desirable things are desirable, it can never intersect the set T − , so this leaves us with the collection of the closed sets of desirable things that we'll call coherent. 3 We'll use the generic notation D for such coherent sets of desirable things. It's a standard result in order theory, and easy to check, that they constitute an intersection structure: the intersection of any non-empty family I = / 0 of them is still coherent: Proof of Equation (2). Let D := i∈I D i for ease of notation. Since, clearly, D ∩ T − = / 0, we need only prove that D ∈ D, or in other words, that Cl D (D) ⊆ D [use C1]. For any i ∈ I, we have that D ⊆ D i , and therefore also that Cl D (D) ⊆ Cl D (D i ) = D i , where the inclusion follows from C2, and the equality from the assumption that Clearly, a set of desirable things S ⊆ T can be extended to a coherent one if and only if Cl D (S) ∈ D, or equivalently, if Cl D (S)∩T − = / 0, and in that case we'll call this S consistent. For any consistent set of desirable things S, it's easy to see that so Cl D (S) is the smallest, or most conservative, or least informative, coherent set of desirable things that the consistent S can be extended to. In this sense, the closure operator Cl D represents conservative inference. ( For the converse inclusion, note that S ⊆ Cl D (S) by C1. But since also Cl D (S) ∈ D by the assumed consistency of S, we find that Cl D (S) ∈ {D ∈ D : S ⊆ D}, whence, indeed,

Proof of Equation
The following result is then a standard conclusion in order theory [5,Chapter 7].
Proposition 1. The partially ordered set D, ⊆ is a complete lattice with bottom 0 D = T + and top 1 D = T. For any non-empty family S i , i ∈ I of elements of D, we have for its infimum and its supremum that, respectively, inf i∈I S i = i∈I S i and sup i∈I S i = Cl D ( i∈I S i ).
Incidentally, the set T + := Cl D ( / 0) = D is the smallest closed set of desirable things.
Proof that T + is the smallest closed set of desirable things. To prove that Cl D ( / 0) ⊆ D, we consider that for any D ∈ D, where the inclusion follows from C2. Hence, indeed, Cl D ( / 0) ⊆ D. For the converse inclusion, simply observe that Cl D ( / 0) ∈ D, by C3.
If T + is coherent, or in other words if the empty set / 0 is consistent, then T + is the smallest, or most conservative, coherent set of desirable things. This will be the case if and only if D3. T + ∩ T − = / 0, or equivalently, D = / 0. We'll from now on also always assume that this 'sanitary' condition is verified. All things in T + are then always implicitly desirable, regardless of any of the desirability statements You might make. 3 It's of course perfectly possible that T − = / 0, and in that case coherence and closedness coincide: D = D.

2.2.
Desirable sets of things. When You claim that a set of things S ⊆ T is a set of desirable things, this is tantamount to a conjunctive statement: You state that "all things in S are desirable". In the formalism described above, there's no way to deal with disjunctive statements of the type "at least one of the things in S is desirable". Let's now look for a way to also allow for dealing with such disjunctive statements. We'll say that You consider a set of things S to be desirable if You consider a least one thing in S to be. In other words, in a set of desirable things, all things are desirable, whereas in a desirable set of things, at least one thing is. As with the desirability of things, You can make many desirability statements for sets of things, and we then collect all of these in a set of desirable sets of things-or for short SDS, plural SDSes-W ⊆ P(T). So W is a set of desirable sets of things for You if all sets of things S ∈ W are desirable to You, in the sense that each of them contains at least one desirable thing.
Sets of desirable sets of things can be ordered by set inclusion too. We take W 1 ⊆ W 2 to mean that W 1 is less informative, or more conservative, than W 2 , simply because a subject with a set of desirable sets of things W 1 finds fewer sets of things desirable than a subject with set of desirable sets of things W 2 .
The inference mechanism for the desirability of things also has its consequences for the desirability of sets of things, as we'll now make clear. Consider any set of sets of things W ⊆ P(T), then we denote by Φ W the set of all so-called selection maps Each such selection map σ ∈ Φ W selects a single thing σ(S) from each set of things S in W, and we use the notation for the corresponding set of all these selected things.
We now call a set of desirable sets of things K ⊆ P(T) coherent if it satisfies the following conditions: K1. / 0 / ∈ K; K2. if S 1 ∈ K and S 1 ⊆ S 2 then S 2 ∈ K, for all S 1 , S 2 ∈ P(T); K3. if S ∈ K then S \ T − ∈ K, for all S ∈ P(T); K4. {t + } ∈ K for all t + ∈ T + ; K5. if t σ ∈ Cl D (σ(W)) for all σ ∈ Φ W , then {t σ : σ ∈ Φ W } ∈ K, for all / 0 = W ⊆ K. The first condition K1 takes into account that the empty set of things can't be desirable, as it contains no desirable thing. The second condition K2 reminds us that if a set of things contains a desirable thing, then of course so do all its supersets. The third condition K3 reflects that things in T − can never be desirable, by D2, and can therefore safely be removed from any set of things without affecting the latter's desirability. And, to conclude, we'll see further on that the last two conditions K4 and K5 do a very fine job of lifting the effects of inferential closure from the desirability of things to the desirability of sets of things. They can be justified as follows. For K4, recall from the discussion above that any element of T + is always implicitly desirable, and so therefore will be any set that contains it. For K5, recall that / 0 = W ⊆ K means that each set of things S ∈ W contains at least one desirable thing, and therefore there must be some selection map σ o ∈ Φ W such that σ o (S) is a desirable thing for all S ∈ W. This implies that all things in σ o (W) are desirable, and therefore so are all things in Cl D (σ o (W)), by D1. Whatever t σ o we chose in Cl D (σ o (W)) will therefore be desirable, which guarantees that the set of things {t σ : σ ∈ Φ W } must also be desirable, because it contains the desirable thing t σ o .
Interestingly, Axioms K4 and K5 can be replaced by a single axiom, obtained from K5 by also allowing W ⊆ K to be empty: Proof that K4&K5 is equivalent to K45. It clearly suffices to show that K45 restricted to the case that W = / 0 is equivalent to K4. But there, Φ W contains only the empty map σ / 0 with σ / 0 (W) = / 0, and therefore Cl D (σ / 0 (W)) = Cl D ( / 0) = T + , so K45 now simply requires that for any t + ∈ T + , also {t + } ∈ K, which is, of course, exactly the purport of K4.
We denote the set of all coherent sets of desirable sets of things by K, and we let K := K ∪ {P(T)}. Observe that P(T) is never coherent, by K1. Since each of the axioms K1-K5 is preserved under taking arbitrary non-empty intersections, the set K of all coherent sets of desirable sets of things constitutes an intersection structure: the intersection of any non-empty family of coherent sets of desirable sets of things is still coherent, or in other and more formal words, for any non-empty family K i , i ∈ I of elements of K, we see that still i∈I K i ∈ K. As explained in Ref. [5,Chapter 7], this allows us to capture the inferential aspects of desirability at this level using the closure operator Cl K : P(P(T)) → K associated with the collection K of closed sets of desirable sets of things, defined by If we call a set of desirable sets of things W consistent if it can be extended to some coherent set of desirable sets of things, or equivalently, if Cl K (W) = P(T), then we find that Cl K (W) is the smallest, or most conservative, coherent set of desirable sets of things that includes W, for any consistent W. Of course, and therefore also K = Cl K (P(P(T))). The following result is then again a standard conclusion in order theory [5,Chapter 7]. Proposition 2. The partially ordered set K, ⊆ is a complete lattice with top 1 K = P(T) and bottom 0 K = Cl K ( / 0) = K = K. For any non-empty family W i , i ∈ I of elements of K, we have for its infimum and its supremum that, respectively, inf i∈I W i = i∈I W i and sup i∈I W i = Cl K ( i∈I W i ).
Interestingly, the smallest coherent set of desirable sets of things 0 K is easy to identify.
Proof of Proposition 3. If we let, for ease of notation, K + := {S ∈ P(T) : S ∩ T + = / 0}, then it follows from K4 and K2 that it clearly suffices to prove that K + is a coherent set of desirable sets of things.
It's a matter of immediate verification that it trivially satisfies K1, K2 and K4. For K3, consider that for any S ∈ P(T), (S \ T − ) ∩ T + = S ∩ (T + \ T − ) = S ∩ T + , where the last equality follows from the fact that T + ∩ T − = / 0 [use D3]. For K5, first observe that if T + = / 0, and therefore also K + = / 0, then the condition is satisfied vacuously. Otherwise, consider any non-empty W ⊆ K + and observe that there always is some σ + ∈ Φ W such that σ + (S) ∈ S ∩ T + for all S ∈ W. Then σ + (W) ⊆ T + , and therefore also Cl D (σ + (W)) = T + [to see why this equality holds, observe that / 0 ⊆ σ + (W) ⊆ T + implies Cl D ( / 0) ⊆ Cl D (σ + (W)) ⊆ Cl D (T + ) by C2, and recall that T + = Cl D ( / 0) by definition, and that then Cl If we now choose any t σ ∈ Cl D (σ(W)) for all σ ∈ Φ W , then we must prove that S ∈ K + for the corresponding S := {t σ : σ ∈ Φ W }, or in other words, that S ∩ T + = / 0. But this is obvious, since on the one hand t σ + ∈ S, and on the other hand t σ + ∈ Cl D (σ + (W)) = T + .
2.3. Desirable sets of things: the finitary case. We call a subset K of P(T) a finitely coherent set of desirable sets of things if it satisfies conditions K1-K4, together with the following finitary version of K5: K5fin. if t σ ∈ Cl D (σ(W)) for all σ ∈ Φ W , then {t σ : σ ∈ Φ W } ∈ K, for all / 0 = W ⋐ K; We can replace K4 and K5fin by a single axiom, which is the finitary counterpart of K45: K45fin. if t σ ∈ Cl D (σ(W)) for all σ ∈ Φ W , then {t σ : σ ∈ Φ W } ∈ K, for all W ⋐ K.
We denote by K fin the set of all finitely coherent sets of desirable sets of things, and we let K fin := K fin ∪ {P(T)}.
For this finitary version, the discussion, definitions and the ensuing results about the intersection structure K fin , the complete lattice K fin , ⊆ , and the associated closure operator Cl K fin (counterparts to Propositions 2 and 3) are completely similar, and we'll refrain from repeating them here. Observe nevertheless that K ⊆ K fin and therefore also K ⊆ K fin : since K5 clearly implies K5fin, any coherent K is also finitely coherent, so finite coherence is the weaker requirement. As a consequence, we also find that Cl K fin (W) ⊆ Cl K (W) for all W ⊆ P(T).

2.4.
Conjunctive models. Let's now show that it's possible to order embed the structure D, ⊆ into the structure K, ⊆ , and therefore also into the structure K fin , ⊆ , in a straightforward and very natural manner.
If we consider any set of things S that's an element of the coherent set of desirable sets of things K, then we know from the coherence condition K2 that all its supersets are also in K. But, of course, not all of its subsets will be, as is made clear by the coherence condition K1.
This observation brings us to the following idea. Consider any set of desirable sets of things W-not necessarily coherent-and any element S ∈ W. If there's some finite subsetŜ of S such thatŜ ∈ W, then we'll call S finitary (in W). If, moreover, all the elements S of the set of desirable sets of things W are finitary, then we'll call W finitary as well; so any desirable set in a finitary W has a desirable finite subset. The (finitely) coherent finitary sets of desirable sets of things will be studied in much more detail in Section 8. They are special because they are completely determined by their finite elements.
For the present discussion, however, we restrict our attention to an important special case of such finitary sets of desirable sets of things, where each desirable set has a desirable singleton subset: Definition 1 (Conjunctivity). We call a set of desirable sets of things W ⊆ P(T) conjunct- In the remainder of this section, we'll spend some effort on identifying the conjunctive coherent sets of desirable sets of things.
We begin by introducing ways to turn a set of desirable things into a set of desirable sets of things, and vice versa. Consider, to this end, any S o ⊆ T and any W o ⊆ P(T), and let The following conclusion is immediate.
Proposition 4. Consider any S, S 1 , S 2 ∈ P(T) and any W 1 ,W 2 ⊆ P(T). Then K S is conjunctive, and D K S = S. Moreover, Let's investigate in the following two propositions some conditions under which D W o is a coherent set of desirable things, and K S o is a (finitely) coherent set of desirable sets of things.
Proposition 5. Consider any set of desirable sets of things K. If K is (finitely) coherent, then K D K ⊆ K. Moreover, the following statements hold: (i) if K is coherent, then D K is coherent; 4 In earlier papers [9,10,12] we used the term 'binary' instead of 'conjunctive', because the corresponding preference models are binary relations. But it seems a bit counter-intuitive to call sets based on singletons 'binary'. We now use the term 'conjunctive' because, as we'll see further on in Proposition 7 and Corollary 8, these models are essentially representations of a conjunction of desirability statements for things.
(ii) if K is finitely coherent and the closure operator Cl D is finitary, then D K is coherent.
Proof of Proposition 5. Before we really begin, we can already observe that if K is (finitely) coherent, it follows from K4 that T + ⊆ D K . For the proof of the first statement, consider any S in K D K . Then S ∩ D K = / 0, or, in other words, t o ∈ D K for some t o in S. But then {t o } ∈ K, and therefore K2 guarantees that also To prove the coherence of D K , it is [by C1] enough to show that Cl D (D K ) ⊆ D K and that D K ∩ T − = / 0. To show that D K ∩ T − = / 0, we only need to rely on K satisfying the conditions K1-K3, and therefore not on the finitary or infinitary aspect of the coherence of K, nor on whether the closure operator Cl D is finitary or not. Indeed, assume towards contradiction that T − ∩ D K = / 0, so there's some t − ∈ T − ∩ D K . This implies that {t − } ∈ K, and therefore, (i). In this case, the proof is fairly similar, but let's spell it out anyway, in order to verify explicitly that we don't need the finitary character of the closure operator Cl D here. Again, consider any t o in Cl D (D K ). Since we already know that T + ⊆ D K , we may assume without loss of generality that t o / ∈ T + . But this implies that D K is must be non-empty. If we now consider the special choice t σ o := t o ∈ Cl D (σ o (W)), then K5 guarantees that Proof of Proposition 6. First off, consider the following chain of equivalences, valid for any t ∈ T: which implies that, generally speaking, D K D = D.
(i)⇒(ii). Assume that D is coherent, then we must show that K1-K5 are satisfied for K D . We concentrate on the proofs for K3-K5, since the proofs for K1 and K2 are trivial.
K3. Assume that S ∈ K D , meaning that S ∩ D = / 0. Since D ∩ T − = / 0 by Equation (1), this implies that K4. Since the coherence of D implies that T + ⊆ D, it follows trivially that, indeed, K5. Consider any non-empty W ⊆ K D , meaning that S ∩ D = / 0 for all S ∈ W. This implies that we can always find some σ o ∈ Φ W for which σ o (S) ∈ D for all S ∈ W, and therefore σ o (W) ⊆ D, so we also find that Cl D (σ o (W)) ⊆ D. This guarantees that always t σ o ∈ D, and therefore that, indeed, {t σ : σ ∈ Φ W } ∈ K D .
(ii)⇒(i). Assume that K D is coherent, then Proposition 5(i) guarantees that the set of desirable things D K D is coherent. Since we proved above that D K D = D, this implies that D is coherent.
(iii)⇒(i). Assume that K D is finitely coherent and that the closure operator Cl D is finitary, then Proposition 5(ii) guarantees that the set of desirable things D K D is coherent. Since we proved above that D K D = D, this implies that D is coherent.
So, if we start out with a coherent set of desirable sets of things K, then the coherent set of desirable things D K and the corresponding coherent and conjunctive set of desirable sets of things K D K are conservative approximations of K: going from a model K to its conjunctive part K D K = {S ∈ K : (∃t ∈ S){t} ∈ K} typically results in a loss of information. Similar results hold for finitely coherent sets of desirable sets of things K, provided that the closure operator Cl D is finitary, as then its conjunctive part K D K will be coherent too.
Interestingly, going from a coherent set of desirable things D to the corresponding coherent and conjunctive set of desirable sets of things K D does not result in a loss of information, as we'll see further on in Corollary 8. We're now finally in a position to find out what the conjunctive and (finitely) coherent sets of desirable sets of things look like. Proof of Proposition 7. The 'if' statements follows from Proposition 6, so we turn to the proof of the 'only if' statements. First off, consider the following chain of equivalences, valid for any (finitely) coherent set of desirable sets of things K: where the last equivalence follows from Proposition 5, which tells us that K D K ⊆ K, because K was assumed to be (finitely) coherent. It remains to argue that the set of desirable things D K is coherent. In the case of (i) this follows immediately from Proposition 5(i); and in the case of (ii) this follows immediately from Proposition 5(ii).
where the last equality follows from Proposition 6.
Combining the results of Propositions 4(ii), 6 and 7 leads to a formal statement of what we alluded to in the introduction to this paragraph.
Corollary 8. The map K • is an order embedding of the (intersection) structure D, ⊆ into the (intersection) structure K, ⊆ . Its image is the set of all conjunctive coherent sets of desirable sets of things, and on this set its inverse is the map D • . Similar results hold for finite coherence, provided that the closure operator Cl D is finitary.

TOWARDS A REPRESENTATION WITH FILTERS
It's a well-established consequence of Stone's Representation Theorem 5 [5, Chapters 5, 10 and 11] that filters of subsets of a space constitute abstract ways of dealing with deductively closed sets of propositions about elements of that space, or in other words and very simply put, they allow us to do propositional logic with statements about elements of the space.
Recall that a filter of subsets of a space X -also called a filter on P(X ), ⊆ -is a non-empty subset F of the power set P(X ) of X such that: We call a filter proper if F = P(X ), or equivalently, if / 0 / ∈ F. The particular space we'll be considering in this paper, is the set D of all coherent sets of desirable things. To guide the interpretation of what we're doing, we'll assume that there's an actual (but unknown) set of desirable things D T . This set is assumed to be coherent, and therefore a specific element of the set D. The elements of D T are the things that actually are desirable, and all other things in T aren't. Moreover, each coherent set of desirable things D ∈ D is a possible identification of this actual set D T .
Any non-contradictory propositional statement about D T corresponds to some nonempty subset A ⊆ D of coherent sets of desirable things for which the statement holds true, and this subset A represents the remaining possible identifications of D T after the statement has been made. We'll call such subsets events. The empty subset of D represents contradictory propositional statements.
Any proper filter F of such events A ⊆ D then corresponds to a deductively closed collection of propositional statements-a so-called theory-about D T , where intersection of events represents the conjunction of propositional statements, and inclusion of events represents implication of propositional statements. The only improper filter P(D), which contains the empty event, then represents logical contradiction at this level.
We can interpret the desirability statements studied in Section 2 as statements about such an actual D T . Stating that a 'set of things S is desirable' corresponds to the event as this amounts to requiring that at least one element of S must be actually desirable, and must therefore belong to D T . In other words, the desirability statement is equivalent to 'D T ∈ D S '. As a special case, stating that a thing t is desirable corresponds to the event D {t} := {D ∈ D : t ∈ D}, as it amounts to requiring that t must belong to D T . Also observe that D / 0 = / 0, the so-called impossible event. More generally, working with sets of desirable sets of things W as we did in Section 2, therefore corresponds to dealing with a conjunction of the desirability statements 'the set of things S is desirable' for all S ∈ W, or in other words with events of the type where, of course, as a special case we find that W = / 0 corresponds to a vacuous assessment, which leads to no restrictions on D T : E ( / 0) = D. 6 Working with the filters of subsets of D-filters of events-that are generated by such collections, then represents doing propositional logic with basic statements of the type 'the set of things S is desirable', for S ∈ P(T). We might therefore suspect that the language of such filters could be able to represent, explain, and perhaps also refine the relationships between the inference mechanisms that lie behind the intersection structures and closure operators in Section 2. Investigating this type of representation in terms of filters of events is the main aim of this paper.
Two complications. There are, however, two particular aspects of the inference mechanisms at hand that tend to complicate-or is it simplify?-matters somewhat.
The first is that not all events in P(D) are relevant to our problem; only the ones that are intersections (and, as we'll see further on, unions) of the basic events of the type D S , S ⊆ T seem to require attention. We'll therefore restrict our focus to these, and as a result, the representing collection of events will no longer constitute a Boolean lattice, but only a specific distributive sublattice. As we'll see in Sections 6 and 7, the effect will be two-fold: we'll broadly speaking be led to a more general prime filter rather than an ultrafilter representation, and this representation will be an isomorphism rather than an endomorphism.
The second complication lies in the ordering of events. Observe first of all that for any non-empty events A, B ∈ P(D) with A ⊆ B, the set A of coherent sets of desirable things can be seen as less conservative, or more informative, than the set B, as A contains fewer possible identifications of D T than B.
But, as we found out in Section 2, the coherent sets of desirable things-the possible identifications for D T -can themselves also be provided with the ordering of set inclusion ⊆. If we have two possible identifications D 1 , D 2 for D T , then D 1 ⊆ D 2 can also be taken to mean that the identification D 2 is less conservative, or more informative, than the identification D 1 , as it contains more desirable things.
Combining these two notions of 'less conservative' leads us to a new type of ordering '⊑' of events, that turns out to be a coarsening of the set inclusion '⊆'. Indeed, for any events A, B ⊆ D we then have that For non-empty A and B, this means that for every possible identification D 1 for D T in the set A, there's some possible identification D 2 for D T in the set B that's less informative, as it contains fewer desirable things, so we can see A as less conservative, or more informative, than B. It turns out that this new ordering of events '⊑' has a very simple characterisation, as we now proceed to show. In order to do this, we need to introduce some new notation, which we'll have occasion to use later in other contexts as well. In any partially ordered set L, ≤ , we let, for any a ∈ L and any A ⊆ L, Further on, we'll also make use of the fact that, with obvious notations, In the special case that L, ≤ = D, ⊆ , we get, for any D ∈ D and any A ⊆ D, Proposition 9. Consider any A, B ⊆ D, then the following statements are equivalent: Proof. It's clear from Equation (7) that (i) and (ii) are equivalent. Since A ⊆ Up D (A), it therefore only remains to prove that (ii) implies (iii). Consider any D ∈ Up D (A), then there's some D ′ ∈ A such that D ′ ⊆ D. If we assume that A ⊆ Up D (B), then this implies that there's some D ′′ ∈ B such that D ′′ ⊆ D ′ and therefore also D ′′ ⊆ D. This implies that By combining these two observations, we're led to the expectation that events of the type Up D (E (W)) for W ⊆ P(T), ordered by set inclusion, will play an important part in our filter representation efforts.

THE BASIC REPRESENTATION LATTICES
Since we expect the events Up D (E (W)) to become important in what follows, let's study them a bit closer.
Basic properties. The following propositions are easy to prove, but they will be instrumental in our discussion further on, so we've gathered them here for easy reference. They link the events D S , E (W) and Up D (E (W)) to our discussion of desirability in Section 2, and summarise their relevant properties. We start with a definition of a type of event that's intricately linked with the production axiom K5 there: The following proposition relates such events to the events E (W) and Up D (E (W)) we came across in Section 3.

Proposition 10. Consider any W
Proof. For a start, consider any D ∈ D, and verify the equivalences in the following chain: where the penultimate equivalence follows from C1-C3 and the assumption that D ∈ D. The proof of the last equivalence goes as follows. The converse implication is immediate, looking at the definition of D(W) in Equation (9). For the direct implication, assume that these is some σ ∈ Φ W such that Cl D (σ(W)) ⊆ D. Since D ∈ D, we see that then necessarily also Cl D (σ(W)) ∈ D, and therefore, by Equation (9), that Cl D (σ(W)) ∈ D(W).
This proves that (i) holds; and (ii) then follows readily from (i) and the already established fact that Up D (Up D (A )) = Up D (A ) for any A ⊆ D.
The next two propositions focus on relevant properties of events of the type D S . Proposition 11. Consider any S, S 1 , S 2 ∈ P(T). Then the following statements hold: Proof. The proofs of (i) and (iii) are trivial, looking at Equation (5). For (ii), consider any D ∈ D, and the following sequence of equivalences: Conversely, assume that D S = D, then since T + = D ∈ D, we find that in particular T + ∈ D S , so indeed T + ∩ S = / 0.

Proposition 12.
For all S ∈ P(T) and D ∈ D, we have that Proof. Simply, consider the following chain of equivalences, valid for any D ∈ D: where the last equivalence follows from C1-C3 and the restriction that D ∈ D.
It turns out that the propositional statement that W is a set of desirable sets of things is never contradictory as soon as W is an element of some coherent set of desirable sets of things. A similar result holds for finitely coherent sets of desirable sets of things, of course, at least for finite W.
Proposition 13 (Consistency). For any coherent set of desirable sets of things K ∈ K, it holds that E (W) = / 0 for all W ⊆ K. Similarly, for any finitely coherent set of desirable sets of things K ∈ K fin , it holds that Proof. We give a proof for the first statement involving coherence. The proof for the second statement involving finite coherence is completely analogous. Assume, towards contradiction, that there's some W ⊆ K such that E (W) = / 0, then Proposition 10 tells us that D(W) = / 0, so we get from Equation (9) The following result is formulated for finitely coherent sets of desirable sets of things, but the infinitary version clearly holds mutatis mutandis-not restricted to finite sets of desirable sets of things-for their coherent counterparts as well. It's on these two simple properties that all our results about filter representation will essentially rest.

Proposition 14.
Consider any finitely coherent set of desirable sets of things K ∈ K fin , then the following statements hold for all W 1 ,W 2 ⋐ P(T): Proof. We only give a proof for the first statement, as the second statement is a trivial consequence of the first. Since always / 0 ⋐ K, it's clear that we may assume without loss of generality that W 2 = / 0. It follows from Proposition 13 that we may also assume without loss of generality that E (W 1 ) = / 0. Using Proposition 10, we can then infer from Now, Φ W 1 can be partitioned into two disjoint sets We infer from Equation (10) that Φ ′ W 1 = / 0 and that Cl D (σ(W 1 )) ∩ S = / 0 for all S ∈ W 2 and all σ ∈ Φ ′ W 1 , and it follows from the definition of Φ ′′ If we now fix any S ∈ W 2 , then this tells us that we can always choose a t σ ∈ Cl D (σ(W 1 )) such that also t σ ∈ S, for all σ ∈ Φ ′ W 1 . Similarly and at the same time, we can always choose a t σ ∈ Cl D (σ(W 1 )) such that also t σ ∈ T − , for all σ ∈ Φ ′′ W 1 . Now let S ′ := {t σ : σ ∈ Φ ′ W 1 } ⊆ S and S ′′ := {t σ : σ ∈ Φ ′′ W 1 } ⊆ T − , then, since we assumed that W 1 ⋐ K, we infer from K5fin that S ′ ∪ S ′′ = {t σ : σ ∈ Φ W 1 } ∈ K, and from K3 that then also S ′ \ T − ∈ K. But since, by construction, S ′ \ T − ⊆ S ′ ⊆ S, we infer from K2 that also S ∈ K. Hence, indeed, W 2 ⋐ K.
The appropriate event lattices. We're now ready to introduce the particular sets of events we'll build our further discussion on. Let's consider the sets and order them by set inclusion ⊆. Interestingly, ordering the events E (W) by ⊆ is equivalent to ordering them by ⊑, as E (W) = Up D (E (W)); see Propositions 9 and 10(ii).
Proposition 15. The partially ordered set E, ⊆ is a completely distributive complete lattice, with union as join and intersection as meet, / 0 as bottom and D as top. Similarly, E fin , ⊆ is a bounded distributive lattice, with union as join and intersection as meet, / 0 as bottom and D as top.
Proof. Consider any non-empty family of where we let W := i∈I W i . That i∈I E (W i ) still belongs to E allows us to conclude, via a standard result in order theory [5,Corollary 2.29], that intersection is indeed the infimum in the poset E, ⊆ . Moreover, taking into account the complete distributivity of unions over intersections, we also find that where Ψ is the set of all choice maps ψ : That i∈I E (W i ) still belongs to E allows us to conclude, again via the same standard result in order theory [5,Corollary 2.29], that union is indeed the supremum in the poset E, ⊆ . This structure is therefore a complete lattice (of sets), since the non-empty index set I was arbitrary in our argumentation. It follows at once that this complete lattice is completely distributive, because any complete lattice of sets is [5,Theorem 10.29]. Finally, we infer from Equation (6) that E ({ / 0}) = D / 0 = / 0, so / 0 ∈ E and therefore / 0 is the bottom of this structure. Similarly, we can infer from Equation (6) that E ( / 0) = D, so D ∈ E and therefore D is the top of this structure.
The proof for the second statement uses the same ideas, but in a simpler, finitary guise, as the index set I and the subsets W i in the argumentation above are now kept finite, and only finite distributivity is required.

A BRIEF PRIMER ON (INFERENCE WITH) FILTERS
The introductory discussion in Section 3 led us to try and represent inference about desirability statements using filters on appropriate lattices of events. After spending some effort on identifying these lattices in Section 4, we're now ready to start looking at how to do inference with filters, and how to use that inference mechanism to represent reasoning about desirability statements. The present section summarises those aspects of filters and filter inference on (bounded distributive) lattices that are relevant to our representation effort.
We begin by recalling the definition of a filter on a bounded lattice L, ≤ with meet ⌢ and join ⌣. It's an immediate generalisation of the definition of a filter of subsets we gave near the beginning of Section 3.
Definition 2 (Filters). A non-empty subset F of the set L is called a filter on L, ≤ if it satisfies the following properties: We call a filter F proper if F = L. We denote the set of all proper filters of L, ≤ by F(L), and the set of all filters by F(L) = F(L) ∪ {L}.
The inference mechanism that's associated with filters is, as are all such mechanisms, based on the idea of closure and intersection structures, which we already brought to the fore in Section 2. Here too, it's easy to see that the set F(L) of all proper filters on a bounded 7 lattice L, ≤ is indeed an intersection structure, meaning that it's closed under arbitrary non-empty intersections: for any non-empty family Again, if we associate with this intersection structure the map Cl F(L) : then this map is a closure operator.
In this language, the filters are the perfect, or deductively closed, subsets of the bounded lattice L, ≤ , and the closure operator can be used to extend any set of lattice elements to the smallest deductively closed set that includes it. If we call a set H filterisable if it's included in some proper filter, or equivalently, if Cl F(L) (H) = P(L), then we find that Cl F(L) (H) is the smallest proper filter that includes H, for any filterisable set H. Of course, and therefore also F(L) = Cl F(L) (P(L)). The following result is then a standard conclusion in order theory [5,Chapter 7]. . For any non-empty family F i , i ∈ I of elements of F(L), we have for its infimum and its supremum that, respectively, inf i∈I F i = i∈I F i and sup i∈I F i = Cl F(L) ( i∈I F i ).
Two special types of filters deserve more attention in the light of what's to come.
Principal filters. In the special case that L, ≤ is a complete lattice, we can replace the finite meets in LF2 by arbitrary, possibly infinite ones, as in LP2p. if A ⊆ F then also inf A ∈ F, for all non-empty A ⊆ L.
In particular, we then find that inf F ∈ F, and it's not hard to show that then F = up L (inf F). Such a so-called principal filter is clearly proper if and only if inf F = 0 L . We see that the set of all principal filters, partially ordered by set inclusion, is trivially order-isomorphic to the complete lattice L, ≤ itself.
Prime filters on distributive lattices. A prime filter G on L, ≤ is a proper filter that also satisfies the following condition: We denote the set of all prime filters on L, ≤ by F p (L). When the lattice L, ≤ is distributive, then any proper filter can be represented by prime filters, as it's the intersection of all the prime filters that include it; see Ref. [5, for more details. Taking stock. Now that we know what the inference mechanism underlying filters is, we can make clearer what we mean by filter representation of other inference mechanisms. Axioms K1-K5 govern the inference mechanism behind the desirability of sets of things, and we've seen in Section 2.2, and in particular in Proposition 2, that its mathematical essence can be condensed into the complete lattice K, ⊆ and the closure operator Cl K . Similarly, the finitary version of this inference mechanism is laid down in Axioms K1-K5fin, and is captured by the complete lattice K fin , ⊆ and the closure operator Cl K fin .
The question raised in Section 3 is then, in its purest form: can we find bounded lattices L, ≤ such that the complete lattice F(L), ⊆ and the closure operator Cl F(L) are essentially the same as-can be identified through an order isomorphism with-the complete lattice K, ⊆ and the closure operator Cl K ; or in the finitary case, the same as the complete lattice K fin , ⊆ and the closure operator Cl K fin ? We'll show in Sections 6 and 7 that, indeed, we can find such bounded lattices: the completely distributive complete lattice of events E, ⊆ and the bounded distributive lattice of events E fin , ⊆ , respectively.
Why bother? What's so special about such representations in terms of filters of events? The answer's twofold.
First of all, there's the issue of interpretation we've already drawn attention to in Section 3. The events in E and E fin represent propositional statements about the desirability of things, and filters of such events represent collections of such propositional statements that are closed under logical deduction-conjunction and modus ponens. The order isomorphisms that we'll identify below then simply tell us that making inferences about desirable things and desirable sets of things based on the Axioms D1-D3 and K1-K5/K5fin is mathematically equivalent to doing propositional logic with propositional statements about the desirability of things.
The second reason has a more mathematical flavour. Since the sets of events E and E fin constitute bounded distributive lattices when ordered by set inclusion, we can make use of the Prime Filter Representation Theorem on such bounded distributive lattices, which states that any filter can be written as the intersection of all the prime filters it's included in. The order isomorphisms we're about to identify in the following sections will then allow us to transport this theorem to the context of (finitely) coherent sets of desirable sets of things, and write these as intersections of special types of them, namely the conjunctive ones. This will lead us directly to the so-called conjunctive representation results for coherent SDSes in Theorem 26 and for finitely coherent SDSes in Theorem 30 below.

FILTER REPRESENTATION FOR FINITELY COHERENT SDSES
We're now first going to consider finitely coherent sets of desirable sets of things, and try to relate them to the filters on the distributive lattice E fin , ⊆ . This will lead to a socalled conjunctive representation result of finitely coherent SDSes in terms of conjunctive ones. We'll then see in the next section that coherent SDSes also have a conjunctive representation result, that turns out to be formally simpler.
We adapt the notations and definitions from Section 5 from a generic bounded distributive lattice L, ≤ to the specific bounded distributive lattice E fin , ⊆ . This leads to the complete lattice F(E fin ), ⊆ of all filters on E fin , with bottom 0 F(E fin ) = F(E fin ) = {D} and top 1 F(E fin ) = E fin ; see also Propositions 16 and 15. We also denote the set of all proper filters by F(E fin ) := F(E fin ) \ {E fin }, and the corresponding closure operator by Cl F(E fin ) .
6.1. The order isomorphism. In order to establish the existence of an order isomorphism between the complete lattices K fin , ⊆ and F(E fin ), ⊆ , we consider the maps Proof. For statement (i), assume that K is a finitely coherent set of desirable sets of things. We must show that ϕ fin D (K) is a proper filter. To see that ϕ fin D (K) is non-empty, observe that always / 0 ⋐ K, and therefore . Then Proposition 14(ii) implies that W 1 ⋐ K, and Proposition 14(i) then guarantees that also W 2 ⋐ K, whence, indeed, also E (W 2 ) ∈ ϕ fin D (K). To show that ϕ fin D (K) is closed under finite intersections [satisfies LF2], consider arbitrary W 1 ,W 2 ⋐ K. Then we infer from Equation (11) that E (W 1 ) ∩ E (W 2 ) = E (W 1 ∪W 2 ), and since W 1 ∪ W 2 ⋐ K, this tells us that, indeed, E (W 1 ) ∩ E (W 2 ) ∈ ϕ fin D (K). To show that the filter ϕ fin D (K) is proper, simply observe that it follows from Proposition 13 that E (W) = / 0 for all W ⋐ K, which ensures that, indeed, / 0 / ∈ ϕ fin D (K). For statement (ii), assume that F is a proper filter. We check that the relevant conditions are satisfied for K to be a finitely coherent set of desirable sets of things.
K1. Since D / 0 = / 0 and F is proper, we find that D / 0 / ∈ F. Hence, indeed, / 0 / ∈ κ fin D (F). K2. Consider any S 1 , S 2 ∈ P(T) with S 1 ⊆ S 2 . Assume that S 1 ∈ κ fin D (F), so D S 1 ∈ F. That S 1 ⊆ S 2 implies, via Proposition 11(i), that D S 1 ⊆ D S 2 . This allows us to infer that also D S 2 ∈ F, using LF1. Hence, indeed, S 2 ∈ κ fin D (F). K3. Consider any S ∈ P(T), and assume that S ∈ κ fin D (F). Infer from Proposition 11(ii) that D S = D S\T − and therefore also D S ∈ F ⇔ D S\T − ∈ F. Hence, indeed, S \ T − ∈ κ fin D (F). K4. We may assume without loss of generality that T + = / 0, because otherwise this requirement's trivially satisfied. Consider then any t + ∈ T + , then we must show that {t + } ∈ κ fin D (F), or in other words that D {t + } ∈ F. Now simply observe that where the equalities hold because the coherence of D implies that / 0 ⊂ T + ⊆ D, and the final statement holds because the smallest filter 0 F(E fin ) = {D} is included in all filters.
K5fin. Consider any non-empty W ⋐ κ fin D (F) and any choice t σ ∈ Cl D (σ(W)) for all the σ ∈ Φ W , then we must prove that S o := {t σ : σ ∈ Φ W } ∈ κ fin D (F), or in other words that D S o ∈ F. It follows from the assumptions that D S ∈ F for all S ∈ W, and therefore also, by LF2 and the finiteness of W, that E (W) = S∈W D S ∈ F, so it's enough to prove that E (W) ⊆ D S o , because then also, by LF1, D S o ∈ F, as required. Consider, to this end, any D ∈ E (W), then it follows from Proposition 10(i) and Equation (9) that there's some selection map σ o ∈ Φ W such that Cl D (σ o (W)) ⊆ D. This guarantees that t σ o ∈ D, and therefore D ∩ S o = / 0. Hence, indeed, D ∈ D S o . For statement (iii), let F ′ := ϕ fin D (K) and let K ′ := κ fin D (F ′ ), then we have to prove that K = K ′ . We start with the following chain of equivalences: where the second equivalence follows from the fact that, by Equation (6), E ({S}) = D S . To prove that K ⊆ K ′ , consider any S ∈ K, then {S} ⋐ K. Using Equation (14) with W := {S}, this implies that, indeed, S ∈ K ′ . For the converse inclusion, consider any S ∈ K ′ , so there's, by Equation (14), some W ⋐ K for which E (W) = E ({S}). Proposition 14(ii) then guarantees that also {S} ⋐ K, or in other words, that, indeed, S ∈ K.
For statement (iv), let K ′ := κ fin D (F) and let F ′ := ϕ fin D (K ′ ), then we have to prove that F ′ = F, or in other words, that E (W) ∈ F ′ ⇔ E (W) ∈ F for all W ⋐ P(T). Now, consider the following chain of equivalences, valid for any W ⋐ P(T): where the third equivalence follows from Proposition 11(v). For the tops, we find that κ fin D (1 F(E fin ) ) = κ fin D (E fin ) = {S ∈ P(T) : E ({S}) ∈ E fin } = P(T) = 1 K fin . Finally, we turn to statement (ix). Consider that the proper filter F is prime if and only if for all W 1 ,W 2 ⋐ P(T), Since we've already argued in the proof of Proposition 15 that E (W 1 ) ∪ E (W 2 ) = E (W), with W := {S 1 ∪ S 2 : S 1 ∈ W 1 and S 2 ∈ W 2 }, it follows from Proposition 14(ii) that the statement (15) is equivalent to Assume now that F is prime, and apply the condition (16) with W 1 := {S 1 } and W 2 := {S 2 } to find the completeness condition (13). Conversely, assume that K is complete, and consider any W 1 ,W 2 ⋐ P(T). To prove that condition (16) is satisfied, we assume that {S 1 ∪ S 2 : S 1 ∈ W 1 and S 2 ∈ W 2 } ⋐ K but at the same time W 1 ⋐ K, and we prove that then necessarily W 2 ⋐ K. It follows from the assumptions that there's some S o ∈ W 1 such that S o / ∈ K, while at the same time {S o ∪ S 2 : S 2 ∈ W 2 } ⋐ K. But then the completeness condition (13) guarantees that S 2 ∈ K for all S 2 ∈ W 2 , whence, indeed, W 2 ⋐ K. 8 The case that W = / 0 is also covered, because E ( / 0) = D as well; see the discussion after Equation (6).

Lemma 19. Let F be any proper filter on E fin , ⊆ and let W ⋐ P(T) be any finite set of desirable sets of things, then W
Consider the following chain of equivalences: where the last equivalence follows from Equation (6) and the fact that F satisfies LF2 and LF1.
6.2. Conjunctive and prime filter representation. An important consequence of the existence of the order isomorphism in Theorem 18 is that it allows us to represent any finitely coherent set of desirable sets of things in terms of coherent but conjunctive models. This is interesting, because by Proposition 7 such coherent conjunctive sets of desirable sets of things are conceptually much simpler, as they represent sets of desirable things-they only represent conjunctive desirability statements.
To see how this representation in terms of conjunctive models comes about, we begin by recalling that the events E (W) for W ⋐ P(T) are sets of coherent sets of desirable things. They are completely determined by the following argument: consider any D ∈ D, then where we recall from Equation (4) that K D := {S ∈ P(T) : S ∩ D = / 0}. This tells us that Let's now consider any proper filter F ∈ F(E fin ) and any finitely coherent set of desirable sets of things K ∈ K fin that correspond, in the sense that K = κ fin D (F) and F = ϕ fin D (K). On the one hand, we infer from K = κ fin D (F) that for any S ∈ P(T): where the second equivalence follows F1. These equivalences tell us that On the other hand, we infer from F = ϕ fin D (K) and Equation (17) We're thus led to the following representation result for finite consistency, finite coherence, and the corresponding closure operator Cl K fin in terms of the conjunctive models K D .

is finitely coherent if and only if K is finitely consistent and
Proof. (i). For necessity, suppose that K is finitely consistent. That means that there's some finitely coherent K o ∈ K fin such that K ⊆ K o . But it then follows from Proposition 13 that E (W) = / 0 for all W ⋐ K o , and therefore in particular also for all W ⋐ K. For sufficiency, assume that E (W) = / 0 for all W ⋐ K. Then we infer from Lemma 21(i) that B K := {E (W) : W ⋐ K} is a filter base. Let's denote by F K the (smallest) proper filter that it generates; see Lemma 21(ii). If we now let K ′ := κ fin D (F K ), then it follows from Theorem 18(ii) that K ′ is a finitely coherent set of desirable sets of things: K ′ ∈ K fin . We're therefore done if we can prove that K ⊆ K ′ . To this end, consider any S ∈ K, then {S} ⋐ K and therefore also D S = E ({S}) ∈ B K . But then D S ∈ F K , and therefore indeed also S ∈ κ fin D (F K ) = K ′ . (ii). If K is not finitely consistent then it follows from (i) and Equation (17) that there's some W ⋐ K for which E (W) = / 0 and therefore D∈D : W⋐K D K D = D∈E (W) K D = P(T), making sure that both the left-hand side and the right-hand side in (ii) are equal to P(T). We may therefore assume that K is finitely consistent. If we borrow the notation from the argumentation above in the proof of (i), and take into account Lemma 21(iii) and Equation (17), then it's clear that we have to prove that Cl K fin (K) = V ∈F K D∈V K D . But if we take into account Equation (18), then we see for the right-hand side that V ∈F K D∈V K D = κ fin D (F K ) = K ′ , so we need to prove that K ′ = Cl K fin (K). But since we infer from the proof of (i) that K ′ is finitely coherent and that K ⊆ K ′ , we find that, on the one hand, Cl K fin (K) ⊆ K ′ , because Cl K fin (K) is the smallest finitely coherent set of desirable sets of things that the finitely consistent K is included in. To prove that, on the other hand, K ′ ⊆ Cl K fin (K), it suffices to consider any K ′′ ∈ K fin such that K ⊆ K ′′ , and prove that then K ′ ⊆ K ′′ . Now we can infer from K ⊆ K ′′ that F K = ϕ fin D (K) ⊆ ϕ fin D (K ′′ ), by Theorem 18(v). But then indeed also K ′ = κ fin D (F K ) ⊆ κ fin D (ϕ fin D (K ′′ )) = K ′′ , where the inclusion follows from Theorem 18(vi) and the last equality follows from Theorem 18(iii).
(iii). The proof's now straightforward, given (ii), since finitely coherent means finitely consistent and closed with respect to the Cl K fin -operator.

Lemma 21. Consider any set of desirable sets of things K ⊆ P(T) such that E (W) = / 0 for all W ⋐ K, then the following statements hold:
(i) the set B K := {E (W) : W ⋐ K} is closed under finite intersections, and therefore a filter base; (ii) the smallest proper filter F K that includes B K is given by Proof. (i). Any collection of non-empty sets that's closed under finite intersections is in particular a filter base; see Ref. [29,Definition 12.1]. So consider any W 1 ,W 2 ⋐ K, then we infer from Equation (11) (ii). See the standard construction of the (smallest) filter generated by a filter base [29, Definition 12.1].
(iii). The second equality's trivial given the definition of B K . For the first inequality, it's clearly enough to prove that V ∈F K D∈V K D ⊆ B∈B K D∈B K D . But this is immediate, since (ii) tells us that for any V ∈ F K there's some B ∈ B K such that B ⊆ V and therefore also D∈V K D ⊆ D∈B K D .
This tells us that a set of desirable sets of things is finitely consistent if and only if any of its finite subsets is included in some conjunctive model, and that any finitely coherent set of desirable sets of things can be written also as a limit inferior of conjunctive models. Even if the representation in terms of such limits inferior is formally somewhat complicated, it has the advantage that the basic representing models are the conjunctive ones, which are easy to identify and 'construct'.
There's, however, another representation result that's formally simpler, but where the representing models are now less easy to 'construct': a representation that's based on the representing role that prime filters play in distributive lattices; see the discussion in Ref. [5, and Section 5 for more details. Let's now, in the rest of this section, explain how it comes about. (13):

Definition 3 (Completeness). We call a set of desirable sets of things W ⊆ P(T) complete if it satisfies the completeness condition
we denote by K fin,c the set of all complete and finitely coherent sets of desirable sets of things, and by K c the set of all complete and coherent sets of desirable sets of things.
Theorem 18(ix) tells us that the complete finitely coherent sets of desirable sets of things are in a one-to-one relationship with the prime filters on the distributive lattice E fin , ⊆ , and the order isomorphism κ fin D identified in that theorem allows us to easily transform the prime filter representation result of Theorem 17 into the following alternative representation theorem for finitely coherent sets of desirable sets of things.

Theorem 22 (Prime filter representation). A finitely consistent set of desirable sets of things K ⊆ P(T) is finitely coherent if and only if K
Proof. Consider, for any finitely consistent K, the following chain of equivalences: As suggested before, a disadvantage of this type of representation is that the complete sets of desirable sets of things are-much like their prime filter counterparts-hard to identify 'constructively'. They include, however, all conjunctive models, which will be very helpful in our discussion of finitary models in Section 8 further on.

Proposition 23. Consider any coherent set of desirable things D ∈ D, then the (finitely) coherent conjunctive set of desirable sets of things K D is complete.
Proof. First off, it follows from Proposition 6 that K D is (finitely) coherent. Now, consider any S 1 , S 2 ⊆ T and assume that S 1 ∪ S 2 ∈ K D , or equivalently, that which clearly implies that, indeed, S 1 ∈ K D or S 2 ∈ K D .

FILTER REPRESENTATION FOR COHERENT SDSES
We now turn to a representation result for coherent, rather than merely finitely coherent, sets of desirable sets of things, and as is to be expected, we'll focus on the filters on the set E in order to achieve that.
It turns out that our representation will only involve the proper principal filters on this set E. where we let, as a special case of the definition in Equation (8), Taking into account Equation (12), it's readily verified that P(E) is closed under arbitrary intersections, and therefore P(E), ⊆ is a complete lattice, with intersection as infimum, and with bottom 0 P(E) = {D} and top 1 P(E) = E.
7.1. The order isomorphism. In order to establish the existence of an order isomorphism between the complete lattices K, ⊆ and P(E), ⊆ , we now consider the maps

iii) if K is a coherent set of desirable sets of things, then
This tells us that ϕ D is an order isomorphism between K, ⊆ and P(E), ⊆ , with inverse order isomorphism κ D .
Proof. For statement (i), assume that K is a coherent set of desirable sets of things.
We must show that ϕ fin D (K) is a proper principal filter. To see that ϕ fin D (K) is non-empty, observe that always / 0 ⋐ K, and therefore D = E ( / 0) ∈ ϕ fin D (K). To show that ϕ D (K) is increasing [satisfies LF1], consider any W 1 ,W 2 ⊆ P(T) such that E (W 1 ) ∈ ϕ D (K) and E (W 1 ) ⊆ E (W 2 ). Then the infinitary version of Proposition 14(ii) implies that W 1 ⊆ K, and the infinitary version of Proposition 14(i) guarantees that then W 2 ⊆ K, whence, indeed, also E (W 2 ) ∈ ϕ D (K). To show that ϕ D (K) is closed under arbitrary non-empty intersections [satisfies LF2p], consider any non-empty family of W i ⊆ K, i ∈ I. Then we infer from Equation (11) that i∈I E (W i ) = E ( i∈I W i ), and since i∈I W i ⊆ K, this tells us that, indeed, i∈I E (W i ) ∈ ϕ D (K). To show that ϕ D (K) is proper, simply observe that it follows from Proposition 13 that E (W) is non-empty for all W ⊆ K. This ensures that / 0 / ∈ ϕ D (K). We conclude that ϕ D (K) is a proper principal filter.
For statement (ii), assume that F is a proper principal filter. We check that the relevant conditions are satisfied for K to be a coherent set of desirable sets of things.
K3. Consider any S ∈ P(T), and assume that S ∈ κ D (F). Infer from Proposition 11(ii) that D S = D S\T − and therefore also D S ∈ F ⇔ D S\T − ∈ F. Hence, indeed, S \ T − ∈ κ D (F).
K4. We may assume without loss of generality that T + = / 0, because otherwise this requirement's trivially satisfied. Consider then any t + ∈ T + , then we must show that {t + } ∈ κ D (F), or in other words that D {t + } ∈ F. Now simply observe that where the second equality holds because the coherence of D implies that T + ⊆ D, and the final statement holds because the smallest principal filter 0 P(E) = {D} is included in all principal filters. K5. Consider any non-empty subset W of κ D (F) and any choice t σ ∈ Cl D (σ(W)) for all the σ ∈ Φ W , then we must prove that S o := {t σ : σ ∈ Φ W } ∈ κ D (F), or in other words that D S o ∈ F. It follows from the assumptions that D S ∈ F for all S ∈ W, and therefore also, using LF2p and Equation (6), that E (W) = S∈W D S ∈ F, so it's enough to prove that E (W) ⊆ D S o , because we'll then also find, by LF1, that D S o ∈ F, as required. This is what we now set out to do. Consider any D ∈ E (W), then it follows from Proposition 10(i) and Equation (9) that there's some selection map σ o ∈ Φ W such that Cl D (σ o (W)) ⊆ D, which guarantees that t σ o ∈ D, and therefore D ∩ S o = / 0. Hence, indeed, D ∈ D S o . For statement (iii), let F ′ := ϕ D (K) and let K ′ := κ D (F ′ ), then we have to prove that K = K ′ . We start with the following chain of equivalences: where the second equivalence follows from the fact that, by Equation (6) For statement (iv), let K ′ := κ D (F) and let F ′ := ϕ D (K ′ ), then we have to prove that F ′ = F, or in other words, if we consider any W ⊆ P(T), that E (W) ∈ F ′ ⇔ E (W) ∈ F. Now, consider the following chain of equivalences, valid for any W ⊆ P(T): where the second equivalence follows from the definition of ϕ D (K ′ ), and the fourth equivalence follows from Lemma 25.
Statements (v) and (vi) follow readily from the definitions of the maps ϕ D and κ D . Before we turn to the proofs of statements (vii) and (viii), it will be helpful to recall from Propositions 2 and 3 that 0 K = {S ∈ P(T) : S ∩ T + = / 0} and 1 K = P(T), together with 0 P(E) = {D} and 1 P(E) = E.
For statement (vii), consider any W ⊆ 0 K and any S ∈ W, 9 then S ∩T + = / 0, and therefore, by Proposition 11(v), also D S = D. Equation (6) For statement (viii), consider the following chain of equivalences, for any S ∈ P(T): where the third equivalence follows from Proposition 11(v). For the tops, we find that Lemma 25. Let F be any proper principal filter on E, ⊆ and let / 0 = W ⊆ P(T) be any non-empty set of desirable sets of things, then W ⊆ κ D (F) ⇔ E (W) ∈ F.
Proof. Consider the following chain of equivalences: 9 The case that W = / 0 is also covered, because E ( / 0) = D as well; see the discussion after Equation (6).
where the last equivalence follows from Equation (6) and the fact that F satisfies LF1 and LF2p.
7.2. Conjunctive representation. We begin by recalling an earlier finitary argument and transporting it to the current infinitary context: consider any W ⊆ P(T) and any D ∈ D, Similarly to what we did for finitely coherent sets of desirable sets of things in Section 6.2, we now consider any proper principal filter F ∈ P(E) and any coherent set of desirable sets of things K ∈ K that correspond, in the sense that K = κ D (F) and F = ϕ D (K). Observe that the principal filter F = ϕ D (K) is completely determined by its smallest element F, which is the subset of D given by: where the last equality follows readily from Equation (6). This leads to the following chain of equivalences, for any D ∈ D: where in the third equivalence we recall the definition of K D from Equation (4). This tells us that, on the one hand, where the first statement can also be recognised as a special case of Equation (21). On the other hand, we infer from K = κ D (F) that for any S ∈ P(T), taking into account the principal character of F, where in the third equivalence we recall Equation (22). So, we can conclude that We're thus led to the following representation result for coherent sets of desirable sets of things in terms of the conjunctive models K D .
Theorem 26 (Conjunctive representation). Consider any set of desirable sets of things K ⊆ P(T), then the following statements hold:

is coherent if and only if K is consistent and K
Proof. (i). The equality between the two sets follows from Equation (21).
For necessity, suppose that K is consistent. That means that there's some coherent K o ∈ K such that K ⊆ K o . But it then follows from Proposition 13 that E (W) = / 0 for all W ⊆ K o , and therefore in particular also for W = K.
For sufficiency, assume that E (K) = / 0. Denote by F K := up E (E (K)) the (smallest) proper principal filter that it generates. If we now let K ′ := κ D (F K ), then it follows from Theorem 24(ii) that K ′ is a coherent set of desirable sets of things: K ′ ∈ K. We're therefore done if we can prove that K ⊆ K ′ . To this end, consider any S ∈ K, then {S} ⊆ K and therefore also E (K) ⊆ E ({S}). But then D S = E ({S}) ∈ F K , and therefore indeed also S ∈ κ D (F K ) = K ′ .
(ii). If K is not consistent then it follows from (i) that E (K) = / 0, and therefore also from Equation (21) that D∈D : K⊆K D K D = D∈E (K) K D = D, making sure that both the left-hand side and the right-hand side in (ii) are equal to D.
We may therefore assume that K is consistent, and use the argumentation and notations above in the proof of (i), starting with the proper principal filter F K := up E (E (K)) and the corresponding coherent set of desirable sets of things K ′ := κ D (F K ) for which we know that K ⊆ K ′ , and, by Theorem 24(iv), that ϕ D (K ′ ) = F K . Applying Equation (23) for the coherent K ′ leads to so Equation (24) tells us that K ′ = D∈D : K⊆K D K D . We therefore need to prove that Cl K (K) = K ′ . But since we already know that K ⊆ K ′ , it's enough to consider any K ′′ ∈ K such that K ⊆ K ′′ , and to prove that then where the last equality follows from applying Equation (23) for the coherent K ′′ . But then indeed also, by The- (iii). The proof's now straightforward, given (ii), since coherent means consistent and closed with respect to the Cl K -operator.

FINITARY SETS OF DESIRABLE SETS OF THINGS
We conclude from the discussion in the previous section that the conjunctive representation for coherent SDSes is remarkably simpler than the one for merely finitely coherent SDSes. But, as we'll explain presently, it turns out that we can recover the simpler conjunctive representation also for finitely coherent SDSes, provided that we focus on finite sets of things. This was already proved by De Bock [8], based on ideas in Refs. [9,10], but we intend to derive this remarkable result here using our filter representation approach, which allows for a different and arguably somewhat simpler proof, based on the Prime Filter Representation Theorem that we recalled in Theorem 17. We call the set of desirable sets of things W finitary if each of its desirable sets has a finite desirable subset, meaning that Interestingly, for any (finitely) coherent set of desirable sets of things K, the coherence condition K2 guarantees that, since K ∩ Q(T) ⊆ K, also fin(K) ⊆ K. This tells us that a (finitely) coherent set of desirable sets of things K is finitary if and only if K = fin(K): a finitary and (finitely) coherent set of desirable sets of things K is equal to its finitary part fin(K), and therefore completely determined by its finite part K ∩ Q(T).
Moreover, it's easy to see that fin(fin(K)) = fin(K) for any (finitely) coherent set of desirable sets of things K, implying that its finitary part fin(K) is always finitary. 10 Does the (finite) coherence of an SDS imply the coherence (finite or otherwise) of its finitary part? The following proposition provides the beginning of an answer, which we'll be able to complete further on in Corollary 31.

Proposition 27. If a set of desirable sets of things K is (finitely) coherent, then its finitary part fin(K) is finitely coherent.
Proof. It's enough to consider any finitely coherent K ∈ K fin , and to check that fin(K) satisfies the relevant axioms K1-K4 and K5fin.
The relation with completeness and conjunctivity. Let's now find out more about how, for a (finitely) coherent SDS, being finitary relates to being complete, and in particular to being conjunctive. All coherent and conjunctive SDSes are, of course, finitary. Proof. It follows from Proposition 6 that K D is (finitely) coherent. It then follows from K2 that fin(K D ) ⊆ K D . For the converse inclusion, consider any S ∈ K D , then S ∩ D = / 0. Consider, therefore, any t ∈ S ∩ D, then {t} ⊆ S and {t} ∈ K D ∩ Q(T), so S ∈ fin(K D ), and therefore, indeed, K D ⊆ fin(K D ).
Coherent SDSes that are conjunctive are always complete; see also Propositions 7 and 23. On the other hand, complete coherent SDSes are not necessarily conjunctive, but we're now about to prove that they necessarily have a conjunctive finitary part. Consequently, the (finitely) coherent conjunctive SDSes are exactly the complete and coherent SDSes that are finitary; see also the diagram below. The following proposition gives a more detailed statement. Proof. We give the proof for the second statement involving finite coherence. The proof for the first statement involving coherence is similar, but slightly simpler. That fin(K D ) = D K follows from Proposition 28. So, assume that the closure operator Cl D is finitary, and consider any K ∈ K fin,c . It suffices to prove that K ∩ Q(T) = K D ∩ Q(T) for some coherent D ∈ D.
Let D := D K , then D is coherent and K D ⊆ K, by Proposition 5, so it only remains to prove that K ∩ Q(T) ⊆ K D ∩ Q(T). So, consider any S ∈ K ∩ Q(T), then there are two possibilities. If S is a singleton {t}, then necessarily t ∈ D K = D, and therefore S = {t} ∈ K D . Otherwise, it follows from the completeness of K that there's some strict subset S ′ of S such that S ′ ∈ K, and therefore also S ′ ∈ K ∩ Q(T). Now replace S with S ′ in the argumentation above, and keep the recursion going until we do reach a singleton {t} for which t ∈ D K = D, or equivalently {t} ∈ K, which must happen after a finite number of steps, because S is finite. But then necessarily also t ∈ S, so S ∩D = / 0, and therefore indeed S ∈ K D ∩ Q(T).
Since the finitary part of a complete and (finitely) coherent SDS is conjunctive, the Prime Filter Representation Theorem results in a representation with conjunctive models. Proof. Due to Theorem 26(iii), it clearly suffices to prove necessity. So, assume that K is finitary and finitely coherent. As suggested above, our proof will rely on the Prime Filter Representation Theorem [Theorem 22], so let's consider any K ′ ∈ K fin,c such that K ⊆ K ′ . Then also fin(K) ⊆ fin(K ′ ), and therefore where the first equality follows from the finitary character of K, the second equality follows from Proposition 29, the last equality follows from the finitary character of the finitely coherent conjunctive K D K ′ [see Proposition 28], and the last inclusion follows from Proposition 5. We've thus proved that Hence, But since D K ′ ∈ D for all K ′ ∈ K fin,c , by Proposition 5, this implies that {K D K ′ : K ′ ∈ K fin,c and K ⊆ K ′ } ⊆ {K D : D ∈ D and K ⊆ K D }, and therefore where the third inclusion follows again from Proposition 5, and the equality follows from Theorem 22.
This leads to the remarkable conclusion that for finitary SDSes there's no difference between finite coherence and coherence, as long as the closure operator Cl D is finitary. Proof. All the conjunctive models have the form K D for D ∈ D by Proposition 7, and are therefore coherent by Proposition 6. So is, therefore, any intersection of them. Now recall Theorem 30.
Corollary 32. If the closure operator Cl D is finitary, then the finitary part fin(K) of any (finitely) coherent set of desirable sets of things K is coherent.
Proof. Assume that K is (finitely) coherent, then Proposition 27 guarantees that fin(K) is finitely coherent. But since fin(K) is finitary, its finite coherence implies its coherence, by Corollary 31.

EXAMPLE: PROPOSITIONAL LOGIC
As a first and very simple example, we consider propositional logic under the standard axiomatisation; see Ref. [5,Section 11.11 onwards] for a more detailed account of the facts we're about to summarise below. We'll assume the reader to be familiar with the basic set-up of this logic using well-formed formulas.
The basic setup. In this context, the things t in T are the well-formed formulas (wffs) in the language; 'desirable' means 'true' or 'derivable'; the set T − contains all contradictions and the set T + all tautologies. We'll denote and-ing by '∧', or-ing by '∨' and negation by '¬'. Moreover, 'closed' means 'deductively closed'; the closure operator Cl D represents the usual deductive closure under finitary conjuction and modus ponens, and the 'coherent' sets D in D are the sets of wffs that are deductively closed and contain no contradictions; see the summary in Table 1.
abstract theory of things propositional logic thing wff desirable thing true, derivable wff T + all tautologies T − all contradictions closed deductively closed consistent set of desirable things logically consistent set of wffs coherent set of desirable things logically consistent and deductively closed set of wffs It will be useful to consider the Lindenbaum algebra associated with this propositional logic. We use the notations 't 1 ⊢ t 2 ' for 't 2 ∈ Cl D ({t 1 })' and 't 1 ≡ t 2 ' for 't 1 ⊢ t 2 and t 2 ⊢ t 1 '. Then the equivalence relation ≡ partitions the set T into classes of logically equivalent wffs, which we collect in the set This set L can be ordered by the partial order ≤, defined by which turns it into a Boolean lattice-or Boolean algebra, the so-called Lindenbaum algebra-with meet ⌢ and join ⌣ defined by and complement operator co given by Its top collects all tautologies, and its bottom all contradictions.
It's important to recall-or observe-that the map •/ ≡ : T → L connects the coherent sets of desirable wffs to the proper filters of the Lindenbaum algebra L: for any coherent set of desirable wffs D ∈ D, the corresponding set of equivalence classes is a proper filter on L, and conversely, for any proper filter F ∈ F(L) on L, the corresponding set of wffs {t ∈ T: t/ ≡ ∈ F} is a coherent set of desirable wffs. Moreover, the closed but inconsistent set of wffs T is mapped to the improper filter P(L).
Towards desirable sets. Let's now bring desirable sets of wffs to the forefront, in order to complete the picture in Table 1. As the closure operator for propositional logic is finitarysince it's based on the finitary conjunction and modus ponens production rules-we'll focus on the finitary aspects of this type of coherence, and rely on the representation results of Section 8.
To allow ourselves to be inspired by interpretation, we recall that a finite set of wffs S ∈ Q(T) is considered to be desirable of it contains at least one desirable (true) wff, or equivalently in this special case, if its disjunction S := t 1 ∨ · · · ∨ t n with S = {t 1 , . . . ,t n } is desirable (true). This simple observation leads us to the following definition, for which we can prove a basic but very revealing proposition. We let, for any set of desirable sets of things K, D(K) := S : S ∈ K ∩ Q(T) .
Proposition 33. Consider any finitary and finitely coherent set of desirable sets of wffs K ∈ K fin , then the following statements hold: Proof. First, fix any D ∈ D such that K ⊆ K D . Consider any S ∈ K ∩ Q(T), then clearly also S ∩ D = / 0. Consider any t S ∈ S ∩ D [which is always possible], then t S ⊢ S, and therefore also S ∈ D, because D is deductively closed. This tells us that D(K) ⊆ D. It therefore suffices to prove (i) in order to also prove (ii). And (ii) then readily leads to (iii), taking into account Theorem 30.
So, let's concentrate on the proof of (i). First, we prove that D(K) is deductively closed. So consider any S ∈ K ∩ Q(T) and any t ∈ T such that S ⊢ t. Then we have to prove that also t ∈ D(K). To this end, let S ′ := S ∪ {t} then still S ′ ∈ K ∩ Q(T) by coherence [K2], and therefore also S ′ ∈ D(K). But we infer from S ⊢ t that S ′ = t, and therefore indeed t ∈ D(K).
It now only remains to prove that D(K) is consistent. Assume, towards contradiction, that it isn't, so D(K) ∩ T − = / 0. Consider any t − ∈ D(K) ∩ T − , so there's some S ∈ Q(T) for which t − = S. This can only happen if all wffs in S are contradictions, so S ⊆ T − . But then K3 implies that / 0 ∈ K, contradicting K1.
We conclude that in the special case of propositional logic, all finitary (parts of) (finitely) coherent sets of desirable sets of wffs are conjunctive, or in other words, that working with desirable sets of wffs does not lead to anything more interesting than simply working with desirable wffs. The reason for this is, of course, that in the case of propositional logic, the language of desirable wffs is already powerful enough to also accommodate for or-ing desirability statements, besides the and-ing that's inherently possible in any language of desirable things.

EXAMPLE: COHERENT CHOICE
As another and final example, we'll consider coherent choice functions, and coherent sets of desirable option sets, on an option space. Our discussion here will be based mostly on earlier work by some of us [9][10][11][12], to which we refer for the full details, and for explicit proofs of the results mentioned below.
The ground layer. We consider a real linear space V , called option space, and whose elements u are called options. These options are ordered by strict vector ordering ≻, called the background ordering. We denote by V ≻0 := {u ∈ V : u ≻ 0} the convex cone of all options that are (strictly) preferred to the zero option 0 under this background ordering. We assume that V is also provided with a norm • V that turns it into a Banach space. With this norm, we can associate a topological interior operator Int(•).
We'll be interested in special types of real functionals on V . We call a real functional Γ : V → R superlinear if it's superadditive and non-negatively homogeneous: [superadditivity] SL2. Γ(λ u) = λ Γ(u) for all u ∈ V and all real λ ≥ 0.
[positivity] We'll from now on fix any u o ∈ Int(V ≻0 )-which we'll assume to be non-empty-and call a real functional Γ : V → R constant additive (with respect to u o ) if CA. Γ(u + µu o ) = Γ(u) + µ for all u ∈ V and µ ∈ R.
[constant additivity] Finally, a real functional Γ : We denote by P the set of all real functionals Λ that are bounded, superlinear, positive and constant additive, and by P the subset of all real functionals Λ that are bounded, linear, positive and constant additive. We'll assume both sets to be non-empty.
The special case of gambles. As a familiar example, consider a variable X whose value it takes in a finite set X is unknown. Any bounded map h : X → R then corresponds to a real-valued uncertain reward h(X), and is called a gamble on X. We'll typically assume this reward is expressed in units of some linear utility scale. The set G (X ) of all such gambles h constitutes a linear (option) space.
In a typical decision problem, You are uncertain about the value of X, and are asked to express Your preferences between several possible decisions or acts, where each such act has an associated uncertain reward, or gamble.
We consider the strict vector ordering >, defined by as a background ordering, reflecting those (minimal) strict preferences we want any subject who's uncertain about X to always have, regardless of their beliefs about X. The supremum norm • ∞ , defined by h ∞ := sup x∈X |h(x)|, turns the option space G (X ) into a Banach space, which puts us squarely in the context of the present section. We'll take as u o the gamble that assumes the constant value 1.
In this special case, P is the set of all so-called linear previsions Λ on G (X ), i.e. the expectation operators associated with the probability mass functions p ∈ Σ X , so Moreover, P is the set of all so-called coherent lower previsions Λ on G (X ), i.e. the lower expectation operators associated with the convex closed M ⊆ Σ X , so For more details about coherent lower and linear previsions, we point to Refs. [1,25,27].
Choice and rejection functions. In the general, abstract setting, the elements u of the Banach space V are intended to represent all the possible options in a decision problem under uncertainty.
In a specific application, there will be a finite subset S ⋐ V containing options that You have to-in some way or other-express preferences between. Such a decision problem may be 'solved' when our subject, specify Your subset R(S) ⊆ S of rejected, or inadmissible, options. We may interpret rejecting an option u ∈ R(S) from S as 'S contains another option that You prefer to S'. The remaining options C(S) := S \ R(S) are then Your admissible or non-rejected options in S.
Initially, You may be uninformed, which will be reflected by Your set R(S) of rejected options being small. You may gather more information, and when You do so, You may be able to additionally identify options that You reject, yielding a larger set R(S) and hence a smaller C(S). It may happen that You are optimally informed, and in this case Your set C(S) of admissible options will be a singleton, or possibly a set of 'best' options that are 'equally good' in the sense that You are indifferent between them. However, we'll not assume that this state of being optimally informed is always attainable: You needn't always be indifferent between the options in C(S). Instead, these admissible options may be incomparable, and as such, the setup we're describing here can deal with partial preferences, and leads to so-called imprecise-probabilistic decision-making approaches. This is one of the features that makes this approach to decision theory more general than the classical idea of maximising expected utility: there, all the admissible options in C(S) have the same highest expected utility, so the subject is indifferent between them.
These ideas formalise to all decision problems-all sets in Q(V )-as follows. The function R : Q(V ) → Q(V ) : S → R(S) ⊆ S that maps any finite option set S ∈ Q(V ) to its subset of rejected options, is called a rejection function, and the dual function C : Q(V ) → Q(V ) : S → C(S) ⊆ S that identifies the admissible options, is called a choice function. As C(S) = S \ R(S) for every S in Q(V ), either function can be retrieved from the other, so they are both equivalent representations of the same information. Choice functions in an imprecise-probabilistic decision-making context were first introduced by Kadane et al. [15], who later [22] also established a representation result for what they called coherent choice functions in terms of sets of probabilities and a decision criterion called E-maximality, going back to Isaac Levi [16]. Some time after that, some of us [9,12] extended these results by establishing, amongst other things, more general representations in more abstract contexts. We intend to briefly report on this work below, and to show how it relates to the desirable sets of things framework.
Recall that the background ordering ≻ for the option space V reflects those strict preferences between options that it is rational for You to have, regardless of any information or beliefs You might have about the decision problem at hand. We'll assume it's reflected by the following requirement on rejection functions: If u ≻ v then v ∈ R({u, v}), for all u and v in V . This will imply, together with Sen's [23] Property α, 11 that v is inadmissible as soon as S contains another option u ≻ v, or in other words, as soon as it is dominated by some option in S; The linearity of the utility scale is reflected in the assumption that any coherent rejection function R satisfies where S 1 + S 2 := {u 1 + u 2 : u 1 ∈ S 1 , u 2 ∈ S 2 } and −S := {−u : u ∈ S} for any S, S 1 , S 2 ⊆ V . As a precursor to the discussion below, we see that to infer whether an option u is rejected from S-whether u ∈ R(S)-it suffices to check whether 0 ∈ R(S − {u}), in which case we'll call S − {u} a desirable option set.
Binary choice and sets of coherent sets of desirable options. Equation (27) We'll call a set of desirable options D coherent [12] (but see also Refs. [4,13,20,28]) when OD1. 0 / ∈ D; OD2. V ≻0 ⊆ D; OD3. if u, v ∈ D and (λ , µ) > 0, 12 then λ u + µv ∈ D, for all u, v ∈ V and λ , µ ∈ R. We see that if we identify options as special cases of the abstract things, where we let the set of things T be the set of options V , and desirable options with desirable things, then we have made a start with identifying the correspondences between things and options in Table 2. Let us now work towards completing this table, beginning at the level of desirable things, where we have to identify the sets T − , T + and the closure operator Cl D .
First of all, observe that a coherent D can't have anything in common with the set V 0 := V ≺0 ∪ {0}. Indeed, assume towards contradiction that D contains some option u ∈ V 0 , then necessarily u ≺ 0 by OD1. Since then −u ≻ 0 because ≻ is a vector ordering, we infer from OD2 that −u ∈ D, and therefore, by OD3, that 0 = u − u ∈ D, contradicting OD1. Hence, indeed, D ∩ V 0 = / 0, indicating that the convex cone V 0 plays the role of the set of forbidden things T − .
To identify the closure operator governing the desirability of options, we observe that OD3 makes sure that coherent sets of desirable options D are convex cones: they satisfy D = posi(D), where posi(S) := n ∑ k=1 λ k u k : n ∈ N, λ k ∈ R >0 , u k ∈ S for any S ⊆ V is the set of all positive linear combinations of elements of S, and therefore the smallest convex cone that includes S. We see that the coherent sets of desirable options D ⊆ V are exactly the convex cones in V that include V ≻0 and have nothing in common with V 0 , 11 Sen's [23] Property α is the requirement that R(A 1 ) ⊆ R(A 2 ), for all A 1 and A 2 in Q(V ) such that A 1 ⊆ A 2 , which is an axiom for choice under uncertainty that is almost always assumed. 12 We'll use the notation (λ , µ) > 0 to mean that λ ≥ 0 and µ ≥ 0 but not both equal to zero. ) takes the role of the closure operator Cl D , and implements the inference mechanism behind the desirability of options, also called natural extension [13]. Interestingly, it is clear from its definition that this closure operator is finitary. Also observe that posi( / 0 ∪ V ≻0 ) = posi(V ≻0 ) = V ≻0 plays the role of the set T + . To conclude, let's check that with these identifications, the desirability axioms D1-D3 in Section 2.1 are verified. For D1, assume that all options in S ⊆ A are desirable to You. Then we infer from OD2 and OD3 that any positive linear combination of elements of S and V ≻0 must also be desirable to You. As these positive linear combinations are precisely the elements of the closure posi(S ∪ V ≻0 ) of the set S, we see that D1 is indeed satisfied. We have already argued that D ∩ V 0 = / 0 for Your set of desirable options D, so D2 is satisfied as well. And finally, for D3, note that, indeed, Coherent sets of desirable option sets. To continue filling out Table 2, we now lift the framework of sets of desirable options to sets of desirable option sets [8][9][10]12]. The underlying idea is that, rather than merely use options as elements that are potentially desirable, we now turn to option sets, instead. In doing so, we'll move from (strict) binary preferences between options to more general preferences that aren't necessarily binary. We'll allow You to state for a finite option set S ∈ Q(V ) that at least one of its elements is desirable to You, but without Your needing to specify which; we'll then say that S is desirable to You, and call S a desirable option set. Your set of desirable option sets K may then contain singletons {u}, reflecting that You find u desirable, but also option sets S that aren't singletons. In fact, it's perfectly possible for K to contain no singletons, apart from the {u} for u ∈ V ≻0 , which result form the background ordering. It would then contain no non-trivial binary preferences.
Generally speaking, a set of desirable option sets K is called coherent [9, 10, 12] when OK1. / 0 / ∈ K; OK2. if S 1 ∈ K and S 1 ⊆ S 2 then S 2 ∈ K, for all S 1 , S 2 ∈ Q(V ); OK3. if S ∈ K then S \ V 0 ∈ K, for all S ∈ Q(V ); OK4. {u + } ∈ K for all u + ∈ V ≻0 ; OK5. if S 1 , S 2 ∈ K then also {λ u,v u + µ u,v v : u ∈ S 1 , v ∈ S 2 } ∈ K, provided that (λ u,v , µ u,v ) > 0 for all u ∈ S 1 and v ∈ S 2 . It's not too difficult to see that these coherence requirements can be reinterpreted as, essentially, Axioms K1-K5fin, after a proper identification of the relevant concepts here with those in the abstract treatment of desirable sets of things, as summarised in Table 2. After this identification, we find that the coherent sets of desirable option sets are the finite parts of the finitely coherent sets of desirable sets of things in K fin , using the specific finitary closure operator posi( • ∪ V ≻0 ), and are thus essentially equivalent to working with finitary and finitely coherent sets of desirable sets of things. As a consequence, the inference mechanism for finitely coherent sets of desirable sets of things expressed by the closure operator Cl K fin , and all concomitant machinery, can be applied to sets of desirable option sets.
Back to choice and rejection functions. We can now easily relate sets of desirable option sets K back to rejection functions R. To do so, we'll consider any finite option set S, and any option u ∈ S for which we want to find out whether it's being rejected. We'll follow Ref. [12] in introducing the corresponding option set S ⊖ u := (S \ {u}) − {u}, which then allows for an efficient connection, taking into account Equation (27) and our interpretation of rejecting an option: This connection allows for a one-to-one correspondence between rejection functions R and sets of desirable option sets K, allowing us to transport coherence notions from the latter to the former.
What about (principal) filter representation? So, now that we know that working with desirable options and desirable option sets fits in the context of the present paper, we also know that it there will be representations in terms of (principal) filters of events. We've seen above that coherent sets of desirable option sets are special instances of the finite parts of finitely coherent sets of desirable sets of things, which are of course in a oneto-one relationship with the finitary and (finitely) coherent sets of desirable sets of things, which puts us squarely in the context of Section 8, and the representation results proved therein. Underlying these results is the lattice set of events E fin = {E (W) : W ⋐ P(V )}, where each E (W) is some subset of D, so some collection of coherent sets of desirable options, which can be interpreted as a set of possible identifications of the actual set of desirable option sets D T ; see the discussion in Section 3.
Every finitary and (finitely) coherent set of desirable option sets K = fin(K) is then represented by a principal filter on E fin , with as smallest element the event E (K) = {D ∈ D : K ⊆ K D }. Therefore, this E (K) can be seen as the set of remaining possible identifications for D T after making all the desirability statements corresponding to all the desirable option sets in K. Now, as discussed in great detail in Ref. [12], it's possible to impose additional (rationality) requirements on finitary sets of desirable option sets K, besides coherence, and it will be interesting to discuss a few of them here, using the superlinear functionals introduced earlier.
It can then be proved [12,Theorem 25] that a finitary set of desirable option sets K is Archimedean if and only if K = {K Λ : Λ ∈ P(K)} = {K Λ : Λ ∈ P and K ⊆ K Λ }.
In the light of the discussion above, this leads to the following conclusion. Rather than saying something about an actual model D T ∈ D, the desirability statements present in an Archimedean K can be interpreted as propositional statements about an actual model Λ T in a set of possible identifications P, and the corresponding 'event' P(K) is then the set of all possible identifications of Λ T that remain after making the desirability statements in K. In fact, this allows us to identify a representation in terms of (principal) filters of subsets of P.
In the special case where the options are gambles on a finite set X , the representation is then in terms of (principal) filters of coherent lower previsions, or equivalently, of closed convex sets M of probability mass functions on X . We can go even further, and impose yet another condition, called mixingness. Recall the posi(•) operator defined in Equation (10), which, for any subset S of V , returns the set of all positive linear combinations of its elements. A finitary and (finitely) coherent set of desirable option sets K = fin(K) ∈ K fin is then called mixing if it satisfies the following condition: K M . if B ∈ K and A ⊆ B ⊆ posi(A), then also A ∈ K, for all A, B ∈ Q(V ). One of us proved in Ref. [12, Corollary 30] that a finitary and (finitely) coherent set of desirable option sets K is Archimedean and mixing if and only if, with now obvious notations, K = {K Λ : Λ ∈ P(K)} = {K Λ : Λ ∈ P and K ⊆ K Λ }.
As above, this leads to an interesting conclusion. Rather than saying something about an actual model D T ∈ D, the desirability statements present in an Archimedean and mixing K can be interpreted as propositional statements about an actual model Λ T in a set of possible identifications P, and the corresponding 'event' P(K) is then the set of all possible identifications of Λ T that remain after making the desirability statements in K. In fact, this allows us to identify a representation in terms of (principal) filters of subsets of P.
In the special case where the options are gambles on a finite set X , the representation is then in terms of (principal) filters of linear previsions, or equivalently, probability mass functions p on X . We therefore recover, as a special case, the filter representation results proved in the seminal work by Catrin Campbell-Moore [2]. Let us, to conclude this section, find out what the representation result (30) tells us about the choice function C and the rejection function R that are associated with an Archimedean and mixing set of desirable option sets K, where we use the correspondence established in Equation (29). First of all, we find that for any Λ ∈ P and any S ∈ Q(V ), and therefore u ∈ R(S) ⇔ (∀Λ ∈ P(K))(∃v ∈ S)Λ(v) > Λ(u), and u ∈ C(S) ⇔ (∃Λ ∈ P(K))(∀v ∈ S)Λ(v) ≤ Λ(u).
Recall that in the case that options are gambles, discussed above, P(K) is a set of expectation functionals associated with a set of probability mass functions-a so-called credal set-, and then Equation (31) tells us that an option u is admissible in an option set S if it is Bayes-admissible for at least one mass function in the credal set, in the sense that it maximises expectation. We therefore recover Levi's E-admissibility criterion [16] in decision making as a special case of our representation results, and in this sense, all representation results in this paper can be seen as generalisations of Levi's E-admissibility.

DISCUSSION AND CONCLUSIONS
Laying bare the exact nature of the conservative inference mechanism behind coherent sets of desirable sets of things has allowed us to prove powerful representation results for such coherent sets of desirable sets of things in terms of the simpler, conjunctive, models which are essentially coherent sets of desirable things. These representation results, in their simplest form (Theorems 26 and 30), are reminiscent of, and in fact formal generalisations of, decision making using Levi's E-admissibility Rule [16].
Indeed, E-admissibility is recovered as a very special case, where the desirable things are desirable gambles, and where in addition to coherence, further requirements of Archimedeanity and mixingness are imposed on sets of desirable gamble sets.
Interestingly, in another interesting special case, where the desirable things are asserted propositions in propositional logic, the additional layer of working with asserted sets of propositions-desirable sets of things-does not add anything new: all coherent sets of desirable sets of things are conjunctive there. This is, of course, not really surprising, as desirable sets of things are introduced to deal with disjunctive statements, which are already present in the language of things themselves as propositions in propositional logic.
The case of things as options, on the other hand, shows that in other inference contexts where disjunctive statements are not already part of the language of things, going from desirable things to desirable sets of things is indeed meaningful and useful.
A more detailed and comprehensive study of these and other special cases is the topic of current research.