1 Introduction
De Finetti [dF51, dF37] proposed the following axiomatization of qualitative probabilistic comparison (presented here based on [Sco64]): for sets , , and coming from the powerset of a nonempty finite set , we have

,

,

or ,

implies , and

if and only if for disjoint from and .
De Finetti conjectured that any binary relation on that satisfies these conditions is realizable by a probability measure on , which means that we have if and only if . While every probability measure realizing a binary relation on satisfies de Finetti’s conditions, these conditions do not in general guarantee the existence of a realizing probability measure: it was shown by Kraft, Pratt, and Seidenberg [KPS59] (presented here as in [Seg71]) that for , the relations
may be extended to a binary relation over that satisfies de Finetti’s conditions and yet has no realizing probability measure. Kraft, Pratt, and Seidenberg (“KPS”) also determined what was missing from de Finetti’s axiomatization; Scott [Sco64] later presented the KPS conditions in a linear algebraic form.
Theorem 1.1 ([Sco64, Theorem 4.1] reformulated with instead of a general Boolean algebra).
Let be a nonempty finite set. Given , let
be the characteristic function of
(i.e., if , and if ). Construe functionsas vectors:
indicates the realnumber value of vector at coordinate . Addition and negation of these vectors is taken componentwise: and . A binary relation on is realizable by a probability measure if and only if it satisfies each of the following: for each and , we have
;

;

or ; and

if for each and , then for each .
Scott’s fourth condition is the most difficult. The algebraic component
(1) 
of this condition says: for each coordinate , the number of ’s that contain is equal to the number of ’s that contain . Intuitively, Scott’s forth condition tells us that if two length sequences of coordinate sets are related componentwise by the relation “is no more probable than” and the occurrence multiplicity of any given world is the same in each of the sequences, then the sets are also related componentwise by the relation “has the same probability.”
Using Scott’s theorem to prove completeness, Segerberg [Seg71] studied a modal logic of qualitative probability. Segerberg’s logic has a binary operator expressing qualitative probabilistic comparison and a unary operator expressing necessity. Gärdenfors [Gär75] considered a simplified version of Segerberg’s logic that, among other differences, eliminated the necessity operator in lieu of the abbreviation , which has the semantic meaning that has probability (and implies that is true at all outcomes with nonzero probability). Both Gärdenfors and Segerberg express the algebraic component (1) of Scott’s fourth condition using Segerberg’s notation
which we sometimes shorten to . This expression abbreviates the formula
where each is the disjunction of all conjunctions
satisfying the property that exactly of the ’s are the empty string, exactly of the ’s are the empty string, and the rest of the ’s and ’s are the negation sign . Intuitively, says that of the ’s are true and of the ’s are true; says that the number of true ’s is the same as the number of true ’s; and says that at every outcome with nonzero probability, the number of true ’s is the same as the number of true ’s. Using this notation, it is possible to express the fourth condition of Scott’s theorem and thereby obtain completeness for the probabilistic interpretation.
In the present paper, we follow this tradition of studying probability from a qualitative (i.e., nonnumerical) point of view using modal logic. However, our focus shall not be on the binary relation of qualitative probabilistic comparison but instead on the unary notions of certainty (i.e., having probability ) and “high” probability (i.e., having a probability greater than some fixed rationalnumber threshold ). That is, our interest is in unary modal logics of high probability.
For convenience in this study, we shall identify epistemic notions with probabilistic assignment, which suggests a connection with subjective probability [Jef04]. In particular, we identify knowledge with probabilistic certainty (i.e., probability ) and belief with probability greater than some fixed rationalnumber threshold . Therefore, instead of the unary operator , we shall use the unary operator and assign this operator an epistemic reading: says that the agent knows , which means she assigns subjective probability . We shall use the unary modal operator to express belief: says that the agent believes , which means she assigns a subjective probability exceeding the threshold (which will always be a fixed value within a given context or theory). Though our readings of these formulas are epistemic and doxastic, we stress that our technical results are independent of this reading, so someone who disagrees with subjective probability or our epistemic/doxastic readings is encouraged to think of our work purely in terms of high probability: says , and says for some fixed . That is, the technical results of our work are in no way dependent on our use of epistemic/doxastic notions or on the philosophy of subjective probability.
Lenzen [Len03, Len80] is to our knowledge the first to consider a modal logic of high probability for the threshold . Actually, his perspective is slightly different than the one we adopt here. First, his reading of formulas is different (though not in any deep way): he identifies “the agent is convinced of ” with and “ is believed” by . More substantially, Lenzen’s conviction (German: Überzeuging) does not imply truth. Technically, this amounts to permitting the possibility that there are outcomes having probability zero. For reasons of personal preference, we forbid this in our study here, though this difference is nonessential, as it is completely trivial from the technical perspective to change our setting to allow zeroprobability outcomes or to change Lenzen’s setting to forbid them. Therefore, we credit Lenzen’s work [Len80] as the first to provide a proof of probabilistic completeness for . As with Segerberg’s and Gärdenfors’ probabilistic completeness results, Lenzen’s proof made crucial use of Scott’s work.
In more recent work, Herzig [Her03] considered a logic of belief and action in which belief in is identified with . This is equivalent to Lenzen’s notion, though Herzig does not study completeness. Another recent work by Kyberg and Teng [KT12] investigated a notion of “acceptance” in which is accepted whenever the probability of is at most some small . This gives rise to the minimal modal logic , which is different than Lenzen’s logic.
We herein consider belief à la Lenzen not only for the case but also for the case . As it turns out, the logics for these cases are different, though our focus will be on the logic for because this is the only threshold for which a probability completeness result is known. In particular, probability completeness for is still open. Thresholds permit simultaneous belief of and while avoiding belief of any selfcontradictory sentence such as the propositional constant for falsehood. This might suggest some connection with paraconsistent logic. However, we leave these logics of low probability for future work, though we shall say a few words more about them later in this paper.
In Section 2, we identify a Kripkestyle semantics for probability logic similar to [EoS14, Hal03] (and no doubt to many others). We require that all worlds are probabilistically possible but not necessarily epistemically so, and we provide some examples of how this semantics works. In particular, we demonstrate that our requirement is not problematic: world can be made to have probability zero relative to world if we cut the epistemic accessibility relation between these worlds.
In Section 3, we define our modal notions of certain knowledge and of belief exceeding threshold , explain the motto “belief is willingness to bet,” and prove a number of properties of certain knowledge and this “betting” belief. For instance, we show that knowledge is and belief is not normal. We show a number of other thresholdspecific properties of betting belief as well. In particular, we see that the belief modality extends the minimal modal logic by way of certain schemes relating knowledge and belief.
We then introduce a formal modal language in Section 4, relate this language to the probabilistic notions of belief and knowledge, and introduce an epistemic neighborhood semantics for the language. We study the relationship between the neighborhood and probabilistic semantics. In particular, we introduce a notion of “agreement” between epistemic probability models and epistemic neighborhood models, the key component of which is this: an event is a neighborhood of a world if and only if the probability measure at satisfies . We use one of Scott’s theorems to prove that epistemic neighborhood models satisfying certain properties give rise to agreeing epistemic probability models for the threshold . This result we credit to Lenzen; however, we prove this result anew in a modern, streamlined form that we hope will make it more accessible. The main remaining open problem is to prove the analogous result for thresholds (i.e., find the additional sufficient conditions on epistemic neighborhood models we need to impose so as to guarantee the existence of an agreeing epistemic probability model for threshold ). Finally, we prove that epistemic probability models always give rise to agreeing epistemic neighborhood models.
In Section 5, we introduce a basic modal theory that is probabilistically sound. We adapt an example due to Walley and Fine [WF79] that shows is probabilistically incomplete. This leads us to add additional principles to , thereby producing the modal theory , our name for our modern reformulation of Lenzen’s modal theory of knowledge and belief (or, in Lenzen’s terminology, his theory of “acceptance” and belief). Using the results from Section 4, we prove that this logic is sound and complete for epistemic probability models using threshold . Regarding the semantics based on our epistemic neighborhood models, we prove that is sound and complete for the full class of these models and that is sound and complete for the smaller class that satisfies the additional Lenzenderivative properties needed to guarantee the existence of an agreeing probability measure for threshold .
Stated in an analogy: is to de Finetti’s axiomatization as is to the KPS/Scott axiomatization. However, do not be misled: de Finetti, KPS, and Scott considered qualitative probabilistic comparison, which is a binary notion based on a binary operator . See also [HI13] for a revival of this tradition. We, on the other hand, consider high probability, which is a unary notion based on unary operators we denote as and .
Another version of our main open question can be restated in the following syntactic form: given a threshold , find the additional principles that must be added to our probabilistically sound but incomplete base logic in order to obtain a probabilistically sound and complete logic for threshold . In our conclusion, we present some additional sound principles that might come up in this work, but we have not been able to find the probabilistically sound and complete axiomatization for thresholds .
Given the link between epistemic neighborhood models and epistemic probability models, our results may be viewed as a contribution to the study connecting two schools of rational decision making: the probabilist (e.g., [Kör08]) and the AIbased (e.g., [KT12]). We also hope that it will be of some use in future work on qualitative probability.
2 Epistemic Probability Models
Definition 2.1.
We fix a set of propositional letters. An epistemic probability model is a structure satisfying the following.

is a finite singleagent Kripke model:

is a finite nonempty set of “worlds” or “outcomes.” An event is a set of worlds. When convenient, we identify a world with the singleton event .

is an equivalence relation on . We let
denote the equivalence class of world . This is the set of worlds that agent cannot distinguish from .

assigns a set of propositional letters to each world .


is a probability measure over the finite algebra satisfying the property of full support: for each .
A pointed epistemic probability model is a pair consisting of an epistemic probability model and world called the point.
The agent’s uncertainty as to which world is the actual world is given by the equivalence relation . If is the actual world, then the probability the agent assigns to an event at is given by
(2) 
In words: the probability the agent assigns to event at world is the probability she assigns to conditional on her knowledge at . Slogan: subjective probability is always conditioned, and the most general condition is given by the knowledge of the agent. This makes sense because the right side of (2) is just , the probability of conditional on . Note that is always welldefined: we have by the reflexivity of and hence by full support, so the denominator on the right side of (2) is nonzero.
Example 2.2 (Horse racing).
Three horses compete in a race. For each , horse wins the race in world . The agent can distinguish between these three possibilities, and she assigns the horses winning chances of . We represent this situation in the form of an epistemic probability model pictured as follows:
When we picture epistemic probability models, the arrows of the agent are to be closed under reflexivity and transitivity. With this convention in place, it is not difficult to verify that ; that is, at , the assigns probability to the event that the winner is horse or horse .
The property of full support says that each world is probabilistically possible. Therefore, in order to represent a situation in which the agent is certain that horse can never win, we simply make the worlds inaccessible via .
Example 2.3 (Certainty of impossibility).
We modify Example 2.2 by eliminating the arrow between worlds and .
At world in this picture, there is no accessible world at which horse wins. Therefore, at world , the agent assigns probability to the event that horse wins: .
We define a language for reasoning about epistemic probability models.
Definition 2.4.
The language of (singleagent) probability logic is defined by the following grammar.
, 
We adopt the usual abbreviations for Boolean connectives. We define the relational symbols , , , and in terms of as usual. For example, abbreviates . We also use the obvious abbreviations for writing linear inequalities. For example, abbreviates .
Definition 2.5.
Let be an epistemic probability model. We define a binary truth relation between a pointed epistemic probability model and formulas as follows.
Validity of in epistemic probability model , written , means that for each world . Validity of , written , means that for each epistemic probability model .
3 Certainty and Belief
[Eij13] formulates and proves a “certainty theorem” relating certainty in epistemic probability models to knowledge in a version of these models in which the probabilistic information is removed. This motivates the following definition.
Definition 3.1 (Knowledge as Certainty).
We adopt the following abbreviations.

abbreviates .
We read as “the agent knows .”

abbreviates .
We read as “ is consistent with the agent’s knowledge.”
Theorem 3.2 ([Eij13]).
is an modal operator:

for each instance of a scheme of classical propositional logic.
Axioms of classical propositional logic are valid.

Knowledge is closed under logical consequence.

Knowledge is veridical.

Knowledge is positive introspective: it is known what is known.

Knowledge is negative introspective: it is known what is not known.

implies
All validities are known.

and together imply .
Validities are closed under the rule of Modus Ponens.
We define belief in a proposition as willingness to take bets on
with the odds being better than some rational number
. This leads to a number of degrees of belief, one for each threshold .Definition 3.3 (Belief as Willingness to Bet).
Fix a threshold .

abbreviates .
We read as “the agent believes with threshold .”

abbreviates .
We read as “ is consistent with the agent’s threshold beliefs.”
If the threshold is omitted (either in the notations and or in the informal readings of these notations), it is assumed that .
This notion of belief comes from subjective probability [Jef04]. In particular, fix a threshold . Suppose that the agent believes with threshold ; that is, . If the agent wagers dollars for a chance to win dollars on a bet that is true, then she expects her net winnings to be
dollars on this bet. This is a positive number of dollars if and only if . But notice that the latter is guaranteed by the assumption . Therefore, it is rational for the agent to take this bet. Said in the parlance of the subjective probability literature: “If the agent stakes to win in a bet on , then her winning expectation is positive in case she believes with threshold .” Or in a short motto: “Belief is willingness to bet.”
Remark 3.4.
Belief based on threshold or is trivial to express in terms of negation, , and falsehood . So we do not consider these thresholds here. Beliefs based on lowthresholds have unintuitive and unusual features. First, lowthreshold beliefs unintuitively permit inconsistency of the kind that an agent can believe both and while avoiding inconsistency of the kind that the agent can believe a selfcontradictory formula such as . (This suggests some connection with paraconsistent logic.) Second, the dual of a lowthreshold belief implies the belief at that threshold (i.e., ), which is unusual if we assign the usual “consistency” reading to dual operators (i.e., “ is consistent with the agent’s beliefs implies is believed” is unusual). Since lowthreshold beliefs have these unintuitive and unusual features, we leave their study for future work, focusing instead on thresholds .
The following lemma provides a useful characterization of the dual .
Lemma 3.5.
Let be an epistemic probability model.

iff .

iff .
We now consider a simple example.
Example 3.6 (Nonnormality).
In this variation, all horses have equal chances of winning and the agent knows this.
}
Recalling that an omitted threshold is implicitly assumed to be , the following are readily verified.

.
The agent believes the winning horse is among the three.
(The agent is willing to bet that the winning horse is among the three.)

.
The agent believes the winning horse is among any two.
(The agent is willing to bet that the winning horse is among any two.)

.
The agent believes the winning horse is not any particular one.
(The agent is willing to bet that the winning horse is not any particular one.)

.
The agent does not believe that both horses and do not win.
(The agent is not willing to bet that both horses and do not win.)
It follows from Items 3 and 4 of Example 3.6 that the present notion of belief is not closed under conjunction. This is discussed as part of the literature on the “Lottery Paradox” [Kyb61].^{2}^{2}2The usual formulation of the Lottery Paradox: it is paradoxical for an agent to believe that one of lottery tickets will be a winner (i.e., “some ticket is a winner”) without believing of any particular ticket that it is the winner (i.e., “for each , ticket is not a winner”). However, there is no reason in general that it is paradoxical to assign a conjunction a lower probability than either of its conjunctions. Indeed, if and are independent, then the probability of their conjunction equals the product of their probabilities, so unless one of or is certain or impossible, the probability of will be less than the probability of and less than the probability of .
We set aside philosophical arguments for or against closure of belief under conjunction and instead turn our attention to the study of the properties of the present notion of belief. One of these is a complicated but useful property due to Scott [Sco64] that makes use of notation due to Segerberg [Seg71].
Definition 3.7 (Segerberg notation; [Seg71]).
Fix a positive integer and formulas and . The expression
(3) 
abbreviates the formula
where is the disjunction of all conjunctions
satisfying the property that exactly of the ’s are the empty string, at least of the ’s are the empty string, and the rest of the ’s and ’s are the negation sign . We may write as an abbreviation for (3). Finally, let
We also allow the use of in a notation similar to (3).
The formula says that the agent knows that the number of true ’s is less than or equal to the number of true ’s. Put another way, is true if and only if every one of the agent’s epistemically accessible worlds satisfies at least as many ’s as ’s. The formula says that every one of the agent’s epistemically accessible worlds satisfies exactly as many ’s as ’s.
Definition 3.8 (Scott scheme; [Sco64]).
We define the following scheme:
(Scott) 
If , then is . Note that (Scott) is meant to encompass the indicated scheme for each positive integer .
(Scott) says that if the agent knows the number of true ’s is less than or equal to the number of true ’s, she believes with threshold , and the remaining ’s are each consistent with her threshold beliefs, then she believes one of the ’s with threshold . Adapting a proof of Segerberg [Seg71], we show that belief with threshold satisfies (Scott).
We report this result along with a number of other properties in the following proposition.
Theorem 3.9 (Properties of Belief).
For , we have:

.
Belief is not closed under logical consequence.
(So is not a normal modal operator.)

.
Belief is not veridical.

.
What is known is believed.

.
The propositional constant for falsehood is not believed.

.
The propositional constant for truth is believed.

.
What is believed is known to be believed.

.
What is not believed is known to be not believed.

.
Belief is closed under known logical consequence.

If , then .
Highthreshold belief is consistent: belief in implies disbelief in .

.
For midthreshold belief, if is consistent with the agent’s beliefs and is consistent with her knowledge, then she believes .

.
Midthreshold belief satisfies (Scott).
Proof.
We consider each item in turn.

Given and integers and such that , we define as the modification of the model of Example 3.6 obtained by changing as follows:
Since , it follows that
Therefore, we have

For defined in the proof of Item 1, we have

implies . Hence .

. Hence .

. Hence .

implies . To show that , we must prove that
To show this, we prove that . So choose . Since is an equivalence relation, we have
which implies . The result follows.

The argument is similar to that for Item 6, though we note that implies .

We assume that and . This means that and . But then it follows that as well, which is what it means to have .

Assume and . Then . So . The result therefore follows by Lemma 3.5.

We prove something more general. Assume and . By Lemma 3.5, it follows that . Let us assume further that . This means
which implies there exists . Since by full support, it follows that
That is, .

Again, we prove something more general. We assume plus the following:
(4) (5) (6) We recall the meaning of (4): for each , the number of ’s true at is less than or equal to the number of ’s true at . It therefore follows from (4) that
(7) Outlining an argument due to Segerberg [Seg71, pp. 344–346], the reason for this is as follows: we think of each world as being assigned a “weight” . A member of the sum on the left of (7) is just a total of the weight of every that satisfies ; that is,
Assumption (4) tells us that for each , the number of totals on the left of (7) to which contributes its weight is less than or equal to the number of totals on the right of (7) to which contributes its weight. But then the sum of totals on the left must be less than or equal to the sum of totals on the right. Hence (7) follows.
Having established (7), we now proceed further with the overall proof. By (5), we have . Applying (6) and Lemma 3.5, we have for each . Hence
That is, the sum of the ’s must exceed . Since each member of this member sum is nonnegative, it follows that at least one member must exceed . That is, there exists such that . Hence . ∎
4 Epistemic Neighborhood Models
The modal formulas and were taken as abbreviations in the language of probability logic. We wish to consider a propositional modal language that has knowledge and belief operators as primitives.
Definition 4.1.
The language of (singleagent) knowledge and belief is defined by the following grammar.
We adopt the usual abbreviations for other Boolean connectives and define the dual operators and . Finally, the formula
and its abbreviation are given as in Definition 3.7 except that all formulas are taken from the language .
Our goal will be to develop a possible worlds semantics for that links with the probabilistic setting by making the following translation truthpreserving.
Definition 4.2 (Translation).
For , we define as follows.
Since we have seen that the probabilistic belief operator is not a normal modal operator (Theorem 3.9(1)), we opt for a neighborhood semantics for [Che80, Ch. 7] with an epistemic twist.
Definition 4.3.
An epistemic neighborhood model is a structure
satisfying the following.

is a finite singleagent Kripke model (as in Definition 2.1). As before, we let
denote the equivalence class of world . This is the set of worlds the agent cannot distinguish from .

is a neighborhood function that assigns to each world a collection of sets of worlds—each such set called a neighborhood of —subject to the following conditions.
 (kbc)

.
 (kbf)

.
 (n)

.
 (a)

.
 (kbm)

.
A pointed epistemic neighborhood model is a pair consisting of an epistemic neighborhood model and a world in .
An epistemic neighborhood model is a variation of a neighborhood model that includes an epistemic component . Intuitively, is the set of worlds the agent knows to be possible at and each represents a proposition that the agent believes at . The condition that be an equivalence relation ensures that knowledge is closed under logical consequence, veridical (i.e., only true things can be known), positive introspective (i.e., the agent knows what she knows), and negative introspective (i.e., the agent knows what she does not know).
Property (kbc) ensures that the agent does not believe a proposition that she knows to be false: if </