2.1. Bilateralism

Bilateralism is a type of inferentialism that has been salient in the last few years (Rumfitt, ²⁰⁰⁰; Restall, ²⁰⁰⁵; Ripley, ²⁰¹⁵). Inferentialism is a philosophical stance that understands the meaning of logical constants in terms of their inferential role (Dummett, ¹⁹⁹¹; Murzi & Steinberger, ²⁰¹⁷). Bilateralism spells out the inferential role of logical constants in terms of their acceptance and rejection conditions and in doing so it opposes to unilateralism by stating that rejecting a statement cannot be defined in terms of accepting its negation. ² That means that given a proposition φ, one can either accept it or reject it. Nevertheless, accepting ¬φ –and here is how it opposes unilateralism– is not necessarily the same as rejecting φ, because there might be cases where one neither wants to accept φ nor reject φ. Rejection, in this view, has its own force, different and irreducible to acceptance.

There are two main branches of bilateralism, the one that focuses on natural deduction calculi (Rumfitt, ²⁰⁰⁰; Smiley, ¹⁹⁹⁶; Incurvati & Schlöder, ²⁰¹⁷) and the one that focuses on sequent calculi (Restall, ²⁰⁰⁵; Ripley, ²⁰¹⁵). They are both concerned with the meaning of logical constants in terms of their acceptance and rejection conditions and they both agree on the fact that both acceptance and rejection must be taken as primitives. The main difference is that while the first branch builds a whole new presentation of classical logic with these two concepts embedded in the proof theory, the second one takes a classic sequent calculus of a logic (or some proof-theoretic presentation with multiple conclusions) and proposes a new way of reading valid inferences. I will focus on this second branch of bilateralism, the one involving sequent calculi.

First, we should start by becoming familiar with the standard sequent calculus for Classical Propositional Logic (CPL) with its operational rules (the rules for logical connectives, where material conditional, φ→ψ, can be defined as ¬φ˅ψ) and its structural rules (the rules where no appeal to any logical connective in particular is made). Let 𝔏 be a propositional language with {¬,˅,˄} representing the usual negation, conjunction and disjunction. Lowercase Greek letters represent formulas of 𝔏 and capital Greek letters represent sets of formulas of 𝔏:

Figure 1 

In order to understand this calculus in a bilateralist fashion we can start by reading David Ripley’s words:

What it is for a bunch of premises to entail a bunch of conclusions is that if you assert the premises and deny the conclusions, then you’re out of bounds. (Ripley, ²⁰¹⁵, p. 28)

This is the main concept of this version of bilateralism. Now, let’s see how we can read this bilateralist slang in relation with concrete inferences. A bilateralist wants us to read a sequent in light of which collection of formulas are incoherent to accept and reject. So, identity can be read as follows:

• Id: It is incoherent to accept φ and reject φ.

Weakening can be read as follows:

• L-Wk: If we already are in an incoherent position by accepting all statements in Γ and rejecting all statements in Δ, then we will still be in an incoherent position if we also accept φ.

• R-Wk: If we already are in an incoherent position by accepting all statements in Γ and rejecting all statements in Δ, then we will still be in an incoherent position if we also reject φ.

On the other hand, the Cut rule can be read as follows:

• Cut: If accepting all statements in Γ and rejecting all statement in Δ as well as rejecting φ is incoherent and accepting all statement in Γ and φ and rejecting all statements in Δ is also incoherent, then the incoherence must rely on accepting all statements in Γ and rejecting all statements in Δ.

We might as well suppress the context in order to facilitate the reading and just focus on the main formulas. For conjunction, disjunction, and negation the rules should be understood in the following way:

• L-¬: If it is incoherent to reject φ then it is also incoherent to accept ¬φ.

• R-¬: if it is incoherent to accept φ then it is also incoherent to reject ¬φ.

• L-˄: If it is incoherent to accept φ (or ψ), it is also incoherent to accept φ˄ψ.

• R- ˄: If it is incoherent to reject φ and it is incoherent to reject ψ, it is also incoherent to reject φ˄ψ.

• L-˅: If it is incoherent to accept φ and it is also incoherent to accept ψ, then it is also incoherent to accept φ˅ψ.

• R-˅: If it is incoherent to reject φ (or ψ), it is also incoherent to reject φ˅ψ.

As it is easy to see, the bilateralist reading is a normative stance. Yet we can strengthen this normative reading by means of an elucidation of what it means to be incoherent (or to be out of bounds, as Ripley paraphrases it). As it was mentioned earlier, the literature leaves this to the reader’s imagination. Does being incoherent mean that the agent is being irrational? Is it somehow forbidden? Does it mean that the agent should revise her beliefs? And how do beliefs enter this picture?

What bilateralism is saying is that there is something wrong with the set of sentences that the agent accepts and the set of sentences the agent rejects. So, there is an agent and there is some mistake that must be avoided or corrected ³ . To understand what this mistake amounts to, it might be useful to know when it is correct to accept or to reject a certain proposition and when it is not. Of course, there are several considerations involved in judging whether a speech act like that is correct or not. There might be pragmatic or semantic considerations, but if we focus only on logical considerations, there are two main aspects. On one hand, there’s entailment: we shouldn’t reject the conclusion of a valid argument if we are certain of its premises. On the other, is whether we are actually certain enough of the premises. Most of the time, certainty is not the place we are at when reasoning. And that is why I think probabilism might be helpful. Because, even if we are not completely certain of our premises, we might have some information on how uncertain we are about the sentences we are stating, which can easily be translated into degrees of belief.

In order to understand the elucidation of bilateralism I am proposing and to understand how to avoid the problems that might stand in the way, we will devote the next section to learn some probabilities. I will explain what probabilism is and what probabilities are. Then I will jump to some epistemic paradoxes that come together with probabilism.

2.2. Probabilism

Probabilism states that it is possible to model degrees of belief by means of probability functions. In order to understand it, I write down here the axioms of probability theory and some theorems that will be relevant for the discussion that will follow. ⁴

This is a possible axiomatization of classic probability theory: ⁵

(Ax1) 0 ≤ P(φ) ≤ 1.

(Ax2) If ⊢ φ, then P(φ) = 1.

(Ax3) If φ and ψ are logically inconsistent, then P(φ˅ψ) = P(φ) + P(ψ).

Some relevant theorems that might come in handy are:

(T1) P(¬φ) = 1- P(φ).

(T2) If φ ⊢, then P(φ) = 0.

(T3) If φ ⊢ ψ then P(φ) ≤ P(ψ).

(T4) P(φ˅ψ) = P(φ) + P(ψ) - P(φ˄ψ)

Finally, conditional probability: ⁶

(C) P(φ|ψ) = P(φ∧ψ)

P(ψ)

Our first axiom (Ax1) states that the range of probabilities is the unit, that is, the real numbers between 0 and 1. (Ax2) states that if a formula is a logical truth, its probability will always be 1. We have two pieces of information about disjunction. (Ax3) states that when two disjuncts are logically inconsistent (or mutually exclusive) the probability of the disjunction equals the sum of the probability of each disjunct. (T4) explains what happens when the disjuncts are not logically inconsistent. In particular, you can get the probability of the disjunction by taking the probability of the first disjunct happening plus the probability of the second disjunct happening minus the probability of both happening together (for the probability of both happening together is already contained in each probability). The probability of conjunction can be extracted by pure algebra from (T4) and conditional probability (C). Conditional probability is read as follows: the probability of φ given ψ can be calculated as the ratio of the probability of φ and ψ happening together and the probability of ψ happening. If you move the terms in the equation, you get that the probability of conjunction can be obtained by making the product of P(φ|ψ) and P(φ).

The probability of a negation in this classical frame is the only compositional probability and it is defined as in (T1). (T1) and (Ax2) entail (T2), which says that if a formula is a logical falsity, its probability will always be 0. Finally, (T3) states that when facing a valid inference with one premise and one conclusion, the probability of our conclusion must be equal to or greater than the probability of the premise.

A natural way to translate these axioms and theorems into an interpretation of how our beliefs should interact is as follows: we interpret our degrees of belief as real numbers between 0 and 1, where believing in degree 1 is having absolute certainty that some formula is true and believing in degree 0 is being absolutely certain that some formula is false, as in tautologies and contradictions. We usually do not believe things in an absolute way. For all those non-absolute cases, we ascribe our beliefs some higher or lower value between 1 or 0. Then we simply interpret our degrees of belief in any formula as respecting the probability axioms.

The axioms stated above entail certain other theorems about the relation between logic and probability. These theorems together with the probabilistic interpretation entail the following classic result stated in Adams (1996), which we will refer to as Adams’ theorem. Take D as a disbelief function defined as D(φ) = 1 - P(φ), then:

If φ1, …, φn ⊢ψ, then D(ψ) ≤ D(φ1) +…+D(φn).

That is, our disbelief in the conclusion of a valid inference must be less or equal to the sum of the disbeliefs of its premises. Field (2015) generalizes this result to inferences with multiple conclusions:

If φ1, …, φn ⊢ψ1, …, ψm, then 1 ≤ D(φ1) + … +D(φn) + P(ψ1) + … +P(ψm).

This means that if the inference φ1,…, φn ⊢ψ1, …, ψm is valid, then the sum of one’s degree of disbelief in φ1,…, φn plus the sum of one’s degree of belief in ψ1, …, ψm has to be greater than or equal to 1.

These results can be understood as constraints on rationality. Every consistent assignment of probabilities for any valid inference is restricted by Adams’ theorem, which means that there are no consistent probability assignments where the number thrown by the disbelief function on ψ (for the case of single conclusions) can be higher than the sum of the disbeliefs of the premises. Or as the multiple conclusions extension result shows, there is no consistent assignment of probabilities where the sum of the disbeliefs of the premises and the probabilities of the conclusion can be smaller than 1. Dorothy Edgington (¹⁹⁹⁷) calls this restriction the constraining property. This property can be of use for our quest, because if we are asking ourselves what it means exactly to be coherent, and we understand coherence as having beliefs that respect the probability axioms, then the set of sentences we accept will be coherent if and only if our degrees of belief in those sentences respect the constraining property. The constraining property will also give us information about invalid inferences. As Edgington puts it:

[...] if an argument is invalid [...] then there is an assignment of probabilities, consistent with the laws of probability, which give its premises probability 1 and its conclusion probability 0. So it does not have the constraining property. ...] An argument has the constraining property if and only if it is valid. (Edgington, ¹⁹⁹⁷, pp. 300-301)

If we follow Field’s proposal, then for every valid inference it will be incoherent to accept all its premises and reject all its conclusions when our degrees of belief are consistent with the probability axioms. Yet, it will be coherent to accept all the premises and reject all the conclusions of an invalid inference because that’s exactly what it means for an inference not to have the constraining property. ⁷ Field is aiming at something different than we are. He is digging into what it means for two agents to differ on what is valid and how every agent determines what is valid and what is not (see Field, ²⁰⁰⁹, ²⁰¹⁵). Yet, his project is related to ours because he chooses to address the problem by searching for some normative principle that restricts our credence when facing valid inferences. ⁸ For that, he analyses two different scenarios: when talking about partial beliefs he uses Adams’ theorem to restrict our degrees of belief. When talking about full belief and full disbelief he proposes to adjust the bilateralist approach to this framework. Field says the following:

The idea is that the sequent A₁, …, A_n ⇒ B₁, …, B_m directs you not to fully believe all the A_i while fully disbelieving all the B_j. (Field, ²⁰¹⁵, p. 49)

So that:

To regard the sequent A₁, …, A_n ⇒ B₁, …, Bm as valid is to accept the consequent of (the Adams’ thesis extended to multiple conclusions) as a constraint on degrees of belief. (Field, ²⁰¹⁵, p. 49) [italics are mine].

Field immediately points out the following:

Restall doesn’t explicitly say ‘fully’, but I take it that that’s what he means: otherwise the classically valid sequent A₁, …, A_n ⇒ A₁˄ … ˄ A_n would be unacceptable in light of the paradox of the preface. (Field, ²⁰¹⁵, p. 49)

We could end our quest here and just conform with these two separate frameworks to understand constraints on validity. But we might want to know as well, when the degree of belief in φ is high enough so as to start acting as if we believed φ in degree 1. That is, when it is rational to have a full belief in φ, and then accept φ. And someone might ask why this bridge between partial and full belief is important when talking about validity, for we know that this bridge is, at least at a first strike, bound to fail because of epistemic paradoxes. Well, here I think the way Adams opens his book, A Primer on Probability Logic (¹⁹⁹¹), might be of use to motivate what is left of the quest. He says:

It is often said that deductive logic is concerned with what can be deduced with logical, mathematical certainty, while if probability has a part in logic at all, it falls into the province of Inductive Logic. This has some validity, but it also overlooks something important. That is that deductive logic is usually supposed to apply to reasoning or arguments whose premises aren’t perfectly certain, and whose conclusions can’t be perfectly certain because of this. (Adams, ¹⁹⁹⁶, p. 1)

So, it would be good to be able to set a bridge between partial and full belief and these two different constraints on validity in order to be able to say something about these contexts in which we are not perfectly certain about our premises. I think this quest is well-motivated when we are asking about bilateralist statements, because, as we saw earlier, these statements have a great normative force in them, and the main use of normative statements is in real contexts of reasoning, which are the contexts where agents tend to be uncertain of the premises they are using. At the same time, if we can actually state when it is, according to our degrees of belief, that we can accept or reject a premise or a conclusion without being incoherent, then the elucidation we were looking for will be in our hands. We can already understand bilateralism in an abstract way. But by elucidating the concepts of acceptance, rejection, and incoherence in terms of degrees of belief, we might be able to actually use bilateralist statements in concrete contexts of reasoning. Let’s not forget that bilateralism states that we get to learn the meaning of our logical constants in usage. If we can actually instantiate bilateralist statements in concrete contexts, then the idea that we learn how to use logical connectives because we get to understand when it is out of bounds to accept them or to reject them takes an interesting new dimension.

The problem, as we said before, is that it is not that obvious how to settle this bridge. A natural first take might be by means of the Locke thesis, the one that states that there is some r for all φ, such that if P(φ) ≥ r then we can accept φ. The problem is that under a few minimal amount of natural assumptions, Locke’s thesis is haunted by epistemic paradoxes.

This is the point where trouble arises for our investigation. Mainly because as Field explains in the quote above, epistemic paradoxes seem to indicate a dead end. Either we take acceptance as full belief and rejection as full disbelief, or we simply forget about the probabilistic interpretation of bilateralism and we just stick to plain probabilism. I will show that we can fully merge these two philosophical proposals without getting to this dead end.

2.3. The paradoxes

As it was explained above, it is not easy to make probabilism compatible with a consistent probabilistic reading of acceptance and rejection. Mainly because under certain minimal conditions, some paradoxes arise. In this section, I will explain why it is not that easy to fix a threshold for acceptance if we understand our certainty or uncertainty in a probabilistic fashion.

The Lottery paradox (Kyburg, ¹⁹⁶¹) is settled in a lottery scenario (as expected) where one is willing to accept that for any given concrete ticket, that ticket will lose, but one is also aware that some ticket will win, or which is the same, one is willing to reject that every ticket will be a losing one. If we think of it as an inference, one is willing to accept every premise that states that each ticket will lose, but one is unwilling to accept the conjunction of every premise as the conclusion.

What the lottery paradox comes to show us is that we cannot hold these three principles about rationality altogether:

(P1) Rational acceptance is closed under logical consequence.

(P2) Degrees of belief must respect the probability axioms.

(P3) Locke’s thesis: There is a threshold 1 > r > 0.5 for every proposition φ, such that you should believe φ if and only if P(φ) ≥ r.

Locke’s thesis tries to capture the reasonable idea that, for example, if we believe that P(φ) = 0.99, we would like to accept φ in any scenario. Unfortunately, the lottery paradox shows us that this is not that simple. In particular, let’s imagine a scenario where we have 100 tickets, we believe that each ticket has a probability of winning of 0.01, which is equivalent to saying that the probability of each ticket of losing is 0.99. ⁹ Nevertheless, even though we have good reasons to accept every sentence of the form “The ticket n will lose”, we also know that some ticket will win (because that is how the lottery works), so we cannot accept the conjunction of every statement, namely every ticket will lose. Thus, we cannot close our beliefs under conjunction introduction. To see it in a more graphic way, even if our degree of belief in every premise is extremely high, we are unwilling to accept the conclusion of this argument that only uses conjunction introduction:

There are only 100 tickets ¹⁰

Ticket 1 will lose

Ticket 2 will lose

.

.

.

Ticket 100 will lose

Therefore, every ticket will lose.

That is the lottery paradox, we cannot have (P1), (P2) and (P3) together.

On the other hand, the preface paradox (Makinson, ¹⁹⁶⁵) is described in the context of a book, where the scholar who wrote it is willing to accept any of its sentences but is also willing to accept there might be some mistake. Again, she is willing to accept each statement, but not all of them together. Just to be clear, the work I am constantly quoting (Field, ²⁰¹⁵) talks about the preface paradox but I will focus on the lottery paradox instead, and I will explain why. The preface paradox falls under the same family as the lottery paradox. These paradoxes happen to be a problem for our approach because when trying to develop an apparatus for partial belief, we keep on finding sets of inconsistent beliefs that we wish would be consistent. Each paradox seemingly happens because one is willing to accept the truth of every statement of a given set but wouldn’t want to accept the conjunction of all of them, or similarly, one is willing to accept that the set of statements is false. Field chooses to talk about the preface paradox. I choose to talk about the lottery paradox. Even if they are not essentially the same phenomenon, I don’t think that the ways in which they differ will matter in what follows, and the lottery paradox is presented in an easier way to work with. That is, it is easier to think of degrees of belief when we know that there are 100 tickets than when we are talking about a book, where one doesn’t necessarily accept each sentence by itself, but might accept statements made of several sentences, and where the degrees of belief in each statement might vary or be codependent.

In the next section, I will provide a survey on different possible thresholds that will fail to work, and then explain how to settle a contextual threshold for acceptance.

3.1. The thresholds that won’t work

When it comes to finding a threshold for acceptance and rejection, some of the most obvious proposals fail to solve the problem. The first proposal would come to be the full belief/full disbelief one, where r = 1. Believing in degree 1 for acceptance and believing in degree 0 for rejection. Then what bilateralism would tell us is that when facing a valid inference, it is incoherent to believe in degree 1 all the premises and believe in degree 0 all the conclusions. That makes sense. Yet, it sounds quite weak for our quest. One of the main motivations of this work, as well as Adams’ and Field’s, was expanding the span of logic for everyday reasoning scenarios in which we don’t necessarily have certainties about our premises. If we accept 1 and 0 as our thresholds, then we end up with a really strong norm for acceptance and rejection that takes us far from our goal of trying to guide us in an actual context of everyday reasoning, because as we said on many occasions, we don’t usually have absolute certainty in the propositions with which we reason.

Maybe, we could try setting our threshold as high as we can imagine without getting to 1 and 0. ¹¹ Then, r = 0.99 should be a natural candidate. But as we have seen, the lottery paradox immediately discards 0.99 candidates. Naturally, we could set our threshold to r = 0.999 and avoid this lottery scenario. Yet again an extra-large lottery will arise, this time with 1000 tickets, and so on. So, r = “extremely high but not exactly 1” as a threshold is also ruled out. What should we do then? Is there any other option to consider?

The problems will arise no matter the threshold we choose. For extremely high cases, the lottery paradox will be the witness of our failure. Even if we want to try some critically low threshold, problems will still emerge. For example, another natural candidate would be to define our threshold as asserting φ when r > 0.5. That is, the lowest threshold possible, yet higher than being indifferent about our propositions. ¹² At first glance, we can see why this fails. Take any two probabilistically independent formulas φ and ψ and take that P(φ) = P(ψ)=٠.٥١. Then, P(φ˄ψ) = 0.2601. That means that it is coherent to accept all the premises of our valid inference (conjunction introduction) yet reject its conclusion, for the degree of belief in our premises is higher than 0.5 and our degree of belief in our conclusion smaller. So, 0.5 is also ruled out.

These failed attempts leave us with some understanding of why Field might have given up on the project of finding a threshold for acceptance and rejection in the context of degrees of belief. But I think it is possible to find a different answer. If there cannot be an absolute threshold that works for every argument, then we might find a contextual threshold for each valid inference. In the next section, I will present the tools needed to mend this failure scenario in which it seems impossible to define a threshold for acceptance and rejection.

3.2. The threshold that will work: a contextualist solution

I will show how to solve the lottery paradox within a contextualist framework, which amounts to a reinterpretation of Locke’s thesis so that it doesn’t fall prey of the lottery paradox.

Contextualism here can be simply understood as the statement that we won’t be looking for a universal threshold for acceptance or rejection, but instead, we can look for a threshold for acceptance or rejection in the context of each argument depending on which are our degrees of belief in the involved propositions. ¹³ That is, the threshold for acceptance might vary between an argument with two premises (take p1,p2 ⊢ p1˄p2) and an argument with a thousand (take p1, p2, …, p1000 ⊢ p1˄p2˄… ˄p1000), even if the argument with a thousand premises contains the two premises of the first argument, as well as it might vary depending on our degrees of belief on the premises and conclusions.

Take P to be the classic probability function that we already know, Bel to be a full belief ¹⁴ function that commands us to believe a sentence φ (or for our purposes to accept φ), and r to be the number between 0 and 1 that will become our threshold. On Leitgeb (²⁰¹⁴)’s proposal for solving the lottery paradox, he states the following:

[...] we need to distinguish a claim of the form ‘there is an r < 1 . . . for all P ...’ from one of the form ‘for all P… there is an r < 1…’ As we are going to see, the difference is crucial: while it is not the case that there is an r < 1; such that for all P (on a finite space of worlds) [...] there is an r < 1 such that the same conditions are jointly the case. (^Leitgeb, 2014, pp. 133-134)

Even if it’s true that there is no threshold r that can satisfy the logical closure of Bel (P1), the probability axioms (P2), and Locke’s thesis (P3), it is also true that by modifying Locke’s thesis a little bit, we can find some threshold r for every valid inference that satisfies (P1) and (P2). The trick is to change the scope of the quantifier in Locke’s thesis. Now we won’t ask for one threshold r for every valid inference, but for every valid inference one threshold r. As we said, we turn to a contextual threshold. If we think about it, contextualism here seems pretty justified, for we cannot be as certain about the conclusion of some inference in which we have little uncertainty about two of its premises, compared to our uncertainty about the conclusion of an inference where we have little uncertainty about a thousand of its premises. To paraphrase Edgington (¹⁹⁹⁷), accumulating uncertainties makes the conclusion inherit all the uncertainty of each premise.

So, how do we set this contextual threshold? ¹⁵ For that we will use conditional probability. We will say a proposition φ is P-stable if and only if learning any other proposition ψ compatible with φ, does not lower the agent’s credence in φ to a degree less than 0.5, that is, P(φ|ψ) > 0.5.

Now, imagine a truth table, where each line of the truth table is a possible world. We can name each line n with w _n. Then, if we have two propositions expressed by the sentences φ and ψ, we get to have a line w₁, which would come to be a world in which the sentences φ and ψ are true, another world, w₂ in which φ is true and ψ is false, and so on. Now imagine you have all the possible worlds given by the set of all the propositions you are opinionated in (this is a huge truth-table, we will stick to two propositions for simplicity). By using the power set operation, you can construct a set of all subsets of possible worlds, you don’t need all of them, but you might want to have some different subsets. For example, you might want to express your degree of belief in the sentence that states that either w₁ or w₂ is true. That is one set that contains two possible worlds. You might want to express your degree of belief in the sentence that either w₁ or w₂ or w₃ is true and so on. That set of subsets of worlds is called a sample space of worlds. Then, over that sample space, you might want to define the strongest sentence you believe, that is the sentence that excludes the greatest number of disjunctions of lines of your truth-table. That sentence will be called φ_w. With all this in mind, we can define P-stability in a more formal way:

Definition 1. Let P be a probability measure on a sample space of words W, and φ ⊆ W: For all φ, φ is P-stable if and only if for all ψ ⊆W, such that φ and ψ are not incompatible and P(ψ) > 0, then P(φ|ψ) > 0.5.

Then, Bel satisfies (P1), P satisfies (P2) and both Bel and P satisfy this contextual version of Locke’s thesis: for all φ, Bel(φ) if and only if P(φ) ≥ P(Bw) > 0.5 (Leitgeb, 2014). This way, what we get is at least one possible threshold for every consistent probability distribution. Let’s consider an example. Take a distribution with two formulas φ and ψ and a probability measure like this:

A way to calculate the threshold is the following: we can start by ordering our probability measures top to bottom from the higher probability to the lowest, as they stand on the table above. Then if P(φ˄ψ) > P(φ˄¬ψ) + P(¬φ˄ψ) + P(¬φ˄¬ψ), then {w ₁} is the first and strongest P-stable set of propositions. We can see in this example that this is not the case, so we should move to the next: if P(φ˄¬ψ) > P(¬φ˄ψ) + P(¬φ˄¬ψ), then, {w ₁ , w ₂} is the strongest set, but again, it’s not the case. The case is that P(¬φ˄ψ) > P(¬φ˄¬ψ), so our strongest set of propositions is the one containing {w ₁ , w ₂ , w ₃}, which sets the threshold at r = 0.9 (the sum of the probabilities of each w _i in our set).

In this sense, Leitgeb’s P-stability theory mends the Lottery problem, because the Lottery scenario is one in which we don’t have P-stable beliefs. Because even if we learn that every ticket except tickets 1 and 2 will lose, our belief in the proposition “The ticket 1 will win” will still be 0.5. That is, the only P-stable proposition is the disjunction that states that some ticket among the one hundred tickets will win, and the probability of that statement must be 1.This means that the only degree of belief that would be coherent to have in order to accept a proposition of our lottery scenario is 1. Hence, we shouldn’t accept any proposition of our lottery if we believe it in a degree lower than 1.

Leitgeb sets a contextual threshold for partial belief that allows us to avoid the lottery paradox. My proposal is to import this theory into the bilateralist framework in order to understand acceptance where Leitgeb understands outright belief. This is quite a subtle move. It’s just adding something else to bilateralism.