From Epistemic Norms to Logical Rules: Epistemic Models for Logical Expressivists

In this paper I construct a system of semantics for classical and intuitionistic propositional logic based on epistemic norms governing belief expansion. Working in the AGM-framework of belief change, I give a generalisation of Gärdenfors’ notion of belief systems which can be defined without reference to a logical consequence operator by using a version of the Ramsey Test. These belief expansion systems can then be used to define epistemic models which are sound and complete for either classical or intuitionistic propositional logic depending on which of the two notions of epistemic validity, identified by Levi and Arló-Costa, is used. Finally, I offer a discussion on how these results can be understood as providing a model theory within the framework of logical expressivism.


Introduction
What I want to do in this paper is to lay the groundwork for how to provide a system of semantics for logic in terms of belief change.This idea has already been somewhat explored by Peter Gärdenfors when it comes to counterfactual conditionals, but that investigation proceeds on the basis of an already assumed base classical logic.My first aim here is to show that we can already apply the basic idea of explaining logical connectives in terms of the dynamics of belief to explain our most basic logical systems.As it turns out, belief expansion provides a natural basis for explaining the connectives of propositional logic.
My second goal is to give a normative interpretation of the structures I construct to interpret logical systems.The idea is that the clauses defining belief expansion systems and their constituent states should be understood as explicit formulations of norms which govern how we ought to update our beliefs in the light of new information.In this way, epistemic semantics of the kind I propose are essentially a model theory which fits well with the ideas of Robert Brandom's brand of logical expressivism.By finding a logic which is sound and complete for a structure constructed from the norms of some epistemic practice we have, essentially, found the logical rules implicit in that practice.
In the first section I will give a brief overview of the celebrated AGM-framework for belief dynamics and briefly explain how Gärdenfors used it to deal with counterfactual conditionals.Then, in the second section, I will take the central idea of his approach and use it to define belief expansion systems.These are intended to capture the dynamics of belief for a propositional language together with expansions of our belief states.The reason why I focus on expansions rather than some more complicated epistemic operation is essentially that it turns out to be all we need to explain the basic logical connectives.And in order for the project of explaining more complicated logical systems in terms of normative practice to get off the ground, I need to show that we can do so for the kind of basic logics that more complicated systems are built upon.
In the third section I will give a brief discussion on the two notions of validity which have cropped up in discussions of models based on the AGM-framework.I show that the common observation that negative validity favours classical logic whereas positive validity favours intuitionistic logic still holds in this more general setting.In the fourth section I prove that the belief expansion structures constructed earlier provide a sound and complete semantics for either classical or intuitionistic logic depending on the preferred notion of validity.Section five consists of a brief comparison to a kind of model for intuitionistic logic based on ideas of belief expansion already constructed by Gärdenfors.Finally, in the last section I outline the philosophical interpretation of these results as explicitly expressing the logic already implicit in the practice governed by the epistemic norms in question.Further, I discuss the connection to logical expressivism which is the second aim of this paper.In short, the point is to provide a proof of concept for the idea that we can construct a model theory in terms of epistemic norms and give a brief argument that this approach is quite congenial to the philosophical aims of logical expressivism.

Belief Systems and Epistemic Semantics
I will begin with a brief exposition of the AGM-framework, named for its originators Alchourrón, Gärdenfors, and Makinson [1] 1 , and the idea of such a model can be used for semantics.Their goal is to provide a theory about the dynamics of belief, in the sense of studying how epistemic states develop when we acquire new information.The fundamental building block of the theory is an epistemic state, or belief state.These are to be modelled as the set of sentences, of some language L, which the subject believes in.In this paper, I will only be considering propositional languages.
In order to model specifically rational belief, Gärdenfors [6, pp. 22-24] distinguishes belief states as those sets of L-sentences which are closed under a consequence operator , encoding some extension of classical logic.Now, given a belief state K , we can ascribe the three fundamental epistemic attitudes towards a sentence P.
(i) P is accepted if and only if P ∈ K .
(ii) P is rejected if and only if ¬P ∈ K .
(iii) P is indeterminate if neither P ∈ K nor ¬P ∈ K .Further, he distinguishes the inconsistent belief state K ⊥ which consists of every sentence of L. A collection K of belief sets is called a belief system and we will assume that every system includes K ⊥ .These correspond to the potential belief states of the person in question.
The key insight of the AGM-framework is how it expresses belief change.When we encounter a new piece of information, encoded as a sentence P, it will transform our current belief state into a new one.In the most basic case, when P is indeterminate at our present state K , this is an expansion of our beliefs into a new state K + P .We can think of this as a function + : K × L → K taking a belief set and a sentence and then mapping them to their corresponding expansion.This idea, that belief change can be thought of as certain functions on belief systems, is the core of the AGM-framework.The other two central operators of the framework are revision, written K * P , which handles cases when P might have been rejected at K and contraction, written K − P , which deals with inputs forcing us to disavow our belief in P.
Fundamental to a system of semantics based on belief dynamics, originating in Gärdenfors [5] 2 , is that we can think of a belief system K as a kind of model of the formal language L. Since the belief states of K consist of sets of sentences, we can use epistemic attitudes to P at K to get a kind of satisfaction or forcing relation which then can be extended to the entire belief system.Now, what Gärdenfors was interested in was to provide a semantics for the language of propositional logic extended with a general, possibly counterfactual, conditional >.Specifically, the idea was to expand the notion of acceptance to sentences of the form P > Q by considering acceptance of Q at the state resulting from revision by P.This was done by imposing a condition that's come to be known as the Gärdenfors Ramsey Test.
As the name indicates, this builds on an idea of Frank Ramsey [8, pp. 154-155], namely that if we're unsure of whether to accept P then the conditional P > Q should be accepted if the result of adding P to our present beliefs would also result in adding Q.Using revision as the operator for hypothetically adjoining P to K , Gärdenfors extends this idea from situations where we're unsure about P to ones where we may hold any epistemic attitude towards it. 3ärdenfors [6, p. 148] then proceeds to present the following semantics for the language L < .Let K be a belief system over the extended language which is closed under expansions and let * be a revision operator on K which satisfies (G RT ).Then a L < -sentence P is valid in (K, * ) if ¬P is not accepted at any consistent belief state K ∈ K.It is valid simpliciter if it is valid in every belief revision system (K, * ).Using this definition, he goes on to prove that this system is sound and complete for a logic of conditionals he calls CM. 4But the key idea for the present paper is that we can explain logical symbols in terms of how they relate to epistemic operations.And using that idea to see how we can remove the dependency on an assumed base logic is the goal of the next section.

Belief Expansion Systems
The first step towards an entirely epistemic semantics for a propositional language L will be to revise some of Gärdenfors setup.If belief states are defined as sets of sentences closed under classical deduction it wouldn't be very interesting to show that they validate propositional logic.In particular, it already rules out a semantics for intuitionistic logic if belief states include every instance of the Law of the Excluded Middle.For these reasons I want to reformulate belief states and systems by using the key insight from Gärdenfors's Ramsey Test: that the meaning of logical symbols can be captured by their belief dynamics, expressed in terms of epistemic operations on belief systems.From here on out, then, a belief state K should be thought of as set of sentences in a propositional language L. Belief states are not assumed to be closed under any logical consequence operator and the connectives of L are not yet assumed to have the meaning assigned to them by any particular logic.
What closure conditions I do want to impose on a belief state are instead to be formulated in terms of epistemic norms about which beliefs we're committed to as a result of other commitments.Given that I'm working with a propositional language L we will need closure conditions corresponding to the logical symbols →, ∨, ∧, and ⊥.The general idea is that these conditions should be formulated in terms of how sentences containing these connectives interact with the norms governing our changes in belief.
In order to make it more clear what I mean, let's begin with looking for a closure condition for the conditional.For the moment, we haven't yet assumed it to have any particular meaning.But the basic situation meant to be encoded by → is some form of hypothetical reasoning, supposing P and seeing if Q follows.But that is exactly the situation originally discussed by Ramsey.As he says: If two people are arguing 'If p, then q?' and are both in doubt as to p, they are adding p hypothetically to their stock of knowledge and arguing on that basis about q; so that in a sense 'If p, q' and 'If p, [not q]' are contradictories.We can say that they are fixing their degree of belief in q given p.If p turns out false, these degrees of belief are rendered void.If either party believes not p for certain, the question ceases to mean anything to him except as a question about what follows from certain laws or hypothesis.[8, p. 155, fn. 1] Now, although Ramsey is clearly discussing hypothetical judgements, it's far from obvious what notion of epistemic change he has in mind.Revision plays this role in the Gärdenfors Ramsey Test, since it is conceived of as the minimal change required to yield a consistent belief state where P is accepted.But this makes the resulting conditional counterfactual, in the sense that the belief state K * P at which we're evaluating the consequent can encode very different epistemic attitudes than K .That is, if K is our present belief state, we're reasoning about Q relative to beliefs which are potentially incompatible with our actual beliefs.
Ramsey's remarks, as Arló-Costa and Levi [2, p. 218] point out, only specify the test fully for the case when the antecedent P is indeterminate at K .Then, some notion of expansion by P is the appropriate epistemic change for the test.Now, the only reason to choose a more general epistemic operator to do the trick is to ensure that the resulting belief state is consistent if ¬P ∈ K .But when trying to determine whether we should accept the P → Q at K , consistency of the hypothetical situation isn't what's important; what matters is that it preserves the epistemic attitudes we have at K .As soon as we reject P, whether now or later, our attitude towards P → Q no longer matters since it becomes inferentially inert.The kind of hypothetical judgements I'm interested in are precisely those which correspond to belief expansion.
Acceptance of a conditional P → Q at a belief state K , then, depends on the dynamics of the belief system as follows: if Q is accepted when K is expanded by P, then the conditional is supported.Conversely, if P → Q is accepted at K and we add P to our beliefs at K , then we ought to accept Q.Hence, the closure condition for → is the following Ramsey Test. 5K →) P → Q ∈ K if and only if Q ∈ K + P .Of course, this still leaves the condition underspecified since I haven't yet characterised belief expansions.They are usually distinguished among functions on belief states by the following postulates [6, pp. 48-50]. 6Although we could very well characterise expansion in terms of another set of conditions, I've chosen these both because they're already in common use within the literature on belief change and because I find them intuitively plausible.
The first of these conditions just states that + is a total function, while (K + 2) encodes the requirement that expansion by P successfully adds it to our beliefs.The next con-ditions are expressions of a principle which Gärdenfors [6, p. 49] calls informational economy.Knowledge is precious and once acquired should only be discarded if we're forced to.Since expansions are the epistemic response to information which doesn't conflict with what we already know, they should preserve knowledge in the sense of (K + 3).Conversely, if P is already accepted at K then expansion by it should leave our beliefs unchanged.Finally, belief expansion is monotonic since, if K ⊆ H , then anything which could be inferred from K and P can also be inferred from H and P.
With the closure clause for conditionals out of the way, I can move on to disjunctions.Here, we can state half of the relevant condition without any reference to belief change: if we accept either P or Q at K then we should also accept But the converse wouldn't be a plausible epistemic condition.It would mean that we could never accept any disjunction P ∨ Q without being able to determine which of the disjuncts we accept.We could, perhaps, swallow that conclusion if we're only concerned with belief states accepting sentences which are known for certain, since then we'd have to know P or Q to come to know P ∨ Q, but that seems like an unnecessary restriction of scope.Surely, we'd want to model belief states where we've supposed some theory is accepted in order to consider its consequences, even if that theory contains indeterminate disjunctions.
It's far more natural to think that we're committed to the result of an argument by cases.If we accept P ∨ Q at K and would accept R when hypothetically expanding with either P or Q, then we're already committed to accepting R at K .That is, Next, we get to conjunctions.Their closure condition can actually be stated without any reference to belief change.Since acceptance of both P and Q at a state K should occur precisely when we accept P ∧ Q at K , we get This closure property is local, in the sense that it doesn't depend on other belief states.In this sense, conjunctions are static; we can fully grasp their function without knowing anything about the dynamics of the belief system.Conditionals and disjunctions, on the other hand, are dynamic.We can only specify them fully by looking at what we would believe under the appropriate changes to our informational state.
Finally, I will give a closure condition for ⊥.The idea here is to ensure that belief states allow disjunctive syllogisms.That is, if P ∨ Q is accepted at K and ⊥ is accepted at K + P then Q ought to already be accepted at K .Now, the easiest way to ensure that this is the case is to simply say that whenever a belief state contains ⊥, then it contains every sentence of the language.Then Q will be accepted at both K + Q and K + P so K ∨ 2 ensures that its accepted at K . 7Let K ⊥ be the set of all sentences of L. Then we have the following closure condition.
All of these conditions together yield a joint definition of a belief expansion system and its constituent belief states.Definition 1 A belief expansion system (K, +) is a pair such that + is an expansion function satisfying (K + 1) -(K + 5) and K is a collection of sets K of L-sentences which satisfy the following conditions.
(K ∧) Now, some of the conditions are clearly similar to natural deduction rules governing the corresponding logical constant.But both the conditional and disjunction are essentially tied to the choice of a particular expansion operator.And by altering which conditions this expansion operator has to satisfy we would get entirely different kinds of conditionals and disjunctions, although still ones which satisfy the intuitions expressed by the Ramsey Test and argument by cases.In this way, belief expansion systems encode the reasoning of epistemic practices governed by the specific norms expressed by (K + 1) -(K + 5).
So, what are the first things we can say about belief expansion systems?At the very least their belief states are closed under Modus Ponens, which is fortunate since it seems a rather reasonable epistemic requirement.
But since P ∈ K it follows from (K + 4) that K + P = K , so Q ∈ K .We can also easily introduce a notion of negation without much fuss by exploiting the usual convention that ¬P is notational shorthand for P → ⊥.Then simply applying (K →) yields the following condition. 8K ¬) ¬P ∈ K if and only if ⊥ ∈ K + P .Further, it immediately yields the following lemma.

Lemma 3
Let K be a consistent belief state.Then K + P is inconsistent if and only if ¬P ∈ K .
Finally, we can see that the unique contradictory belief state is present in every belief system.Lemma 4 Every belief expansion system contains K ⊥ .
Proof Let K ∈ K be a belief state containing ¬P.Then ⊥ ∈ K + P which, by (K + 1), must be an element of K. Hence, by K ⊥ it follows that K ⊥ = K + P ∈ K.With these structures in hand, we can turn them into models by defining the notion of epistemic validity.

Epistemic Models and Validity
The next step in developing epistemic semantics is to endow belief expansion systems with a notion of validity.Following the terminology established by Horacio Arló-Costa and Isaac Levi discussing models for L < , we can distinguish two kinds of validity based on belief systems [2, pp. 232-233].
To know that P is positively valid in K is to know that every consistent belief state accepts it, whereas to be negatively valid it's sufficient that no consistent belief state could reject it.This idea that the lack of a counterexample is sufficient for validity is very similar to the rule RAA, which distinguishes classical and intuitionistic logic.This similarity isn't just a coincidence.In fact, it can be proven that the Law of the Excluded Middle is negatively valid in every belief expansion system.9 Lemma 7 Let (K, +) be a belief expansion system.Then K N V P ∨ ¬P.
In fact, the relationship between the two notions of validity can be seen to correspond to the one between intuitionistic and classical provability.
Proof Let K be a belief state in K such that ¬ϕ is accepted at K .Then K can't be consistent since, if it were then K PV ϕ tells us that ϕ is also accepted at K and so ⊥ is accepted at K .Hence, ¬ϕ isn't accepted at any consistent K ∈ K so K N V ϕ.
Proof Assume for contradiction that K is a consistent belief state in K such that ¬¬ϕ / ∈ K .Now, by (K + 1), it follows that K + ¬ϕ ∈ K which, by Lemma 3, is consistent.But since ¬ϕ ∈ K + ¬ϕ by (K + 2) there is a consistent K ∈ K where ¬ϕ is accepted which contradicts the assumption that ϕ is negatively valid.This last proof, by the way, only works by using the very principle that PV is designed to make invalid.What it shows is that if K ϕ then there can't be a counterexample to the claim that K PV ¬¬ϕ.That is, this proof isn't intuitionistically valid.
Since I find it rather more natural to consider a sentence to be valid when it must be accepted, my official epistemic models are defined in terms of positive validity.
Definition 10 Let ϕ be a L-sentence, is a set of L-sentences, and (K, +) be a belief expansion system.Then we say that

Soundness and Completeness
With the semantics constructed, the goal is now to show that it's adequate.I will begin by showing that my official epistemic models are sound and complete for intuitionistic logic.A short argument, based on the lemmas relating positive and negative validity, then establishes the same for classical logic with respect to negative validity.
Proving soundness over epistemic models is relatively straightforward.As always, the argument proceeds via induction over all intuitionistic derivations D whose conclusion is ϕ and every undischarged assumption is in .Writing I for intuitionistic provability in a standard system of natural deduction we have the following.

Proof Since
I ϕ there is a derivation D with conclusion ϕ and all undischarged assumptions in .I will prove by induction on the height of D the slightly stronger statement that if ⊆ K then ϕ ∈ K .For the base case, D is simply an assertion of ϕ as an undischarged assumption in .Hence, ϕ ∈ and so ϕ ∈ K .Now, assume for induction that the theorem holds for all derivations whose height is at most k.
(∨I) As D ends with an application of ∨I we know ϕ is of the form ψ ∨ χ and that there is a derivation D of ψ with height k whose undischarged assumptions are all in .We know, by the inductive hypothesis ψ ∈ K for every K ∈ K such that ⊆ K .By (K ∨ 1 ) it follows that ψ ∨ χ ∈ K for every such K .(∨E) In this case D has the form: Since D is a derivation with height at most k whose undischarged assumptions are in we know that ψ ∨ χ ∈ K for every K ∈ K such that ⊆ K .Now, applying the inductive hypothesis to D 1 , D 2 tells us that for every K ∈ K such that ⊆ K we have that both ψ ∈ K and χ ∈ K imply that ϕ ∈ K .Since (K + 2) tells us that ψ ∈ K + ψ and χ ∈ K + χ it follows that ϕ ∈ K + ψ and ϕ ∈ K + χ for every K such that ⊆ K .Hence, by (K ∨ 2 ), it follows that ϕ ∈ K for every such K .
(∧I) As D ends with an application of ∧I we know ϕ has the form ψ ∧ χ .That means there are derivations D 1 , D 2 of ψ and χ respectively with height at most k and all undischarged assumptions in .By the inductive hypothesis it follows that ψ, χ ∈ K for every K such that ⊆ K .Hence, by (K ∧), ψ ∧ χ ∈ K for every K such that ⊆ K .(∧E) In this case D has the form: Applying the inductive hypothesis to D we have that ϕ ∧ ψ ∈ K for every K such that ⊆ K .Hence, by (K ∧), ϕ ∈ K for every such K .(→I) As D ends with an application of →I we know that ϕ has the form ψ → χ and that there is a derivation D of χ whose only undischarged assumptions are ψ or in .By the induction hypothesis applied to D we have that if ∪{ψ} ⊆ K then χ ∈ K .Now, ψ ∈ K + ψ , by (K + 2), so it follows that χ ∈ K + ψ for every K such that ⊆ K .Finally, (K →) tells us that ψ → χ ∈ K for every such K .(→E) In this case D has the form: Applying the inductive hypothesis to D 1 and D 2 respectively we have that ψ → ϕ, ψ ∈ K for every K such that ⊆ K .Then Lemma 2 implies that ϕ ∈ K for every such K .(⊥E) In the case where the last rule applied in D is ⊥E we know from the inductive hypothesis that ⊥ ∈ K for every K such that ⊆ K .In other words, every K containing is inconsistent so it follows from K ⊥ that ϕ is accepted in every such K .
Finally, let K be an epistemic model of .Then ⊆ K for all K ∈ K so it follows that ϕ ∈ K for all K ∈ K. Hence, E ϕ.
To prove a Completeness Theorem is only a little harder.The basic approach is a Henkin-style proof via a Model Existence Lemma.The basic idea of the Model Existence Lemma is, as usual, showing that a collection of theories forms a model.In some ways this is easier for epistemic models than for Kripke models of intuitionistic logic.Since belief states aren't required to be disjunctive there's no need to construct disjunctive extensions before making them into a model.
What we have to do is construct an expansion operator on these theories.Fortunately, its not hard to prove that we can use intuitionistic consequence to construct the expansion operator for a suitable belief expansion system.
To get this proof sketch started, all that's needed is to find a suitable set of intuitionistic theories.Since it's supposed to be a model of it's quite sensible to start there.Now, since it needs to satisfy (K + 1), we need our system to contain at least + P for every sentence P. Further, it has to contain every extension ( + P ) + Q of these sets as well.Now, as it turns out, these double extensions are already themselves simple extensions of as follows from two results shown by Gärdenfors [6,p. 50], generalised here to the present framework.

P∧Q
With these lemmas, all that's left to do is define the extension operator in terms of I and go through the straightforward verification that it satisfies the postulates.
Lemma 14 (Model Existence Lemma) If I ϕ then there exists a belief expansion system (K, +) such that K E and there is a minimal belief state K ∈ K such that ϕ / ∈ K .
Proof Define a function + on intuitionistic theories by Before showing that this is a model of , I need to prove that (K, +) is a belief expansion system.First, I will check that this definition satisfies (K + 1) -(K + 5).
(K + 2) Since K , P I P it follows that P ∈ K P then there is a deduction of Q all of whose undischarged assumptions are in K ∪ {P} ⊂ H ∪ {P}.Hence, Q ∈ H + P so K + P ⊆ H + P .(K + 1) Having shown that + satisfies (K + 2) -(K + 5) I can now use Lemma 13.
Every K ∈ K is, by definition, of the form + P for some P ∈ L. Hence, by Lemma 13, we have for all Q ∈ L that The next step is to check the that K also satisfies the closure conditions from Definition 1.
We also know from (K + 2) that P ∈ K + P so, since K + P is closed under intuitionistic logic, it follows that Q ∈ K + P .For the other direction, let Q ∈ K + P .Then K , P I Q so, since → I is a rule of intuitionistic logic, we have that Then, since ⊥E is a rule of intuitionistic logic, and K is an intuitionistic theory we know that it contains every sentence of L. Hence, K = K ⊥ .Conversely, if K = K ⊥ then K trivially contains the sentence ⊥.Now, we want to see that K is a model of .By (K + 3) it follows that ⊆ K for all K ∈ K so every sentence of is accepted in every belief state in K. Consequently, K E and is a minimal belief state in it.Since I ϕ, and is deductively closed, it follows that ϕ / ∈ .
Theorem 15 (Completeness Theorem) If E ϕ then I ϕ.Proof Assume for contradiction that I ϕ.By Lemma 14 there is an epistemic model K of with a minimal belief state K such that ϕ / ∈ K .But since K E and and E ϕ it follows that ϕ ∈ K which is a contradiction.Hence, I ϕ.As you'll recall from above, we know that the Law of the Excluded Middle is negatively valid in every belief expansion system.Since adding it to intuitionistic logic is sufficient to recover classical logic, it's easy to show that classical propositional logic P L is sound and complete with respect to negative validity.

Corollary 16
P L ϕ if and only if N V ϕ.
Proof For soundness, it is well-known that extending intuitionistic logic by the Law of the Excluded Middle yields classical logic so if P L ϕ then , L I ϕ, where L is the set {P ∨ ¬P | P ∈ L}.By Theorem 11 it follows that , L PV ϕ which, in turn, means that , L N V ϕ by Lemma 8.But since Lemma 7 tells us that K N V L for every belief system K, it follows that N V ϕ.Now, onto completeness.Since N V ϕ we know by Lemma 9 that PV ¬¬ϕ.Hence, by Theorem 15, it follows that I ¬¬ϕ so by appending a use of RAA to that proof we have that P L ϕ.

Gärdenfors' Propositional Models
The system of epistemic semantics based on belief expansion systems is sound and complete for intuitionistic logic.This result is similar to one obtained by Gärdenfors [6, pp. 132-141] on what he calls belief models.There the idea is to begin with an abstract notion of epistemic states and define propositions as functions between them.That is, a proposition is identified with the changes in belief which are caused by its acceptance.
Formally, a propositional model (K, Prop) consists of a set K of epistemic states and a set Prop of propositions, understood as functions from K to K. A proposition P is accepted at a state K if and only if P(K ) = K .(P1) For every P, Q ∈ Prop there is a function P ∧ Q ∈ Prop such that P ∧ Q(K ) = P(Q(K )) for all K ∈ K. (P2) P ∧ Q = Q ∧ P for all P, Q ∈ Prop.(P3) P ∧ P = P for all P ∈ Prop.(P4) There is a function ∈ Prop such that (K ) = K for all K ∈ K. (P5) For every P, Q ∈ Prop there is a function These propositional models, Gärdenfors has shown, generate Heyting algebras in such a way that if P = then P is Heyting-valid.Hence, the logic of his propositional models is intuitionistic.It's not coincidence that propositional models and belief expansion systems validate the same logic.As Gärdenfors [6, p. 141] himself notes, the idea of defining propositions through their epistemic effects essentially comes down to identifying them with the expansion they generate, that is P(K ) = K + P .This identification does result in a propositional model, but since he defines expansions through logical consequence this shouldn't be much of a surprise.Now, the point of introducing belief expansion systems was to generalise Gärdenfors ideas away from their reliance on an underlying logic.Using the identification above, we can also see how they generalise propositional models in the sense that every such model generates a belief expansion system.If (K, Prop) is a propositional model we can take Prop to be our language.Then for every epistemic state K ∈ K let H K be the set of P ∈ Prop such that P(K ) = K .That is, H K is the set of propositions accepted at K .If we let H = {H K | K ∈ K} then we have our set of belief states.To define expansions on these sets, let A simple but tedious verification of (K + 1) -(K + 5) and the closure conditions shows that (H, +) is a belief expansion system.

Epistemic Semantics and Logical Expressivism
What is it, then, that I've done here?Fundamentally, the idea has been to demonstrate that we can provide a system of semantics for a logic based on the norms governing a particular kind of belief change.Doing so is a two step process: First, we introduce a logical symbol in terms of how it interacts with some particular kind of belief change.
In the semantics I've described here, we connect conditionals and disjunctions to belief expansion through the respective clauses:10 The second part of the process is to characterise the relevant belief change operator in terms of some conditions it must satisfy.For expansion, I've here used the first five Gärdenfors postulates (K + 1) -(K + 5).While I've only constructed an epistemic semantics for the very simple language of propositional logic, in principle this very same process can be applied to many other logical operators.As such, the results herein are intended mostly as a proof of concept that the construction of semantic systems in terms of belief dynamics is a viable project; a starting point on which to build more complicated structures with multiple belief change operators which can model more complex systems of logic.An example of how that can be done is Gärdenfors' semantics for counterfactual conditionals > by way of a Ramsey Test using belief revision as the operator in question.
As he shows, with the right conditions characterising belief revision, these structures provide models for Lewis logic VC for counterfactuals [6, p. 151].11Similarly, we can look to belief revision as an operator to provide epistemic models for modal logics through clauses along the following lines: (K ) P ∈ K if and only if P ∈ K * Q for all Q. (K ♦) ♦P ∈ K if and only if P ∈ K * Q for some Q.That is, P is necessary if it is accepted under every revision and possible if it is accepted under some revision.I hope to have shown, then, that this style of epistemic semantics can be plausibly devised for a much wider and more interesting collection of logics.Now, the reason why I think we should care about that is because of how it connects to some wider philosophical issues.Namely, it points toward a distinctively pragmatist answer to the question of what is the correct logic.The first step is to understand the conditions defining epistemic structures, both the ones characterising the logical symbols in terms of operators for belief change and the ones governing the belief operator itself, to be normative rather than descriptive.That is, we should think of, using belief expansion systems as an example, (K ∧), (K →), (K ∨ 1 ), (K ∨ 2 ), and (K + 1) -(K + 5) as expressing norms we ought to follow when organising our beliefs.Of course, these are not normally explicitly stated rules which we're aware of, but more akin to what Robert Brandom [3, pp. 30-46] calls rules implicit in practice.That is, they are patterns which are enforced through sanctions against those who are taken to violate them.This practice of normatively assessing and enforcing patterns of how one ought to update ones beliefs is what institutes these unspoken norms for belief change. 12n this kind of view, AGM-style systems for the dynamics of belief provide a language for stating the norms which govern our actual practice of updating our commitments in the light of new information.Given such a characterisation of a given epistemic practice, what soundness and completeness for a particular logic over those systems show is that this deductive system corresponds precisely with the demands implicitly imposed by the norms of that epistemic practice.For example, by applying the rules of intuitionistic logic you will never reach a conclusion which violates the epistemic practice whose norms are expressed through belief expansion systems.Similarly, beliefs which the norms expressed by belief expansion systems says we ought to accept at every belief state are always intuitionistically provable.In this sense, intuitionistic logic is the correct logic for an epistemic practice with these norms.It gives the most general rules of reasoning which satisfies the norms without going beyond them.But it is correct only in sense that it gets the corresponding epistemic practice right.A sufficient change to the underlying normative practice would topple its status.
This kind of answer to the correctness question is broadly within the outlines of the logical expressivism which Brandom espouses.The idea is that the role of logical vocabulary is to make explicit the features argument already present in our pre-logical reasoning practice [4, p. 70-71].On the view I share with Brandom, informal reasoning is explanatorily prior to deductive systems of logic.The latter are concise and systematic ways of expressing the rules which the already existent practice implicitly enforces.So, any violation of the rules of the logic in question would be a violation of the epistemic norms already in force.In this way, the order of explanation of logical rules as emerging from a pre-existing practice provides an answer to where their normative force comes from.It's inherited from the epistemic norms they make explicit.
In fact, Brandom [4, has, in collaboration with Ulf Hlobil and Dan Kaplan, developed these ideas in a proof-theoretic direction by developing a sequent calculus on the basis of a version of the Ramsey Test in the form of a rule: P → Q if and only if , P Q.This impressive proof-theoretic version of the expressivist approach to logic tries instead to analyse the implicit structure of reasoning directly into logical rules.13 What the present kind of epistemic semantics can bring to the table, then, is a parallel model theory which is congenial to the project of logical expressivism.
Funding Open access funding provided by Lund University.
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