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Classicality. Given the semantic clauses for the connectives at possible worlds and the lack of accessibility relation on the alethic modal operators in combination with the fact that entailment is defined as ‘truth-preservation atpossible worlds’ and va- lidity as ‘truth at all possible worlds’, it is quite easy to see that these all behave classically. By definition, impossible worlds only make their mark inside the scope of an intentional operator. Hence, this semantics, as with Priest’s (2005) semantics, provides the ‘standard’ semantics when sentences are evaluated at possible worlds.

Impossible Belief. We want our semantics to allow for reports of agents that hold beliefs that are impossible. As, for example, the ancients believing that Hesperus is

not Phosphorus or the fact that one might believe a conjunction without believing any individual conjunct (or the other way around, e.g., the preface paradox), or believe of one object that it does and does not have a property (e.g., Superman can fly and Clark Kent cannot fly).

We will show that it is possible to have each of these beliefs in our model by constructing a model on which it is true. First, the belief of the ancients that Hesperus is not Phosphorus (which, if we believe Kripke, is a necessary falsehood, i.e., an impossibility). Consider a belief-relation for an ancient such that RJaK

w

v =

{hw1, w2i}, where w1 is the actual world and w2 an impossible world. Then, let ‘H = P’ be true atw1 and ‘H6= P’ be true atw2. Then we get that

JH6= PK

w1 _{= 0, while}

Jabelieves that H6= PK

w1 _{= 1}

Now consider a version of the preface paradox, where an agents believes that, on a whole, there is at least one false sentence in her book (i.e.,¬a v0s(ϕ1∧ · · · ∧ϕn)). However, she has checked each sentence individually and believes of each sentence that it is true (i.e.,a v0s ϕi, for eachi < n). Consider a belief-relation for this agent such thatR={hw1, w2i}, wherew1 is the actual world andw2 an impossible world. Now, letϕ1∧ · · · ∧ϕn be false atw2 and ϕi be true atw2 for each i < n. Then we get that:

Jabelieves thatϕ1∧ · · · ∧ϕnK

w1 _{= 0, while}

Jabelieves thatϕiK

w1 _{= 1, for eachi < n.}

Finally, the contradictory belief-ascription. That is, say Lois does not know that Clark Kent is Superman and believes of the latter that he can fly, while she does not believe that of the former. We do both know that Clark Kent is Superman and I want to let you know that, unbeknownst to her, Lois holds a contradictory belief. So I say to you ‘Lois believes that Superman can and Lois believes that he cannot fly’. Consider a belief-relation for Lois such thatRLois

believe ={hw1, w2i}, where w1 is the actual world and w2 an impossible world. Let ‘ϕ’ represent ‘Superman can fly’ and letϕand ¬ϕbe true at w2. Then, we get that:

JLois believes thatϕK

w1 _{= 1 and}

JLois believes that¬ϕK

w1 _{= 1}

Consistent Actuality. We have shown above that it is possible to report contradictory beliefs in our model, however, we do not want that holding such beliefs lead to true contradictions at possible worlds. Specifically, we do not want that bothϕ and¬ϕ

are true at any w ∈ P. We will prove this by induction on the complexity of ϕ. Note that this will be fairly trivial for all the regular connectives, for we have shown above that these are classical. I will therefore not do it for all the connectives, but only for a few. The real trouble is the case where we access impossible worlds, i.e., the case whereϕis of the forma v0s ψ.

Base case: Take an arbitrary w ∈ P and let ϕ be an atomic sentence. Then, by definition, ¬ϕ can only be true if ϕ is false. So, if ϕ is an atomic sentence,ϕand ¬ϕcannot both be true atw.

Impossible Worlds Semantics in Action|55

Negation: Take an arbitrary w ∈P and let ϕat w be of the form ¬ψ, where ψ and ¬ψ cannot both be true (induction hypothesis). Then, ϕ

can only be true, when ψis false and for¬ϕto be true, by definition of ‘¬’ at possible worlds, ψ has to be true. Hence, ϕand ¬ϕ cannot both be true at w.

We can check by the truth-conditions of the other connectives, that ϕ

and its negation cannot both be true at a possible world if ϕ is of the form: ψ∨χ,ψ∧χ,ψ→χ,∀xψ,∃xψ,ψ,♦ψ. Let us now turn to the most important step.

Intentional operator: Take an arbitrarya,v, andw∈P and let ϕatw

be of the forma v0s ψ. Ifa v0s ψis true, then allv-accessible-worlds,w0, fora are such that ψ is true at w0. Yet, if ¬a v0s ψ is true, then there exists av-accessible-world, w00, forasuch thatψis not true atw00. But, it cannot be the case that there exists a world inW such that a sentence is both true and not true at that world.3 Hence, these two can never be true simultaneously.

This shows that, for anyϕ,ϕand its negation cannot both be true at a

possible world.

Informally, we can also argue for why contradictory beliefs never lead to contra- dictions at a possible world. For, consider why we introduced the framework as it is. We wanted to model beliefs without, necessarily, modelling beliefs that logically follow from them. For example, we want to be able to say that a v0s(ϕ∧ψ), while it might not be the case that a v0s ϕ or that a v0s ψ. Or, similarly, we want to be able to say that a v0s((ϕ→ψ)∧ϕ), while it might not be the case that abelieves

ψ. And this is what the above framework does.

As it seems that there are counterexamples to beliefs that ‘should’ follow from believing certain sentences containing a connective, it seems only natural that we also have thata v0s¬ϕ, while it might not be the case that¬(a v0s ϕ).