It is well known that it seems possible to have a situation in which there are two propositions p and q which are logically equivalent and yet are such that a person may believe the one but not the other. If we regard a proposition as a set of possible worlds then two logically equivalent propositions will be identical, and so if "x believes that' is a genuine sentential functor, the situation described in the opening sentence could not arise. I call this the paradox of hyperintensional contexts. (Cresswell 1975, p.25)

The following application of impossible worlds involves the analysis of belief and knowledge. That is, an analysis of contexts where hyperintensional distinctions arise

naturally, and where failing to give an adequate account of such distinctions can lead to very bizarre (incorrect) consequences. Exploiting the intuitive analogies between epistemic and modal propositional attitudes β knowledge and necessity in particular β had opened the door to allowing epistemic (and doxastic) logic enjoy the same intuitive semantics that modal logic had.

Hintikka (1962) pioneered an intuitive and successful Kripke-style interpretation of epistemic
language, one in which epistemic space (set of epistemic alternatives) is identified with
logical space (possible worlds) and epistemic operators are interpreted in a way analogous to
modal operators of the modal language. Knowledge for an agent π is interpreted as truth at all
πβs epistemic alternatives, i.e. truth at all worlds epistemically possible for π. Consequently,
since only possible worlds are the available epistemic alternatives for any agent, then all
*logical truths are epistemically necessary for any agent on the above interpretation. That is, *
on this interpretation all agents know all logical truths, which is certainly not the case.
Moreover, since among logical truths there are entailments, any agent will know all the
*logical consequences of what they know. This predicament of logical omniscience is a direct *
outcome of the above interpretation. Belief is analyzed analogously, and is burdened with
analogous issues.

That is, Hintikkaβs (1962) analysis sanctioned the following principles, where π²π΄ is read as βit is known that π΄β and π©π΄ is read as βit is believed that π΄β:46

(C1) If π²π΄ and π΄ β¨ π΅, then π²π΅ (Closure under entailment) If π΄ is known, and π΄ entails π΅, then π΅ is known.

If π©π΄ and π΄ β¨ π΅, then π©π΅

If π΄ is believed, and π΄ entails π΅, then π΅ is believed.

(C2) If β¨ π΄, then π²π΄ (Knowledge of all valid formulae) If π΄ is a necessary truth, then π΄ is known.

46_{ Hintikka (1962) relativizes knowledge and belief to agents, i.e. πΎ}_{ππ΄ and π΅}_{ππ΄ read as βagent π knows π΄β and }
βagent π believes π΄β, respectively. But we can simplify the discussion by depersonalizing the analysis, since that
is not where the relevant issues are, i.e. although the epistemic/doxastic accessibility relations π
πmay be
relativized to agents π, and for any to agents π and π, π
πβ π
π, nevertheless both π
π(π€) β π and π
π(π€) β
π, where π is a set of possible worlds (a modelβs domain) and π
π₯(π€) = {π’ β π: π€π
π₯π’}, i.e. the image of π€
under π
π₯.

If β¨ π΄, then π©π΄ (Belief in all valid formulae) If π΄ is a necessary truth, then π΄ is believed.47

Both principles do not seem right, as people are neither omniscient, nor do they know all logical consequences of what they know. Similarly, people do not believe all necessary truths or logical consequences of their beliefs. For example, Giuseppe Peano, surely did not know all the theorems of arithmetic (logical consequences of PA axioms), even if he claimed to have justified belief for claiming the truth of PA axioms, i.e. the self-evident nature of their truth. Similarly, he would not believe all conjectures of arithmetic that would turn out to be theorems. The same holds for any other person.

Hintikkaβs (1975) key insight was to correctly identify the crux of the problem by observing
that the theoretical responsibility for logical omniscience was not due to the method of
*possible world analysis per se, but rather the underlying assumption β which he had himself *
*previously endorsed β that βevery epistemically possible world is logically possibleβ.*48 It had
*been precisely the assumption of such a close analogy between necessity and knowledge that *
gave rise to erroneously burdening epistemic logic with logical omniscience. So, if

knowledge is not something that is closed under entailment, then perhaps for a more accurate
world-analysis of epistemic and doxastic propositional attitudes epistemic and doxastic
spaces should be modelled accordingly, by including worlds that fail to be closed under
entailment. This is precisely what Hintikka (1975) proposed. Hintikkaβs suggestion to get
around the logical omniscience problem, although retrospectively rather straightforward,
marked a revolutionary direction in possible world analysis of propositional attitudes. By
abandoning the problematic assumption that all epistemically possible worlds are logically
*possible, he posited worlds that are not logically possible, i.e. βsome epistemically possible *
worlds are not logically possible worldsβ. 49_{ The main motivation for adopting impossible }

worlds as a means to refine the analysis of belief is the now retrospectively obvious observation that human beings are not perfectly (ideal) rational agents:

The way to solve the problem of logical omniscience is hence to give up the assumption [that every epistemically possible world is logically possible]. This means admitting 'impossible possible worlds', that is, worlds which look possible

47_{ Wansing (1990, p.526) Pietarinen (1998, pp.8-9), Berto & Jago (2019, Β§5.3). }
48_{ Hintikka (1975, p.476). }

and hence must be admissible as epistemic alternatives but which none the less are not logically possible. Admitting them solves our problem for good.

(Hintikka 1975, p.477)

Within a decade of Kripkeβs non-normal semantics for S3 and S2, Hintikka (1975) and
Rantala (1975) had extended Kripkeβs model-theoretic βtrickβ employed in non-normal
*models, and had developed a semantics for epistemic and doxastic logics that model non-*
*ideal agents. Introducing and employing impossible worlds gave a semantics that invalidated *
epistemic and doxastic versions of problematic closures (C1) and (C2), thereby doing away
with omniscience and omnidoxasticity. On the impossible-world semantics this is done by
having the valuation function, for each model, assign values directly to formulae at

impossible worlds. Effectively, this technical move gives the set of impossible worlds the capacity to violate any closures, including entailment.

For the doxastic logic we expand the propositional language by an epistemic operator symbol π©, where the intended reading of βπ©π΄β is βit is believed that π΄β. Let the language of basic propositional doxastic logic be {~,.β§,.β¨, β, π©}. Let ππ = {ππ: π β β} be the set of

propositional variables Finally, let πΉππ be the smallest set closed under the following formation rules:

B: All propositional variables are wffs, i.e. ππ β πΉππ. R1: If π΄ β πΉππ then {~π΄, π©π΄ } β πΉππ.

R2: If {π΄, π΅} β πΉππ then {π΄ β§ π΅, π΄ β¨ π΅, π΄ β π΅} β πΉππ.

I present simplified Rantala models, which suffice to illustrate the role of impossible worlds in this context, for the present, introductory purposes. Multimodal systems for multiple agents are generally given, where belief is agent-relative and modelled by the corresponding accessibility relation, but for the introductory illustration purposes I only use a single

accessibility relation for simplicity. The idea can be easily generalized to accommodate multiple agents.50