## 1. Reasons for Introducing Impossible Worlds



Why might one believe in impossible worlds? One argument is the
so-called "argument from ways" (Vander Laan 1997), which
is related to the first definition of impossible world given above.
This draws on the analogy with
 David Lewis's
 notorious argument concerning our quantifying over ways things could
have been (see Lewis 1973: 84). The world could have been different in
so many ways: Hilary Clinton could have won the 2016 US election, I
could be dancing on the ceiling, and Fermat's Last Theorem could
have remained without proof. Our belief in possible worlds is just a
paraphrase of our belief that there are many ways the world could have
been.



Aren't there also ways the world could *not* have been?
Some authors endorse the claim that *anything* is possible
(e.g. Mortensen 1989). However, the majority of philosophers believe
that not everything is possible, in the sense that some things just
*can't* happen. If I tell you that my college has a
cupola which is both fully round and fully square, you are likely to
reply, "it can't be *that* way!". So it seems
that "'ways' talk goes both ways" (Beall and
van Fraassen 2003: 86). If quantification on ways the world could have
been should be taken at face value as providing evidence for possible
worlds, then quantification on ways the world could not have been
should be taken at face value as providing evidence for impossible
worlds.



The argument as such is hardly convincing. Firstly, one author's
*modus ponens* is another's *modus tollens*. Some
have used similar considerations to argue against Lewis's modal
realism (see Skyrms 1976; Naylor 1986): if one believes in possible
worlds (of the Lewisian kind) as ways things could have been, then by
parity of reasoning one should believe in impossible worlds (ditto) as
ways things could not have been. But impossible worlds are too much to
swallow, so (by *modus tollens*) one should not believe in
Lewis's modal realism.



Secondly, taking quantification over any kind of entity whatsoever at
face value, just because it is embedded in ordinary language,
doesn't look like a promising general strategy. Lewis's
case for accepting commitment to possible worlds did not consist just
in an argument from *ways*. He also provided independent
motivation for taking quantification over possible worlds at face
value. A non-reductive account of possible worlds, according to Lewis,
brings net theoretical utility. The ontological cost is compensated by
a theoretical gain, given the variety of ontological, semantic, and
conceptual explanations allowed by our taking the notion of possible
world seriously. This is likely to be the main motivation for
believing in impossible worlds. As we will see below, defenders of
impossible worlds claim that they are are theoretically useful.



Another argument on behalf of impossible worlds, quite pervasive in
the literature, comes from *counterpossible reasoning* (e.g.
Beall and van Fraassen 2003, Chapter 4; Nolan 1997; Restall 1997;
Brogaard and Salerno 2013). This is reasoning from suppositions,
assumptions, or conditional antecedents which are not only false, but
impossible. We can reason non-trivially from impossible suppositions,
by asking what *would* be the case, were (say) the Law of
Excluded Middle false. To say that we reason *non-trivially*
from an assumption means just that we accept some conclusions but
reject others on the basis of that assumption. If we hypothetically
suppose the Law of Excluded Middle to be false, for example, then we
would likely conclude that intuitionistic logic would be preferable to
classical logic, given that supposition. We are unlikely to conclude
that classical logic would be a satisfactory logic, or that scarlet
would be a shade of green, given that supposition. The point readily
generalizes to reasoning about entire theories and to serious
philosophical and logical debates. We often reason from suppositions
about the truth of certain logical, mathematical, or metaphysical
theories which, if in fact false, are necessarily false, because of
the very nature of their subject matter.



This kind of reasoning is related to our assessment of certain
conditional statements with impossible antecedents, often called
*counterpossible conditionals* or, more simply,
*counterpossibles.* These include:



(1.1)  If Hobbes had
squared the circle, then mathematicians would have been
amazed.



Let's call a conditional like this *trivially true* when
it is true and the conditional with the same antecedent and opposite
(negated) consequent is also true. (1.1)   is intuitively
true, and yet



(1.2)  If Hobbes had
squared the circle, then mathematicians would not have been
amazed.



is intuitively false. If that's correct, then there are
non-trivially true counterpossibles. These considerations impact on
our preferred semantics for such conditionals, for the possible worlds
semantics for conditionals has trouble accommodating this position.
Yet the analysis in terms of worlds does a good job for conditionals
with possible antecedents. This motivates a semantics for
counterpossibles in terms of impossible (as well as possible) worlds.
We will get into more detail in
 section 2.5.



These kinds of argument highlight the usefulness of impossible worlds
as devices for analyzing particular linguistic, logical, and
philosophical issues. The point can be expanded into the general
"argument from utility" mentioned above: we should believe
in impossible worlds because they are useful tools for logicians and
philosophers. Whether that general argument is acceptable depends on
how persuasive the specific impossible worlds analyses are.
Let's look at some.


## 2. Applications of Impossible Worlds



This section briefly describes various applications of impossible
worlds, which collectively provide the main motivation for introducing
them.


### 2.1 Intentional States



Modelling intentional states, such as knowledge and belief, is a
prominent motivation for introducing impossible worlds. The intuitive
idea is that one gains knowledge or belief by ruling out would-be
possibilities (the *epistemic/doxastic possibilities* for that
agent). An agent's knowledge is whatever is true according to
all epistemically possible worlds accessible to that agent, i.e., the
worlds which represent ways things could be, for all the agent knows
(and similarly for belief).



Impossible worlds are useful within this approach because these
would-be possibilities often turn out to be impossible. Our beliefs
are often (covertly) inconsistent with one another. Moreover, our
knowledge and belief is not closed under (classical) logical
consequence: we do not know or believe all consequences of what we
know or believe. It is hard to accommodate these features using only
possible worlds. Possible worlds models usually generate the problem
of *logical omniscience* (see
 epistemic logic),
 which we will discuss in
 section 5.3.



One feature of rational agents' intentional states is that they
typically reject *obvious* impossibility or absurdity whilst
being subject to *subtle* inconsistency (Lewis 2004). One
attempt to capture this feature is developed in Jago 2006, 2007, 2009,
2014a. The idea is that only genuine possibilities and non-obvious
impossibilities should be epistemically accessible to rational (but
imperfect) agents such as us. For some agents, it may be epistemically
possible that Fermat's Last Theorem's is false, but 0
being 1 shouldn't be epistemically possible for anyone.
That's why we know the latter but may fail to know the former,
says Jago. One worry with this approach is that it is partly
proof-theoretic (invoking proof rules as relations between worlds),
whereas it isn't clear that proof length correlates well with
obviousness. Bjerring (2013) presents other objections to the view. A
strategy similar to Jago's is adopted in Berto 2014, 2017.



A quite different application of impossible worlds to epistemic states
has been proposed by JC Beall 2009, in relation to the
 Church-Fitch knowability paradox.
 The Church-Fitch reasoning is supposed to show that is contradictory
to suppose that any truth can be known (as some anti-realists claim).
The reasoning is this. Suppose all truths are knowable but some are
not in fact known. Then some truth \(A\), is not known: \(A \wedge
\neg KA\). By assumption, it's possible to know this: \(\Diamond
K(A \wedge \neg KA)\). Seemingly good reasoning (in classical
epistemic logic) implies that it's possible both to know and not
know that \(A\): \(\Diamond(KA \wedge \neg KA)\), which of course
isn't possible at all. The usual moral to draw is that not all
truths are knowable. Beall's idea, by contrast, is that the
knowability principle can be maintained (and the corresponding kind of
anti-realism along with it). His suggestion is to deny
*distribution of knowledge over conjunction*: the principle
that knowing that \(A \wedge B\) implies knowing that \(A\) and
knowing that \(B\). He achieves this using impossible worlds in which
conjunctions may be true even if their conjuncts are not.


### 2.2 Inconsistent Information



Closely connected with inconsistent information (e.g., information
including or entailing contradictions) is the issue of modelling
*inconsistent databases* (see Belnap 1977a,b, Barwise 1997).
These may consist, for instance, in sets of data supplied by different
sources which are inconsistent with each other, such as incompatible
evidence presented by different witnesses in a trial. Intuitively, we
are allowed to draw the logical consequences of data fed in by a
*single* source, but should not conjoin data from distinct
sources which may be inconsistent with each other. The database is
"compartmentalized": occasional inconsistencies are placed
in separate sectors and should not be asserted conjunctively (see
e.g., Belnap 1977a,b, Hyde 1997, Brown and Priest 2004). Impossible
worlds are useful in such models--particularly
*non-adjunctive* worlds, where a conjunction may be false even
if both conjuncts are true. (We discuss such worlds further in
 section 5.2.)


### 2.3 Fiction



Inconsistent information is at issue also in certain works of fiction.
Lewis's classic 1978 paper proposed an analysis of the
expression "true in such-and-such fiction" in terms of
possible worlds. What holds in a certain fictional work is what holds
at a set of possible worlds, properly selected via a series of (quite
subtle and complex) clauses. But fiction can be occasionally
inconsistent. Sometimes, this happens unintentionally: Conan
Doyle's *The Sign of the Four* describes Watson as
limping because of a war wound at his leg. In *A Study in
Scarlet*, however, Watson has no wound at his leg (for his wound
is in his shoulder and he doesn't limp). One may claim that the
set of worlds that make such stories true has to be split into
disjoint subsets, making true consistent fragments of the fiction.
This strategy won't always work, however, for inconsistencies in
fiction may be intentional (as stressed in Proudfoot 2006). Suppose we
write a novel, and in its first chapter we have the Mad Mathematician
produce a round square. If the intentional inconsistency is excised,
the fact that mathematicians all over the world are amazed by this
result in the second chapter becomes unexplainable. A natural
treatment of these cases, then, is obtained by admitting
(appropriately selected) impossible worlds in the set of situations
that realize what is told in the story (see e.g. Priest 1997b; Woods
2003, Chapter 6; Berto 2012, Chapters 7 and 8; Badura and Berto
2019).


### 2.4 Propositional Content



Closely connected to belief is the notion of *propositional
content*. Within possible worlds semantics,
 propositions
 can be defined as functions from worlds to truth values, or as sets
of worlds: a proposition is the set of worlds at which it is true. The
account is notoriously too coarse-grained (Barwise 1997). Intuitively
distinct impossible propositions (that swans are blue and not blue;
that Fermat's Last Theorem is false; that Charles is a married
bachelor) all hold at precisely no possible worlds. And we have a dual
problem with (unrestrictedly) necessary propositions, which are all
identified with the set of all possible worlds. Treating propositions
as set-theoretic constructions out of possible worlds leads to a very
coarse individuation of propositions, and because of this it has been
subject to seemingly devastating attacks, for instance, by Scott
Soames 1987. However, impossible worlds allow for fine-grained
distinctions unavailable in standard possible worlds semantics. An
impossible proposition need not be equated with the empty set of
worlds, for it may be a set which includes (only) impossible worlds.
We can have an impossible world \(w\_1\) with impossibly coloured
swans, a distinct impossible world \(w\_2\) at which Fermat's
Last Theorem is false, and a further impossible world \(w\_3\) at which
bachelors are married (but at which swans and Diophantine equations
behave correctly). Ripley (2012) argues that an account along these
lines is a better strategy for addressing the coarse-grainedness
problem than resorting to
 structured propositions.



One other application of impossible worlds concerns
*perceptual* impossibilities. When we see an Escher drawing or
a Penrose triangle, our experience has content. But that content is
impossible: such structures cannot be realised. The content of our
experience in such cases is naturally captured using impossible
worlds. Splitting that content into smaller internally consistent
parts would lose the essential feature of the whole. This issue is
explored in Mortensen 1997.


### 2.5 Counterpossible Reasoning



Perhaps the most important application of impossible worlds has to do
with counterpossible reasoning, understood as counterfactual reasoning
from impossible antecedents. As we saw in
 section 1,
 this kind of reasoning is often taken to provide independent
motivation for believing in impossible worlds. (For a recent overview
of the literature on counterpossibles, see Kocurek 2021.) In
Lewis-Stalnaker theories of counterfactuals, a conditional of the
form, "if it were the case that \(A\), then it would be the case
that \(B\)" is true if and only if, at the closest world (or
closest worlds) at which \(A\) is true, \(B\) is also true. (This is a
simplification of the truth conditions provided in the full-fledged
semantics of Lewis 1973.)



While the standard conditional logics based on this idea have been
quite successful in the treatment of counterfactuals, the approach
entails that any counterfactual whose antecedent is impossible is
vacuously true. For if there are no possible worlds at which \(A\) is
true, then trivially, all closest \(A\)-worlds (worlds where \(A\) is
true) are \(B\)-worlds. This is unsatisfying in many respects, for we
often need to reason nontrivially about theories that (perhaps
unbeknownst to us) cannot possibly be correct; and we often need to
reason from antecedents that may turn out to be not only false, but
necessarily so. (Compare the conditionals (1.1) and (1.2) in
 section 1
 for an example.) Recent defences of the view that counterpossibles
are not all vacuously true include Sendlak 2021 and McLoone
2021.



Three contexts in which theories of this kind show up are discourses
on (1) alternative logics, (2) mathematical conjectures, and (3)
metaphysical views. We will now say a few words on each of them.



(1) A famous Quinean motto has it that "to change the logic is
to change the subject": apparently disagreeing logical parties
are actually speaking of different things. So when intuitionists deny
that the Law of Excluded Middle holds in non-finitary contexts, they
are actually changing the meaning of logical operators; when
paraconsistentists claim that some formula can be true (in some weird
circumstances) together with its negation, they are not talking of
*negation* anymore (see e.g., Berto 2008).



But this does not make good sense of many disputes between
intuitionists, classical logicians, paraconsistentists, quantum
logicians, etc. It is more fruitful to assume that each party
generally understands the rival logics as intelligible, albeit
necessarily false, theories. Even if classical logic actually is the
one true logic, one can reason counterpossibly about what would be the
case if a certain non-classical logic were the correct one (e.g.,
"if intuitionistic logic were correct, then the Law of Excluded
Middle would fail" is true and "if intuitionistic logic
were correct, then the Law of Explosion would fail" is false).
One can take into account situations in which the Law of Excluded
Middle fails and argue about what would and would not be the case in
them. These situations are, by classical standards, just impossible
worlds (of the third kind: Classical Logic Violators).



(2) Similar claims can be made for mathematical conjectures. Different
set theorists have different views on controversial subjects such as
non-well-founded sets, the
 Continuum Hypothesis,
 the Axiom of Choice, the set/(proper-)class distinction, etc. If one
embraces the Platonic view (subscribed to, at least implicitly, by
many set theorists) that there is One True Universe of sets, then at
most one of the alternative set theories can be correct: the others
are wrong, and necessarily so. But people can work under the
hypothesis that a necessarily false basic mathematical principle
holds, and reason coherently from this:



> 
> It is doubtless true that nothing sensible can be said about how
> things would be different if there were no number 17; that is largely
> because the antecedent of this counterfactual gives us no hints as to
> what alternative mathematics is to be regarded as true in the
> counterfactual situation in question. If one changes the example to
> "Nothing sensible can be said about how things would be
> different if the axiom of choice were false", it seems wrong
> ... : if the axiom of choice were false, the cardinals
> wouldn't be linearly ordered, the Banach-Tarski theorem would
> fail and so forth. (Field 1989: 237-8)
> 



Field takes this as an argument to the effect that mathematical
necessity is not coextensive with logical necessity. But we can turn
the tables around: mathematical necessity is unrestricted and false
mathematical theories are just impossible theories.



(3) The third area in which counterpossible reasoning comes into play
are metaphysical disputes (and more broadly, any philosophical dispute
whose subject matter is necessarily true or necessarily false). Much
metaphysical talk is made with our quantifiers "wide
open", that is, aiming at stating truths about all that there
was, is, or could possibly be. This is evident in modal ontology, when
people advance a theory on the totality of worlds and on their nature.
But other metaphysical debates easily come to mind. Suppose a
philosopher wants to evaluate metaphysical theories which she
considers wrong (say, in order to draw unpalatable consequences by way
of criticism), such as Spinoza's monism or Hegel's
metaphysics of the Absolute. She must envisage situations where such
metaphysics are correct and wonder what would be the case according to
them: situations at which there is only one substance, or at which the
Absolute *Geist* necessarily shapes the teleological
development of history. These situations will be, under the hypotheses
we have made, impossible worlds.



Counterpossible reasoning may also show up in philosophical analyses
of various kinds. For example, Boris Kment (2014) has proposed an
account of modal notions which grounds them in explanatory reasoning,
in particular of the counterfactual kind. To account for non-trivial
counterpossibles, Kment uses impossible worlds taken as as collections
of structured Russellian propositions.



Semantic structures for counterfactual conditionals involving
impossible worlds were first introduced by Routley 1989, and have been
proposed e.g. by Read 1995, Mares and Fuhrmann 1995, Mares 1997, Nolan
1997, Brogaard and Salerno 2013, Bjerring 2014, Berto et al. 2018.
Most of these are natural extensions of Lewis's 1973 semantics
for counterfactuals and capture several intuitions about
counterpossible reasoning. The main task for such theories consists in
accounting for the concepts of closeness and qualitative similarity
between worlds once impossible worlds enter the stage. How to
fine-tune these notions is not a trivial matter (for an extensive
discussion, see Vander Laan 2004; we will say more in
 section 4.2).



Non-trivial treatments of counterpossibles require the failure of
several logical principles which hold in the standard Lewis-Stalnaker
approach to counterfactuals (Williamson 2007 chapter 5, Brogaard and
Salerno 2013). (Williamson uses these failures to argue that
counterpossibles are always trivially true.) One important principle
that fails is the entailment from a strict conditional, "if
\(A\) then-strictly \(B\)", to the corresponding counterfactual,
"if it were the case that \(A\) then it would be the case that
\(B\)". Normally, the former entails the latter. A strict
conditional is true when all the (accessible) possible worlds where
the antecedent is true also make the consequent true. If all the
possible \(A\)-worlds are \(B\)-worlds, then in particular all the
closest possible \(A\)-worlds are \(B\)-worlds. In an account which
admits impossible worlds, however, we can have closest impossible
worlds where \(A\) obtains and \(B\) fails, making the counterpossible
false even though the corresponding strict conditional is true.



The ensuing anarchy can be mitigated to some extent, e.g., by assuming
what Nolan 1997 calls the *Strangeness of Impossibility
Condition* (SIC): any possible world, however weird, should be
closer to any possible world \(w\) than any impossible world is to
\(w\). Reality will be turned upside down before logical laws or
mathematical truths abandon us. Then it is plausible that the
Lewis-Stalnaker principles will still hold whenever the relevant
antecedent is *possible*. For then we will consider only the
closest antecedent-worlds when we evaluate the conditional, all of
which will be possible worlds: the impossible ones will be too far
away (Berto et al. 2018).


## 3. The Metaphysics of Impossible Worlds



Supporters of impossible worlds disagree over their metaphysical
nature, just as supporters of possible worlds do. If one accepts
ontological commitment to worlds of any kind, then one faces the
follow-up question: just what are they, metaphysically speaking?



The two main options among *modal realists* (philosophers who
accept possible worlds in their ontology) are David Lewis's
*extreme* or *genuine modal realism* and
*ersatzism* (or *actualism* or *abstractionism*:
these terms all have slightly different connotations, which
we'll ignore here). It is a common thought among impossible
worlds theorists that impossible worlds should just inherit the
ontological status of their possible mates: whatever your favorite
metaphysics of possible worlds is, impossible worlds are of the same
kind. This has been called the *Parity Thesis* (see Rescher and
Brandom 1980). As Graham Priest puts it:



> 
> As far as I can see, any of the main theories concerning the nature of
> possible worlds can be applied equally to impossible worlds: they are
> existent nonactual entities; they are nonexistent objects; they are
> constructions out of properties and other universals; they are just
> certain sets of sentences. ... There is, as far as I can see,
> absolutely no cogent (in particular, non-question-begging) reason to
> suppose that there is an *ontological* difference between
> merely possible and impossible worlds. (Priest 1997b: 580-1)
> 



Yagisawa's *extended modal realism* proposes a
Lewis-inspired realist account of impossible worlds and
*impossibilia* (the objects exemplifying absolute
impossibilities which inhabit impossible worlds). On this view,
impossible worlds are concrete mereological sums of real individuals,
which are causally and spatiotemporally interrelated within each world
but never across worlds (see Yagisawa 1988). Yagisawa exploits the
"argument from ways" we met above: if quantification on
ways the world might be or have been commits us to possible worlds,
then, by parity of reasoning, quantification on ways the world might
not be commits us to impossible worlds. The argument is backed by
Yagisawa's considerations on the additional logical and
philosophical applications allowed by impossible worlds, which are not
available, in his view, to traditional Lewisian modal realism.
Extended modal realism is a strong position: concrete impossible
worlds represent absolute and logical impossibilities directly, by
*instantiating* them. So impossibilities and, in particular,
logical inconsistencies, are "out there" in reality.



In his 2010 book, Yagisawa is more distant from Lewisian modal
realism. He still admits impossible worlds and *impossibilia*,
and he rejects ersatz accounts of them. However, he now takes worlds
to be points in modal space. Worlds are modal indices for truth, just
like times are temporal indices for it; and modal matters are treated
in a way similar to how four-dimensionalist philosophers, who believe
in
 temporal parts,
 treat temporal matters. According to four-dimensionalists, material
objects are like temporal worms extended across time: an object has a
property at time \(t\) by having a temporal stage at time \(t\) which
has that property. Analogously, for Yagisawa an object has a modal
property, a property at world \(w\), by having a modal stage at world
\(w\) which has that property.



More moderate (Yagisawa would say: *too* moderate) realists
treat impossible worlds as *ersatz* constructions: abstract
entities on a par with ersatz possible worlds (see e.g. Mares 1997,
Vander Laan 1997). Modal ersatzism comes in various shapes (Divers
2002, Part III, is by far the best critical evaluation in the
literature). If one takes possible worlds as maximally consistent sets
of propositions (as per Adams 1974), impossible worlds could be sets
of propositions that are inconsistent and/or incomplete. Similarly,
Plantingan ersatzism (possible worlds are particular states of
affairs) or Stalnakerian ersatzism (possible worlds are world-natures
or maximal properties) could be easily extended to accommodate
impossible worlds. All hands agree that such worlds come at no great
ontological or theoretical cost, once one has accepted ersatz possible
worlds. After all, ersatz worlds are abstract: they account for
impossibilities, not by instantiating them as Lewisian worlds do, but
by representing them in some way or other. Jago (2012) takes both
possible and impossible worlds to be constructions out of positive and
negative facts, such as Barack Obama's not being French (see the
entry on
 facts).



The extension of ersatzism from possible to impossible worlds appears
to be particularly straightforward for *linguistic* ersatzism.
On this approach, possible worlds are *world-books*: sets of
sentences of a special "worldmaking" language.
(Carnap's (1947) state-descriptions and Jeffrey's (1983)
complete consistent novels are examples of this strategy.) It is easy
to admit impossible worlds of the same kind, that is, world-books
which are locally inconsistent or incomplete, which fail to comply
with some logical law or to be closed under some notion or other of
logical consequence.



However, there may be reasons to reject the Parity Thesis. If
Lewis's criticisms of ersatzism in *On the Plurality of
Worlds* are right, then each ersatz account of impossible worlds
inherits the limits of ersatz theories of possible worlds: each of
these theories has to resort to intensional entities taken as
primitive (such as propositions or states of affairs) in its
explanation of what ersatz worlds are, or to primitive modal notions
(most often, to both). Suppose that, instead, one wants to retain the
advantages of both worlds (no pun intended), ersatz and genuine, when
it comes to impossibilities. Suppose, that is, that (a) one wants to
employ a modal framework including both possible and impossible worlds
to retain the theoretical benefits provided by the latter; and (b) one
wants to stick to Lewis's project of a reductive account of
intensional and modal notions to fully extensional ones
(*contra* ersatzism); but also (c) one wants to avoid the
unwelcome consequences of concrete impossible worlds instantiating
impossibilities, such as having true contradictions "out
there" in reality (*contra* Yagisawa's extended
modal realism). One could then try the following hybrid solution: (1)
go realist about possible worlds, and (2) exploit the set-theoretic
machinery of modal realism to represent different impossible worlds as
distinct ersatz, abstract constructions.



To fulfil these desiderata, Berto 2010 sketches an intermediate
account, labeled as *Hybrid Modal Realism* (HMR), which
dispenses with the Parity Thesis. The account follows suggestions from
Divers 2002, Chapter 5, and is similar to a strategy pursued in
Kiourti 2010, Chapter 3. On this view, genuine, concrete possible
worlds are the basic stuff. Atomic propositions are taken as sets of
possible worlds. Distinct impossible situations can then be
represented by distinct world-books, taken as set-theoretic
constructions from atomic propositions. Krakauer 2013 gives a similar
account in terms of
 structured propositions
 built out of ordinary possible worlds. Jago 2012 and Sendlak
2015 criticise Berto's approach on the basis that it cannot
distinguish the proposition *that Hesperus is the second planet
from the sun* from the proposition *that Phosphorus is the
second planet from the sun*. Reinert 2018 attempts to do better by
combining Lewisian possible worlds with an ersatz situation-based
account of impossible worlds. Fouche 2022 develops
Berto's hybrid account into a full-fledged hyperintensional
theory of content.



A metaphysical account of impossible worlds, alternative both to
ersatzism and to Lewisian realism, has been proposed in Zalta 1997.
Zalta's powerful theory of abstract objects is based upon his
logic of *encoding*, whose core idea consists in postulating an
ambiguity in the copula of predication: "\(x\) is \(P\)"
can mean that object \(x\) exemplifies property \(P\), as per ordinary
predication; but it may also mean that \(x\) *encodes* \(P\),
encoding being a special mode of predication. Abstract objects encode
properties, besides exemplifying them; in particular, they can encode
properties they do not exemplify (see Zalta 1983). Within this theory,
*situations* are defined as abstract objects that encode states
of affairs (taken as 0-ary properties); and impossible worlds are
taken as maximal situations that are not possible, that is, such that
it is not possible that all the states of affairs encoded by them
simultaneously obtain.



Zalta claims that, despite treating worlds as abstract objects, this
is not an ersatz conception of worlds. A given state of affairs
\(p\)'s obtaining at world \(w\) (no matter whether \(w\) is
possible or impossible) is analyzed as:


* (Z)  \(w\) encodes the
property *being-such-that-p*,



and so *being-such-that-p* is ascribed, in the encoding sense,
to \(w\). As such, \(w\) *is* (in the encoding sense of the
copula, at least) such that \(p\). Thus, according to Zalta's
theory of encoding, worlds are in some sense metaphysically
characterized or determined by such states of affairs. And according
to Zalta nothing of the sort can be claimed of ersatz conceptions of
worlds.



All the ontological accounts of impossible worlds presented so far are
in a broad sense realist. They all accept that sentences referring to
or quantifying over impossible worlds can be literally true, and take
the entailed ontological commitment at face value, although they
disagree with each other about the metaphysical status of worlds. A
deeply anti-realist alternative to modal metaphysics has also been
developed:
 modal fictionalism.
 The view is fictionalist (or anti-realist) about *worlds*. Its
key claim is that talk of and quantification over worlds ought to be
understood as literally false: it is only true within a "worlds
fiction". We make-believe in the fiction because it delivers
useful results in the explanation of modal notions. Modal fictionalism
promises the theoretical benefits of modal realism without the
ontological costs. We should not include worlds (other than the actual
world) in our ontological catalogue. But talking *as if* there
were worlds is useful. Gideon Rosen (1990), a major proponent of the
view, takes Lewisian modal realism to be the relevant fiction. But it
is relatively easy to extend such modal fictionalist accounts into
fictional treatments of possible *and* impossible worlds,
taking e.g. Yagisawa's extended modal realism as the fiction
which we make-believe. JC Beall (2008) proposes an approach to
impossible worlds (see
 section 5.1),
 which can be motivated by the idea that these are worlds where
"logical fictions" take place.


## 4. The Structure of Impossible Worlds



Another issue which theories of impossible worlds disagree on concerns
the amount of logical *structure* such worlds have. This issue
affects impossible worlds specifically: there is no correlative issue
concerning possible worlds (beyond our choice of logic). Various
classes of impossible worlds display different degrees of anarchic
logical behavior: as we shall see, non-normal worlds for non-normal
modal logics
 (section 5.1),
 for instance, are such that only modal sentences behave in a
non-standard fashion at them, whereas sentences that include just the
Boolean operators of classical logic get the standard treatment. Such
worlds appear to be logically more structured than fully anarchic
"open" worlds
 (section 5.3).
 For, we shall see, even principles of classical logic involving only
the extensional, truth-functional connectives can fail at open worlds:
their openness consists in their not being closed under any
non-trivial logical consequence principle.



Should we require impossible worlds to comply with *any*
logical rules? And if we allow different classes of impossible worlds,
each exhibiting different degrees of logical structure, can these
classes be ordered in a meaningful way? This section focuses on these
two issues.


### 4.1 The Granularity Issue



Are there any logical principles which impossible worlds must obey?
More precisely, is there any logical inference such that, for any
(impossible) world \(w\), if the premises are all true according to
\(w\), then so is the conclusion? There is at least one such
inference: the trivial inference from \(A\) to \(A\). (For if \(A\) is
true at world \(w\), then \(A\) is true at world \(w\)! An impossible
world may represent *that it is not the case that A entails A*;
but unless there are true contradictions, it cannot both represent and
not represent that *A.*) Are there any others? This is the
*granularity issue*.



In addressing the issue, a good starting place is the *Nolan-Zalta
Principle* (Nolan 1997: 542; Zalta 1997: 647):


* (NZ)  If it is
impossible that \(A\), then there's an impossible world which
represents that \(A\).



(This is not an 'if and only if', since the converse is
clearly false: some impossible world represents your reading this
article, yet that's not at all impossible. Impossible worlds
represent possible *and* impossible situations. The things they
represent make for an impossible bunch, but might each be possible
when taken individually.)



The principle has some intuitive force. Nolan thinks of it as a kind
of unrestricted 'comprehension principle' for
impossibilities. It tells something about which impossible worlds
there are. There will be worlds which represent that water is not
H\(\_2\)O, that \(2+2=5\), and that snow both is and is not white. One
might think that (NZ) entails that 'anything goes' with
regard to impossible worlds: that any logical principle (except \(A
\vDash A)\) will be broken by some impossible world. If so, then (NZ)
delivers the open worlds mentioned above. However, to apply (NZ), we
need a single object-language sentence \(A\) which describes an
impossibility. Logical laws, by contrast, are stated as relationships
between *multiple* object-language sentences. So it is not
clear that (NZ) does its intended work.



Priest (2016) adopts two principles that are similar to, but stronger
than, (NZ): 'everything holds at some worlds, and everything
fails at some worlds' (Priest 2016, 5) and, for any distinct
\(A\), \(B\), 'there are worlds where \(A\) holds and \(B\)
fails' (Priest 2016, 7). More specifically, in our
terminology:


* (4.1)  For any \(A\),
there is a world which represents that \(A\) and a world which does
not represent that \(A\).
* (4.2)  For any distinct
\(A\) and \(B\), there is a world which represents that \(A\) but does
not represent that \(B\).



Priest calls these the 'primary directive' and
'secondary directive' on impossible worlds, respectively.
The latter implies the former, which in turn implies (NZ), but neither
converse holds.



To illustrate the extra power (4.2) gives us (over (4.1) and (NZ)),
consider *Simplification*, the inference from \(A \wedge B\) to
\(A\), or *Disjunction Introduction*, from \(A\) to \(A \vee
B\). (4.2) directly entails that there are worlds where these rules
fail. So, if we find (4.2) plausible, we can infer that impossible
worlds are not, in general, governed by standard
 paraconsistent logics.
 A *paraconsistent logic* is any one in which cont radictory
premises \(A\), \(\neg A\) do not entail arbitrary conclusions. But
standardly, paraconsistent logics maintain the principle that
conjunctions are true just in case both conjuncts are; disjunctions
are true just in case at least one disjunct is; and double negations
\(\neg \neg A\) are true just in case \(A\) is. If we accept (4.2),
then these relationships will break down in some impossible
worlds.



Yet even with (4.2) in play, it doesn't follow that
'anything goes' with impossible worlds. No principle so
far entails that some impossible world breaks the Adjunction rule,
from \(A\) and \(B\) to \(A \wedge B\), simply because 4.2
doesn't apply to inferences with multiple premises. (We discuss
Adjunction-violating worlds further in
 section 5.2
 below.) To infer the 'anything goes' conclusion, that for
any logically valid inference there is some impossible world that
breaks it, we'll need this principle:


* (NZ\(^+\)) If it is
impossible that \(A\_1, A\_2,\ldots\) but not \(B\), then there's
an impossible world which represents that \(A\_1, A\_2,\ldots\) but not
\(B\).



However, we can hardly claim that we've derived the
'anything goes' picture of impossible worlds from this
principle, for it is, in effect, an explicit statement of that very
view. Whilst the original (NZ) has a good deal of intuitive force,
it's much harder to feel that way about (NZ\(^+)\).



There may be no completely general, intuitively motivated principle
(along the lines of (NZ)) from which we can ascertain just how
fine-grained impossible worlds should be. Nevertheless, there are
arguments which support the 'anything goes' picture, on
which there exist open worlds (those closed under no valid inferences
except \(A \vDash A)\). We'll briefly consider three such
arguments.



The first is simple. If impossible worlds can break some logical rule,
then why can't they break all of them? Suppose we fix on
standard proof rules for the connectives. Each such rule is as closely
tied to the meaning of the associated connective as the others are (to
their associated connectives). Yet, as each logically impossible world
breaks at least one of those rules, what's to stop some other
impossible world from breaking any of the other rules? This argument
has some intuitive force, but is clearly far from conclusive.



The second argument is from epistemic states. When we consider
real-world finite and fallible epistemic agents, there seem to be no
rules of the form: if someone believes \(A\_1, A\_2,\ldots\), then they
must believe (distinct) \(B\) (that's why the logical
omniscience problem discussed in 2.3 is hard!). If we model their
epistemic states in terms of worlds, then at least one of the worlds
must break the inference from \(A\_1, A\_2,\ldots\) to \(B.\) So each
logical inference (except \(A \vDash A)\) is broken by some world
(Jago 2014a; Priest 2016).



The third argument is from counterpossible reasoning. Suppose, in a
class on alternative logics, we consider what would happen if Excluded
Middle \((\vDash A \vee \neg A)\), Double-Negation Elimination \((\neg
\neg A \vDash A)\), or the Law of Explosion \((A, \neg A \vDash B)\)
were to fail. If classical logic is the one true logic, and logical
necessity is absolute, we're then reasoning counterpossibly. If
we want to analyse counterpossibles in general using impossible
worlds, then we'll need worlds where these principles fail. But
we seem to be able to reason this way, non-trivially, for any kind of
logical principle, and so our analysis will require open worlds
(Priest 2016).



Why have we spared \(A \vDash A\) so far? Well, it seems that in order
to break this, an impossible world would simultaneously have to
represent that \(A\) and not represent that \(A\). Such a world would
itself be an impossible object, one with inconsistent features. Since
most impossible worlds theorists maintain that impossible worlds
(actually) exist and that actuality is not inconsistent, this position
is ruled out. It is available to *dialetheists* and others who
allow reality to have inconsistent features. There seems to be no
theoretical cost in requiring \(A \vDash A\) to hold at all worlds
(impossible or otherwise). Going back to epistemic states, for
example, we want to represent inconsistent beliefs, such as
someone's believing both that \(A\) and that \(\neg A\) at the
same time; but it would be a mistake to infer that she both believes
and does not believe that \(A\).


### 4.2 The Closeness of Impossible Worlds



If impossible worlds display different degrees of logical structure
(or lack thereof), it may make sense to order them. A natural way to
do this is via an extension of the traditional "closeness"
relations between possible worlds. How to spell out the ordering in
detail, though, is far from straightforward. Within standard
conditional logics, and in the treatment of counterfactual
conditionals in terms of possible worlds due to Robert Stalnaker
(1968) and David Lewis (1973), worlds stand in similarity relations;
and similarity comes in degrees. This is usually represented by having
each possible world, \(w\), come with a system of
"spheres". If \(W\) is the set of all worlds, let \(\$\)
be a function from worlds to sets of subsets of \(W\), so that \(\$w =
\{S\_1, S\_2 , \ldots \}\), with \(w \in S\_1 \subseteq S\_2 \subseteq
\ldots = W\). Worlds within a given sphere \(S\_i\) are more similar to
\(w\) than worlds outside it.



If we take the special case in which \(w =\) the actual world (call it
"@"), we get a natural arrangement of possible worlds in a
system of spheres that mirrors their degree of (dis)similarity with
respect to @, according to the different kinds of possibilities and
(relative) impossibilities they represent. For instance, a world which
is exactly like @, except that Franz wears a white t-shirt instead of
the black one he's actually wearing while writing these lines,
is, intuitively, closer to @ than a world at which the laws of physics
are turned upside down. Some people have a general, intuitive
depiction of such closeness relations, and set out a hierarchy of
modalities accordingly: possible worlds where the laws of physics are
different from ours are naturally seen as more eccentric than worlds
where only biological, but not physical, laws are different; and these
are more eccentric than possible worlds with minimal factual changes
with respect to @, such as the white t-shirt world.



Can such a natural view be extended to impossible worlds? First, it is
intuitive to claim that some impossible worlds are more similar to the
actual world @ than others. For instance, the "explosion"
world (call it \(e)\) at which everything is the case (every sentence
is true) seems to be as far from @ as one can imagine, if one can
actually imagine or conceive such an extremely absurd situation. Now,
pick the impossible world, \(t\), at which everything is as in @,
except that Franz wears an impossible t-shirt which is white all over
and black all over. Intuitively, \(t\) is closer to @ than \(e\).



Next, some authors (e.g. Mares 1997) favor Nolan's SIC principle
(introduced in
 section 2.5).
 This implies that any possible world is closer to @ than any
impossible world is to @. A system of spheres for impossible worlds
centered on @ will just extend the intuitive possible worlds spheres
described above, by adding further, larger spheres where worlds
outside (logical, or more generally unrestricted) possibility stand.
But how are these latter to be internally ordered?



One very general option is the following. Even though we subscribe to
some unrestricted comprehension principle for impossible worlds, we
may admit that worlds where only the intensional operators, e.g., the
box and diamond of necessity and possibility, behave in a non-standard
fashion are less deviant than worlds where also the extensional
operators, like classical conjunction and disjunction, do. Let us call
worlds of the former kind *intensionally* impossible and worlds
of the latter kind *extensionally* impossible. This picture
(inspired by Priest 2005, Chapter 1) has some intuitive force to
recommend it. Kripkean non-normal worlds, where only the behaviour of
the modal operators is non-standard (see
 section 5.1),
 are intuitively less deviant than open worlds, where all formulas may
behave arbitrarily. Generalizing, the view would entail arranging the
respective spheres in such a way that any intensionally impossible
world is closer to @ than any extensionally impossible one.



This very general ordering of impossibilities, albeit intuitive, may
not be fully satisfying. A general qualm concerns the SIC principle
itself. For one may claim that, intuitively, some slightly deviant
impossible worlds may be *more* similar to the actual world @
than some possible but very weird worlds. For instance, the impossible
world \(t\) above, which is like @ except for Franz's wearing an
inconsistent T-shirt, may look more familiar than a world which is
logically possible, but where the laws of physics are turned upside
down. Several authors (Nolan 1997, Vander Laan 2004, Bernstein 2016)
have proposed putative counterexamples to SIC along these lines.



Although we cannot pursue this topic further within the limits of this
entry, the discussion developed so far should show that the issue of
the structure, closeness, and ordering of impossible worlds is quite
open.


## 5. The Logic(s) of Impossible Worlds



This section is a bit more technical than the others. None of the
other sections presuppose this material.


### 5.1 Impossible Worlds in Non-Normal Modal Logics



Possible worlds semantics
 is celebrated for providing suitable interpretations for different
axiomatic systems of
 modal logic,
 such as
 C.I. Lewis's
 systems \(\mathbf{S4}\) and \(\mathbf{S5}\) (Lewis and Langford
1931). In each model, sentences are evaluated as true or false
relative to a possible world. Modal sentences \(\Box A\) and
\(\Diamond A\) (usually read as 'it is necessary that
\(A\)' and 'it is possible that \(A\)',
respectively) are evaluated in terms of \(A\)'s truth at all or
some of the *accessible* worlds. By placing various conditions
on the accessibility relation between worlds, different modal logics
can be accommodated. (Readers unfamiliar with modal logic are advised
again to read the entities on
 possible worlds
 and
 modal logic
 before going further in this section.)



This approach validates the *Necessitation* inference rule:



If \(A\) is valid, then so is \(\Box A\). (In symbols: if \(\vDash A\)
then \(\vDash \Box A\).)



To see why, suppose \(A\) is a logical truth. Then in any model, it is
true at all possible worlds. So given any world \(w\) in an arbitrary
model, \(A\) is true at all worlds accessible from \(w\), hence \(\Box
A\) is true at \(w\), and so \(\Box A\) too is valid.



Logics in which the Necessitation rule is valid are called *normal
modal logics*. But historically, not all of the modal logics of
interest are normal logics. C.I. Lewis's systems \(\mathbf{S2}\)
and \(\mathbf{S3}\) (Lewis and Langford 1931) are non-normal logics,
for example. These logics are *weaker* than normal modal
logics, in the sense that they support fewer valid inferences. To give
a worlds-based semantics for such logics, we need to look beyond
possible worlds.



In 1965, Saul Kripke introduced a special kind of world, the
*non-normal worlds*, in order to provide semantics for
non-normal modal logics. Let us introduce some simple semantic
machinery for propositional modal logic. Take a non-normal
interpretation of a propositional modal language, \(\langle W, N, R,
v\rangle\), where \(W\) is a set of worlds; \(N\) is a proper subset
of \(W\), the set of *normal* worlds; \(R\) is a binary
accessibility relation between worlds; and \(v\) is a valuation
function assigning truth values to formulas at worlds:
"\(v\_w(A)\)" denotes the truth value of \(A\) at world
\(w\). Worlds in \(W - N\) are the non-normal worlds. The truth
conditions for the extensional logical vocabulary (negation,
conjunction, disjunction, the material conditional) are given the
usual way. The same holds for the modal operators of necessity
\(\Box\) and possibility \(\Diamond\), but only at *normal*
worlds. If \(w\) is non-normal, the truth conditions for the
modalizers go as follows:

 
\[\begin{align}
v\_w (\Box A) &= 0 \\
 v\_w (\Diamond A) &= 1 
\end{align}\]


where 1 stands for *true*, 0 for *false*. At non-normal
worlds, formulas of the form PS\(A\), with PS a modal, are
not evaluated depending on the truth value of \(A\) at other
(accessible) worlds, but get assigned their truth value directly: all
box-formulas are false and all diamond-formulas are true. In a sense,
at non-normal worlds nothing is necessary, and anything is possible.
These worlds, however, are deviant only in this respect: their
behavior, as far as the extensional connectives are concerned, is
quite regular.



In some (though not all) worlds semantics, logical validity and
consequence are defined relative to just the *normal* worlds.
The idea is that the normal worlds behave 'appropriately'
with respect to the logic in question, whereas the non-normal worlds
do not. For example, \(\Box(A \vee \neg A)\) is valid in
\(\mathbf{S2}\) and \(\mathbf{S3}\), but (by definition) it is false
at the non-normal worlds. So we need to ignore the non-normal worlds
when defining validity and consequence, which we do as follows:



\(A\) is *valid* \((\vDash A)\) if and only if, for all
*normal* worlds \(w\) in all models, \(v\_w (A) = 1\).



Premises \(S\) entail \(A (S \vDash A)\) if and only if, for all
*normal* worlds \(w\) in all models: if \(v\_w (B) = 1\) for all
premises \(B \in\) S, then \(v\_w (A) = 1\).



Even though we have ignored the non-normal worlds in this definition,
they still play a role in invalidating Necessitation. For take any
classical propositional tautology, say \(A \vee \neg A\). This holds
at all worlds of all models, so \(\Box(A \vee \neg A)\) holds at all
normal worlds and hence is valid. But \(\Box(A \vee \neg A)\) holds at
no non-normal world. Now suppose \(w\) is a normal world that has
access to any non-normal world. Then \(\Box \Box(A \vee \neg A)\) is
false at \(w\) and so (since \(w\) is normal) \(\Box \Box(A \vee \neg
A)\) is not valid. That's a counterexample to Necessitation.



In these semantics for non-normal modal logics such as \(\mathbf{S2}\)
and \(\mathbf{S3}\), the valuation function assigns the same truth
value to all box formulas (false) and all diamond formulas (true) at
non-normal worlds. But we can do things differently. In
Cresswell's (1966) semantics for the modal system
**S0.5** (due to E.J. Lemmon 1957), sentences beginning
with a modality are assigned *arbitrary* truth values. The
valuation function \(v\) treats modal sentences as if they were atomic
sentences. (Interpretations for \(\mathbf{S2}\) or \(\mathbf{S3}\) are
thus special cases of the interpretations for **S0.5**.)
The idea of considering impossible (non-normal) worlds as worlds at
which complex formulas are treated as atomic one is a popular one, as
we will see below.



Kripke introduced non-normal worlds as a technical device in order to
treat C.I. Lewis' non-normal modal logics; the question of the
*interpretation* of such structures (particularly, of the
ontological status of impossible worlds), then, makes perfect sense
-- and the answer is not straightforward, as have seen in
 section 3.


### 5.2 Nonadjunctive and Nonprime Impossible Worlds



In 1980, Nicholas Rescher and Robert Brandom published *The Logic
of Inconsistency. A Study in Non-Standard Possible-Worlds Semantics
and Ontology*. They introduced a modal semantics including,
besides ordinary possible worlds (taken as maximally consistent
collections of states of affairs), also non-standard worlds that are
locally inconsistent (such that, for some \(A\), both \(A\) and \(\neg
A\) hold at them), and incomplete (such that for some \(A\), neither
\(A\) nor \(\neg A\) hold at them). These are obtained
combinatorially, via two recursive operations having standard worlds
as their base, and called *schematization* \((\cap)\) and
*superposition* \((\cup)\). Given two worlds \(w\_1\) and
\(w\_2\), a schematic world \(w\_1 \cap w\_2\) is one at which all and
only the states of affairs obtain, which obtain both at \(w\_1\) and at
\(w\_2\). Dually, a "superposed" or inconsistent world
\(w\_1 \cup w\_2\) is one at which all and only the states of affairs
obtain, which obtain at \(w\_1\) or at \(w\_2\). Rescher and
Brandom's inconsistent-superposed worlds are, therefore,
impossible worlds of the fourth kind: Contradiction Realizers making
both \(A\) and its negation true, for some \(A\) (just superpose, for
instance, a possible world, \(w\_1\), at which I am 1.70m tall, and
another possible world, \(w\_2\), at which I am 1.90m tall).



The assignment of truth values at such worlds is not (obviously)
compositional with respect to conjunction. The standard semantic
clause interprets the '\(\wedge\)' symbol using our world
'and':


* (S\(\wedge\))  \(v\_w (A
\wedge B) = 1\) if and only if \(v\_w (A) = 1\) and \(v\_w (B) = 1\)



But the right-to-left direction will have to go, if \(w\) is one of
Rescher and Brandom's impossible worlds. These worlds are
*nonadjunctive*: they allow two sentences to be true even
though their conjunction is not true. (*Adjunction* is the
principle that the truth of a conjunction follows from the truth of
its conjuncts.) Rescher and Brandom's worlds can also be
*nonprime*: a disjunction may hold at them even though neither
disjunct does. They may also make some \(A\) and its negation \(\neg
A\) both true. But the corresponding conjunction, \(A \wedge \neg A\),
doesn't follow. These impossible worlds retain a certain amount
of logical structure. They are closed under any classically valid,
essentially single-premised inference (such as *Disjunction
Introduction*); but they are not closed under essentially
multiple-premised inferences (such as *Adjunction*).



Rescher and Brandom's approach falls in the nonadjunctive
tradition (see Berto 2007, Chapter 6) of
 paraconsistent logics:
 a tradition started by Jaskowksi's *discussive logic*
\(\mathbf{D}\_2\) (also labeled as \(\mathbf{J}\) in the literature,
Jaskowski 1948), and based on the idea of rejecting or limiting
the Adjunction principle. Such an approach has been revived in works
by Hyde (1997), and Varzi (1997 and 2004).


### 5.3 Impossible Worlds in Epistemic Logic



To model a concept such as knowledge, we can use a modality
'\(K\)' for 'knows that', with semantics along
the lines of '\(\Box\)', quantifying over all
epistemically possible worlds: worlds which are ways things could be,
for all one knows, or given the information or evidence one has
available. This is *epistemic modal logic*. (See the entry on
 epistemic logic
 for background material.)



This approach has proved to be very useful. However, when
epistemically possible worlds confirm to the rules of logically
possible worlds, the following principle is valid:



(Closure) If \(KA\) and \(A\) entails \(B\), then \(KB\)



This principle says that one knows all the logical consequences of the
things one knows. A special case of this principle says that all valid
formulas are known:



(Validity) If \(A\) is valid, then so is \(KA\)



But these principles seem false. You don't know all the logical
and mathematical truths, and there are truths which follow from what
you know which you don't know. (You might be indifferent to
those truths; you might even disbelieve them.) This is the problem of
*logical omniscience* (see
 epistemic logic
 again). There is a rich literature on the problem (Alechina et al.
2004, Duc 1997, Hintikka 1975, Jago 2014a,b, Rantala 1982a), including
some who defend these seemingly false principles (Stalnaker 1991,
1999).



Exactly the same holds for belief (treated with a modality
'\(B\)' along the lines of '\(K\)'). The modal
epistemic approach tells us that, as well as believing all
consequences of what we believe, we must hold a perfectly consistent
set of beliefs:



(Consistency) \(\vDash \neg(BA \wedge B\neg A)\).



This is a hard principle to defend, as anyone who has reflected on
their own beliefs will appreciate.



A popular method of avoiding these principles (beginning with
Cresswell 1973 and Hintikka 1975) is to allow impossible worlds into
the account. Consider again Rescher and Brandom's nonadjunctive
and nonprime worlds, at which conjunction and disjunction behave
anarchically. Rantala (1982a) takes the idea further, introducing
worlds at which *any* connective may behave anarchically.



Rantala's approach divides the worlds into *normal* and
*non-normal* ones. Normal worlds behave like possible worlds
whereas at non-normal worlds, *every* sentence is assigned an
arbitrary truth value. In effect, complex sentences \(\neg A, A \vee
B\), and so on, are treated as if they were atomic sentences. The
truth value of \(\neg A\) is independent of \(A\); the value of \(A
\vee B\) is independent of the values of \(A\) and \(B\) at a
non-normal world; and so on for the other complex sentences. These
non-normal worlds are thus a very anarchic form of impossible world.
Priest (2005, 2016) calls them *open* worlds, since they are
not closed under any rule of inference (other than the trivial rule
allowing one to infer \(A\) from \(A\); see also Jago 2014a).



Logical consequence and validity are defined with respect to possible
(normal) worlds only. Impossible worlds come into play only when
evaluating knowledge claims, \(KA\). So, ignoring \(K\)-sentences, the
logic is classical. But the logic of \(K\)-sentences is not closed
under any non-trivial rule of inference, thereby dispensing with
Closure, Validity, and (in the case of belief) Consistency. An agent
is modelled as having inconsistent beliefs, for example, simply by
treating an impossible world where both \(A\) and \(\neg A\) are true
as being epistemically accessible (from the actual world) for that
agent.



The approach has been generalized to quantified modal logics (Rantala
1982b) and developed into a unified framework for epistemic logics
(Wansing 1989, 1990). Wansing has shown that various logics for
knowledge and belief developed in Artificial Intelligence can find
equivalent models in structures including impossible worlds. Further
equivalence results in this area have been obtained in Sillari 2008,
where it is shown that impossible worlds structures using binary
epistemic accessibility relations are equivalent to structures using
Montague-Scott neighborhood semantics.



This kind of approach faces problems, however. If there's no
logical structure to impossible worlds, then we might do as well to
model an agent's knowledge using an arbitrary set of sentences,
as in Konolige 1986. The worry is that unconstrained impossible worlds
semantics makes no real progress over this purely syntactic approach
(Jago 2007, 2009).



One may instead adopt impossible worlds that retain *some*
logical structure, e.g., worlds closed under some
weaker-than-classical logical consequence. One approach of this kind
is found in Levesque 1984 (see also Cresswell 1973). This employs
impossible worlds of the kind used in paraconsistent relevant logics,
which can be locally inconsistent and incomplete but are well-behaved
with respect to conjunction and disjunction, that is, they are
adjunctive and prime. Laws of classical logic fail at them, and by
accessing them a cognitive agent can have inconsistent beliefs.
However, we still have a weakened form of logical omniscience: the
beliefs of an agent are closed under the weaker
paraconsistent-relevant logic at issue. This seems incorrect as an
attempt to model finite agents.



Rasmussen (2015) and Bjerring and Skipper (2019) present a
*dynamic* impossible worlds solution to the logical omniscience
problem. Agents' beliefs evolve over time due to *epistemic
actions*, on this approach (see the entry on
 dynamic epistemic logic
 for background). Bjerring and Skipper focus on *deductive*
actions. Agents count as competent insofar as they unfold the
consequences of their beliefs, up to a certain depth of reasoning.
Their operator "\(\langle n\rangle KA\)", says:
"After some \(n\)-step chain of logical reasoning, the agent
comes to know that \(A\)". The agent can update its epistemic
state by kicking out choices of impossible worlds which were epistemic
possibilities before the deduction took place. One can show that if a
formula \(A\) follows from formulas \(A\_1, \ldots, A\_n\) in \(n\)
steps of reasoning, then \(KA\_1, \ldots, KA\_n\) together entail
\(\langle n\rangle KA\).



Impossible worlds are starting to be used not only in formal, but also
in mainstream and Bayesian epistemology. Modal accounts of knowledge
invoking possible worlds, whether based on the notion of
 safety
 or on that of
 sensitivity,
 make all necessary or logical truths trivially sensitive and safe.
Melchior 2021 proposes to address the issue by using impossible
worlds. Probabilistic-Bayesian accounts of credences or degrees of
belief can be formulated using logically possible worlds and sets
thereof, to which probabilities are assigned by credence functions
obeying the Kolmogorov axioms of probability. As noted by Pettigrew
2021, this models logically omniscient, idealised agents who assign
probability 1 to all logical truths and never have higher credences in
any given premises than in their logical consequences. To model
rational but more realistic agents, Pettigrew introduces a setting
similar to that Berto and Jago 2019 had for all-or-nothing beliefs. He
resorts to "personally possible worlds" which make true
what a rational but cognitively limited agent believes and has not
(yet) ruled out using its bounded cognitive resources. Such worlds are
in fact impossible worlds of the open kind.


### 5.4 Impossible Worlds in Relevance Logic



Relevance logic
 (or relevant logic) is an attempt to capture the idea that good
reasoning requires a genuine condition between premises and
conclusions. This should go beyond a mere guarantee of truth
preservation. For when a conclusion \(A\) is guaranteed to be true
(for example, when it is a logical truth), *any* argument
concluding in \(A\) will preserve truth from premises to conclusions.
But those premises may have no genuine connection whatsoever to what
\(A\) is about. The same consideration applies when taking a
conditional \(A \rightarrow B\) to be valid: there should be a genuine
connection between \(A\) and \(B\).



With this in mind, relevance logic attempts to avoid the
'fallacies of relevance' (also called 'paradoxes of
the material conditional'). These are conditionals that are
valid in classical and modal logic, simply because the antecedent is a
necessary falsehood or the consequent is a necessary truth, but
without a guarantee of any real connection between them. Examples
include \(A \wedge \neg A \rightarrow B\) (*ex contradictione
quodlibet*, or the Law of Explosion), \(A \rightarrow B \vee \neg
B\), and \(A \rightarrow (B \rightarrow B)\) (*verum ex
quolibet*).



Contradiction-realizing impossible worlds can help us avoid *ex
contradictione quodlibet*, if they are worlds where some
contradiction \(A \wedge \neg A\) is true but some other \(B\) is not
true. For that reason, various systems of relevant logic have been
given semantics which include (things naturally thought of as)
impossible worlds.



A *Routley-Meyer* interpretation (see Routley and Routley 1972;
Routley and Meyer 1973, 1976; Routley 1979) for relevant
(propositional) logics is a structure \(\langle W, N, R, {}^\*,
v\rangle\), where \(W\) is a set of worlds; \(N\) is a proper subset
of \(W\) including the normal or possible worlds (the remaining worlds
are the non-normal or impossible worlds); \(R\) is a *ternary*
accessibility relation between worlds, and \({}^\*\) (the *Routley
star*) is a function from worlds to worlds. \({}^\*\) and \(R\)
figure prominently in the truth conditions for negation and the
(relevant) conditional. Their task is precisely to provide a semantics
for negation that allows for the truth of \(A\) and \(\neg A\) at some
worlds, and a semantics for the conditional that frees it from the
fallacies of relevance.


#### 5.4.1 The Relevant Conditional



In order to get rid of such entailments as \(A \rightarrow (B
\rightarrow B)\), we need some world at which \(A\) holds but \(B
\rightarrow B\) fails. One way to achieve this may be to admit
"partial" or incomplete situations of the kind studied in
 situation semantics,
 at which \(A\) holds but \(B \rightarrow B\) fails to hold, just
because they carry no information about \(B\). Another way is via
impossible worlds: an understanding of such worlds, as we have seen,
is as scenarios where logical laws may fail, and the Law of
(propositional) Identity, stating that any formula entails itself, is
one of them. At possible worlds, we still require for the truth of
conditionals \(A \rightarrow B\) that at every accessible world where
\(A\) holds, \(B\) holds, too. Consequently, \(A \rightarrow(B
\rightarrow B)\) is not logically valid. Technically, when \(w\) is an
impossible world, we state the truth conditions for the conditional,
by means of the ternary \(R\), as follows:


* (\(\rightarrow\)) 
\(v\_w (A \rightarrow B) =\) *true* if and only if, for all
worlds \(w\_1\) and \(w\_2\) such that \(Rww\_1 w\_2\), if \(v\_{w1}(A) =\)
*true*, then \(v\_{w2}(B) =\) *true*.



The key difference between \((\rightarrow)\) and the standard modal
clause for the strict conditional (which is true at a world if and
only if, at all accessible worlds where the antecedent is true, the
consequent is true), is that the worlds of the antecedent and the
consequent have been "split". Specifically, \(B
\rightarrow B\) fails at impossible worlds \(w\) when there are worlds
\(w\_1\) and \(w\_2\) such that R\(ww\_1 w\_2, B\) holds at the former,
but fails at the latter. Since we do not want \(B \rightarrow B\) to
fail at normal/possible worlds, we can add a *Normality
Condition*, saying that the accessible worlds \(w\_1\) and \(w\_2\)
are one and the same:


* (NC)  For normal worlds
\(w, Rww\_1 w\_2\) only if \(w\_1 = w\_2\).



Using the ternary relation \(R\), one can build models for different
relevant logics. Starting with the basic relevant system
\(\mathbf{B}\), one obtains models for stronger logics such as
\(\mathbf{R}\), the system of relevant implication, by adding
additional conditions on \(R\). (This is similar to the way we may
move on from basic modal logic \(\mathbf{K}\) to the systems
\(\mathbf{T}, \mathbf{S4}\), and \(\mathbf{S5}\), by adding extra
conditions on the accessibility relation.) The constraints to be added
to the ternary \(R\) are more complex than those of standard modal
logic and some involve the star operator \({}^\*\) (which we'll
address shortly).



It isn't easy to provide an intuitive reading for the ternary
relation \(R\). The basic idea is that the truth of an entailment \(A
\rightarrow B\) at a world \(w\) depends on \(w\)'s
"seeing an accessibility" (Bremer 2005: 67) between two
other worlds \(w\_1\) and \(w\_2\), such that if \(A\) is true at the
former, \(B\) is true at the latter. But what does this mean? This is
perhaps the most important philosophical issue facing semantics for
relevant logics. One approach is the information-based account of
Mares (2004, 2009, 2010) and Restall (1995), based on
 situation semantics.
 Another approach draws on various interpretations of
*conditionality*, such as those found in the literature on
conditional logics (Beall et al. 2012). (See the entry on
 relevance logic
 or Jago 2013c for further discussion.)


#### 5.4.2 The Routley Star



Given a world \(w\), the Routley star function outputs a world \(w^\*\)
which is, in a sense, its "reverse twin". The truth
conditions for negation within the Routley-Meyer semantics are:


* (\(\neg\))   \(v\_w
(\neg A) =\) *true* if and only if \(v\_{w^\*}(A) =\)
*false*.



So \(\neg A\) is true at a world \(w\) if and only if \(A\) is false,
not at \(w\) itself (as it happens with standard negation), but at its
twin \(w^\*\). Relevant negation is therefore an *intensional*
operator: in order to evaluate negated sentences at \(w\), we may need
to check the goings on at some other world.



Adding appropriate constraints provides this negation with many
intuitive inferential features. If \(w^{\*\*} = w\) for all worlds
\(w\), for example, Double Negation introduction and elimination is
valid. This is often called *De Morgan* negation, for De
Morgan's Laws hold of it. But it does not validate the Law of
Explosion (that a contradiction entails any sentence). For a
counterexample, take a model where \(A\) holds at \(w, B\)
doesn't hold at \(w\), and \(A\) doesn't hold at \(w^\*\).
Then, both \(A\) and \(\neg A\) hold at \(w\), whereas \(B\)
doesn't: \(w\) is an inconsistent but non-trivial world.



What is the intuitive connection between \(w\) and \(w^\*\)? The idea
is that the twins are "mirror images one of the other reversing
'in' and 'out'" (Dunn 1986: 191). If
\(w\) is \(A\)-inconsistent (both \(A\) and \(\neg A\) hold), then
\(w^\*\) is \(A\)-incomplete (neither \(A\) nor \(\neg A\) hold), and
vice versa. The \({}^\*\) takes local inconsistency into local
incompleteness and vice versa. It may also be the case that \(w =
w^\*\): the twins are in fact one. In that case, \(w\) must be a
maximal and consistent world, where negation behaves classically:
\(\neg A\) is true if and only if \(A\) is false *there*.


## 6. Objections to Impossible Worlds



This last section discusses some difficulties for impossible worlds
theories.


### 6.1 The Exportation Principle



Suppose that the expression 'at world \(w\)' works as a
restricting modifier: its main task consists in restricting the
quantifiers within its scope to parts of \(w\) (Lewis 1986). If so,
then it should distribute through the truth-functional connectives.
This means in particular that

 
\[ \text{At } w: (A \wedge \neg A) \]


will entail the contradiction

 
\[ (\text{At } w: A) \wedge \neg (\text{at } w: A). \]


This is the *exportation principle*. It is disastrous for any
theory of impossible worlds. It implies that an inconsistency at some
impossible world will spill over into an overt inconsistency. A true
contradiction at some \(w\) implies that there are true
contradictions, full stop. This is hard to swallow, unless one is a
dialetheist. (And even
 dialethists
 may want to reject the exportation principle for impossible worlds:
see Jago 2013b.)



Avoiding the exportation principle is not difficult, however (Kiourti
2010, Chapter 4; Jago 2013b). For it to be valid, it seems to require
genuine worlds, as in Lewis's (1986) genuine modal realism or
Yagisawa's (1988, 2010) extended modal realism. Adopting an
ersatz conception of worlds blocks the principle. If a world
represents that \(A\) (say, by containing a sentence or proposition
expressing *that A*), but is not in itself such that \(A\),
then the principle is blocked. 'At \(w: (A \wedge \neg
A)\)' will be interpreted as: \(w\) contains the sentence (or
proposition) *that A* \(\wedge \neg A\), whereas 'at \(w:
A\)' will mean that \(w\) contains the sentence (or proposition)
*that A*, and 'not at \(w: A\)' will mean that
\(w\) does not contain the sentence (or proposition) *that A.*
But an impossible world may contain both, or neither, of \(A\) and
\(\neg A\), so that containing one implies nothing about containing
the other. An impossible world may (depending on how fine-grained we
take them to be) also contain the conjunction \(A \wedge \neg A\)
independently of whether it contains both \(A\) and \(\neg A.\) So the
inference from 'at \(w: (A \wedge \neg A)\)' to '(at
\(w: A) \wedge \neg\)(at \(w: A)\)' is blocked (possibly twice
over).



It is controversial whether any genuine account of worlds can block
the exportation principle. Jago 2013a,b argues not; Yagisawa 2015
responds in defence of genuine impossible worlds.


### 6.2 Defining Possibility



If there are impossible worlds, then we cannot accept the simple
clause for possibility:


* (P)   It is possible
that \(A\) if and only if there's a world \(w\) such that, at
\(w\), \(A\).



Once impossible worlds enter the stage, (P) becomes false from right
to left. We therefore need a principle that restricts the
quantification in the right half of the biconditional to
*possible* worlds. How to do that, without appealing to modal
notions, is not straightforward. A hybrid account (Berto 2010) can
answer the objection, by taking possible worlds to be all and only the
genuine ones, with impossible worlds as ersatz constructions of some
kind. We can then capture the intent of (P) as:


* (P\({}^\*\))  It is
possible that \(A\) if and only if there's a genuine world \(w\)
such that, at \(w\), \(A\).



Note that it doesn't matter if the hybrid account accepts the
existence of ersatz possible worlds in addition to her genuine
possible worlds: (P\({}^\*\)) will still give the right result.



The ersatzist may respond by biting the bullet. Even without admitting
impossible worlds, most possible worlds accounts do not aim at
providing a reductive and complete analysis of modality.
(Lewis's modal realism does. But it may not succeed, when
so-called 'alien' properties, not instantiated by anything
at the actual world nor obtainable as constructions out of actually
instantiated properties, enter the stage. See Divers 2002, Chapter 7;
Divers and Melia 2002.)


### 6.3 The Usefulness of Impossible Worlds



Stalnaker (1996) argues that, whilst there is nothing wrong in
admitting impossible worlds, not much explanatory work can be expected
from them. For instance, if one takes worlds as sets of propositions,
one cannot then analyze propositions as sets of worlds. But an
advocate of impossible worlds may respond that the same point can be
made against any account of possible worlds (such as Adams's)
that takes them to be maximally consistent sets of propositions. And,
just as ersatz possible worlds needn't be constructed in this
way, nor need ersatz impossible worlds. Impossible worlds have found
many uses in the recent literature, as we have abundantly seen. So we
don't find Stalnaker's worry convincing.


### 6.4 The Semantics of Negation



The standard semantic clause for negation has it that \(\neg A\) is
true if and only if \(A\) is not true. So there could not be worlds at
which both or neither of \(A\) and \(\neg A\) are true, unless we
revise the semantics of negation. Stalnaker (1996) argues that
negation is such a basic operator, whose semantics is 'learned
in a first logic class' that it had better be left alone.



To this, the impossible world theorist can reply that it is in fact
the case that \(\neg A\) is true if and only if \(A\) is not true; for
this concerns truth *simpliciter*, that is, truth at the actual
world. She may also agree that the same holds for any possible world.
But it is precisely *im*possible worlds that we are talking
about here; how negation works at any possible world need not be
affected by the fact that, at some impossible world or other, some
sentence can hold together with its negation: this is one of the
things that makes them impossible, after all.


### 6.5 Counterpossible Reasoning



Timothy Williamson (2007, Chapter 5) has objected to non-trivial
treatments of counterpossibles, in particular (though not perforce)
when they resort to impossible worlds. Consider the claim:


1. If \(5 + 7\) were 13, then \(5 + 6\) would be 12.



*Prima facie*, this is a non-trivially true counterpossible.
However, Williamson argues, other non-trivial consequences of the
supposition would then be that \(5 + 5 = 11\), and \(5 + 4 = 10\), and
... , and \(0 = 1\). Therefore,


2. If the number of answers I gave to a given question were 0, then
the number of answers I gave would be 1.



But (2) is clearly false.



Brogaard and Salerno (2007) pose a dilemma for Williamson: either we
hold the context fixed in this kind of counterpossible reasoning, or
we don't. If we don't, then (2) does not follow from (1).
In particular, the context at which (2) comes out false is one at
which the closest antecedent worlds are possible and, to be sure, at
those worlds, 0 is not 1. But if we hold the context fixed, then what
does follow is just the following counterpossible:


3. If 0 were 1 and the number of right answers I gave were 0, then
the number of right answers I gave would be 1.



Now this is intuitively true, and non-trivially so.



Williamson (2007) also argues that non-trivial treatments of
counterpossibles create opaque contexts, in which the substitutivity
of co-referential terms fails. Supporters of non-trivial
counterpossibles will take the following conditional as false:


4. If Hesperus had not been Phosphorus, then Phosphorus would not
have been Phosphorus.



Given the necessity of identity, (4) is a counterpossible: as Hesperus
is Phosphorus, its antecedent can only be true at impossible worlds.
That's false, supposedly, because even if Hesperus and
Phosphorus had been distinct, Phosphorus would have remained
self-identical. However, all accept that


5. If Hesperus had not been Phosphorus, then Hesperus would not have
been Phosphorus



since it is an instance of 'if it had been that \(A\), then it
would have been that \(A\)'. Yet substituting
'Phosphorus' for the first occurrence of
'Hesperus' in the consequent of (5) gives (4). Since (5)
is true and (4) is (supposedly) false, substitution of identicals has
failed. This failure of substitutivity, Williamson claims, is a bad
result, for counterfactuals should *not* create opaque
contexts.



Brogaard and Salerno (2013) accept that counterpossibles do create
opaque contexts. They argue that the impossible worlds similarity
semantics for counterpossibles should be "partially
epistemic" (Brogaard and Salerno 2013: 654), and this epistemic
component explains the failure of substitution.



An alternative reply to Williamson is that the objection is
question-begging. It is clearly false that counterfactuals allow
substitution of identical terms (of arbitrary type). 'Had
Aristotle never taught, Aristotle would still have been
Aristotle' is true, and Aristotle is the teacher of Alexander,
yet 'had Aristotle never taught, Aristotle would still have been
the teacher of Alexander' is clearly false. The substitution
principle must (at the least) be restricted to *rigid
designators*: terms that denote the same entity in all
*metaphysically possible* worlds. But if the definition of
'rigid designator' is to be restricted to
*possible* worlds, then the application of the corresponding
substitution principle should likewise be restricted to contexts which
do not invoke impossible worlds. Viewed from this perspective, it is
question-begging to insist at the outset that the substitution
principle (for rigid designators) is valid for all counterfactuals
(including counterpossibles).



Berto et al. 2018 is an extensive discussion of a number of
Williamsonian objections to non-trivial treatments of
counterpossibles.


### 6.6 Compositionality



*Compositionality* is the principle that the meaning or content
of a complex expression is a function of the meanings of its
constituent expressions. It's commonly taken to be a mandatory
feature of any adequate theory of meaning and content. The argument is
that, as competent speakers of a language, we are in principle capable
of grasping the meanings of a potentially infinite number of
sentences. And since we've learnt the meanings of a limited
number of words, this is possible only if the meanings of complex
sentences are obtainable recursively from the meanings of their
constituent parts (Davidson 1965).



The worry is that a theory of meaning or content which includes
impossible worlds will not be compositional. Consider what we said
above about the exportation principle (Section 6.1) and the semantics
of negation (Section 6.4). An impossible world may represent that
\(\neg A\) independently of whether it represents that \(A\). But
then, for such worlds, the truth-value of \(\neg A\) is not a function
of the truth-value of \(A\). So, the worry goes, the content or
proposition that \(\neg A\), understood as a set of possible and
impossible worlds, will not be a function of the proposition that
\(A\). Certainly, the latter is not the set-theoretic complement of
the former, as it is on the possible worlds account. The same goes for
all the other logically complex sentences. This is perhaps the most
serious objection to the impossible worlds approach. If it can't
be met, it may well be fatal.



To address the worry, defenders of impossible worlds must show that
their notion of content is compositional, even if it does not provide
uniform truth-conditions across all worlds. In other words, they must
specify a way to calculate complex contents from constituent contents,
but which does not go via the usual truth-at-a-world clauses for
connectives. As far as we know, the only attempt to achieve this is in
Berto and Jago 2019, Chapter 8. They view contents as sets of
(possible and impossible) worlds, which are themselves sets of
sentences of some 'worldmaking' language. They argue that
grammatical structures can thus be recovered from semantic contents,
via the syntactic structure of the worldmaking sentences involved.
This provides a functional map between grammatical structure and
semantic content which in turn, they argue, provides a means for
calculating complex contents from their constituents. Whether this
provides an acceptable notion of compositionality (as they claim)
remains to be seen.



The compositionally objection also appears in Williamson 2020, a book
mostly devoted to a defense of the extensional material conditional as
giving the meaning of the indicative "if", but also
dealing with counterfactuals. Williamson accuses hyperintensional
approaches to conditionals and to content in general -- whether
based on impossible worlds, or of other kinds, e.g., based on
truthmakers -- of "overfitting": complicating the
semantics in order to account for the variability and systematic
inconsistencies of speakers' judgments, with the result of
incorporating "noise" in their models. At the extreme of
fine-graining, open impossible worlds approaches "must
individuate meanings so finely as to restrict synonymy to
self-synonymy, and thereby render the conception of meaning
theoretically useless, because it filters nothing out"
(249).



Of course, hyperintensionalists (with impossible worlds sympathies, or
of other kind) may equally accuse extensional or merely intensional
semantics of "underfitting": conflating contents we may
have good reasons to keep distinct. We seem to need some principled
way to mark the proper boundaries of semantics. Williamson proposes
one: we should pay more attention to a cognitive or epistemic level,
intermediate between semantics and pragmatics. Many intuitive
distinctions hyperintensionalists try to force into the semantics are
better explained as belonging to competent speakers'
cognitive-epistemic heuristics. He grants, for instance, that the view
that counterpossibles are not all trivially true speaks to our
intuitions; but objects that "sometimes, the robustly shared
verdicts of native speakers on a sample sentence will simply be false,
the predictable output of a fallible human heuristic."
(265).



Some authors, e.g. Rothschild (2021), argue that the Williamsonian
stance is at odds with standard practice in semantics. The early
chapters of any textbook introduction to semantics (e.g., Chierchia
and McConnell-Ginet 1990; Heim and Kratzer 1997) tell us that one key
task of semantics is to account for (rather than explain away)
competent speakers' intuitions and judgments of synonymy,
antonymy, entailment, etc. Williamson's endorsement of the
extensional material conditional gives, of course, a very simple
semantics for the indicative "if"; but, says Rothschild,
"the simplicity comes at the cost of failing to integrate
conditionals with the compositional semantics of natural
language" (22). And once the heuristic procedures are developed
in order to carry out the aforementioned key task, "it is not
clear that Williamson's overall system will still look
simple". (Ibid.)



The compositionally objection is also raised by Fine 2021, who
proposes his truthmaker semantics as a hyperintensional account of
content in place of possible and impossible worlds-based accounts.
Fine's semantics (2017a,b) uses *states* in place of
worlds, where states are things that may fail to be maximal (making
some sentences neither true nor false), and they may be inconsistent.
Inconsistent states may be formed as mereological sums of consistent
states, e.g., summing the state in which the table is round with one
in which it is square gives an impossible situation in which there is
a round square table. (To handle conjunction, Fine requires each set
of states to have a mereological sum, so he is committed to
inconsistent states like this.) Impossible worlds, too, may be
inconsistent and incomplete. One live issue is thus whether truthmaker
semantics is a notational variant of a kind of impossible worlds
semantics, rather than an alternative to it. Ontologically, they seem
on a par and, in this sense, truthmaker semantics can be seen as a
form of impossible worlds semantics. However, it is probably more
appropriate to view truthmaker semantics as a rival to standard
impossible worlds accounts. Central to truthmaker semantics is the
notion of *exact truthmaking*, whereby the state in question is
wholly relevant to the sentence's truth. Truthmaker semantics
stakes its reputation on the utility of this notion, whereas standard
impossible worlds semantics does not.



Fine's approach is appealing with respect to the compositionally
worry: it provides uniform clauses for the Boolean connectives at a
state, irrespective of whether the state is consistent. But the same
can be said of some impossible worlds semantics, e.g. the worlds
semantics for First Degree Entailment (FDE), a simple four-valued,
paraconsistent logic (Dunn 1976, Belnap 1977a-b) which has been
provided with a worlds semantics including worlds where formulas can
be both true and false, or neither true nor false. But whether uniform
clauses can be given for all operators is unclear. In Fine's
2012 semantics for counterfactuals, for instance, counterfactuals may
be evaluated only with respect to possible worlds (which are, in
effect, maximal possible states), with the consequence that embedded
counterfactuals are not permitted. It is currently unclear whether
evaluation of counterfactuals at impossible states would require
non-uniform clauses.



Another compositionally-based worry concerns negation: are the
negations of equivalent sentences themselves equivalent? This will not
be the case in general within impossible worlds semantics. The
situation with truthmaker semantics is not straightforward: it holds
in some but not all systems (e.g. not in Fine and Jago 2019).
Fine's response is not to banish those weaker systems, but to
prefer certain systems for certain purposes. That response is also
open to friends of impossible worlds.



Undoubtedly, other objections to impossible worlds can and are likely
to be raised. The current debate on impossible worlds appears to be at
the same stage as the one on *possible* worlds was, some forty
years ago. At that time, people struggled to make sense of the concept
of a possible world. Many declared it meaningless. Nowadays, the
variety of its applications has placed the notion firmly at the core
of much philosophical and logical practice (see e.g., Divers 2002,
Chapter 4). Impossible worlds may undergo the same fate, should they
prove as useful as they appear to be in the treatment of
impossibilities of various kinds.