Impossible Worlds and Content Inclusion
In this paper, I will argue that the relevant logician
general, a relevant logician is one which has requirements on the material
conditional. They aim to eliminate paradoxes of the material conditional found
in classical logic. The primary requirement is that the antecedent and
consequent share some propositional parameter or variable. Then, for an
interpretation v, vw(A → B) = 1 iff for all
x, y ∈ W such
that vx(A) = 1, vy(B) = 1. This evaluation requires a
ternary relation, R. This is a necessary condition for relevance logics. For
more, see Priest (1992, 2008). ] is required to accept philosophically and
metaphysically intractable positions in order to preserve the Law of Excluded
Middle (LEM). I will do this by first explaining the interpretations of a
relevant logic in terms of the ternary relation R, the worlds W, the Routley
Star, and content inclusion. I will consider two possible understandings of
worlds, one being in the sense used by classical modal logic, and another in
terms of information states from authors like Restall. I will explain what the
semantics of R in a relevant logic mean with respect to each of these concepts
of worlds, and what the Routley Star and its characterization of star worlds
offers to the relevant logician in each of these cases. Then, I will discuss
what content inclusion means for both understandings of worlds and how it is
required for LEM to be valid. All of these things come together in the
interpretations I of a relevant logic, with these interpretations being some
structure , with W being the set of all worlds and N
the set of all normal worlds, R the ternary relation, ∗ the Routley Star
operation on worlds in W, ⊑ being the binary operation on worlds called
content inclusion, and ν as the interpretation function. Something is valid in
a relevant logic [I say “a relevant logic” here not
because I am confused, but because there are no fewer than 10 relevant logics to
be considered. The primary targets here are those like BX, DWX, and TWX, which
accept LEM (see Priest (2008) page 203 for more information on their
differences). ] with this structure just in case truth is preserved in
normal worlds. That is to say, it is impossible for the premises to be true and
the conclusion false in all normal worlds of our interpretations. Thus, in
introducing the binary relation of content inclusion to keep LEM as a logical
truth, the relevant logician is required to accept the claim that logically
impossible worlds inform our understanding of logical truths in a normal world.
That is to say, for LEM to be valid, the truths of logically impossible
worlds must be appealed to.
The most distinguishing characteristic of a logic is its interpretation.
Demonstrations of validity for the relevance logician will rely heavily on the
relation in the interpretation. Thus, one of the most important aspects of a
relevant logic is its ternary relation. There are two primary things that this
relation can be between; that of worlds, or that of information states. The case
in which the relation is between worlds is rather simple and is what one would
usually expect. However, the relation changes slightly to accommodate additions
to the interpretation like ∗ or ⊑. The ternary relation primarily
changes the evaluation of the material conditional, and the change from a binary
relation to that of a ternary one is mainly motivated by the requirement of
relevance between the antecedent and the consequent in a material conditional
(see footnote 1, above). Thus, R is impacted on how it bears out in evaluating
the material conditional, and thus only matters in such cases with respect to
normal worlds, and otherwise behaves as we would usually expect it to.
I will use “worlds”
to include both senses of worlds outlined above, but will distinguish the two
However, there is an alternative view to what this relation
could be between. One such position is that of possible information states.
See Priest, (2008), page
For instance, we can have what are
called “states”, or ways in which the world could be. These states are
information in parts of worlds, and behave as we may expect states of
information to behave. Thus, the way in which a relevant logician could read the
ternary relation R is as follows. Rxyz will hold when the information given by x
(in a particular state), when considered with respect to another information
state y, also holds in z. As Restall puts it, “you are to think of the
information supported by x as ‘data’, and that supported by y as
information to be applied to the data. If the results are all supported by z,
then Rxyz holds”.
See Restall, (1999) for a more
interesting discussion of states.
This is the best that I can do for an intuitive
reading on what R is. Although R resembles the modal relation due to Kripke and
others in modal semantics, it is by no means the same and the two should not be
(1999) page 7.
discussion on R will be important later in discussing worlds, most notably for
Let us now turn to W, the set of worlds in our interpretations. We have both
normal and nonnormal worlds in this interpretation.
See Berto (2007) for a
more detailed discussion on R’s intuitive readings.
Normal worlds are the worlds in which validity is determined and
the conditional may be evaluated.
recall that our interpretation includes N, the set of all
normal worlds, a subset of W. Nonnormal worlds are the worlds in W –
Our collection of worlds also include a
curious set of counterparts to the worlds we usually have. These worlds are star
(∗) worlds, and the ∗ operation is a function between worlds. This
function produces a new understanding of negation. With this new operation,
vw(¬p) = 1 if vw*(p) = 0 and is otherwise zero. Thus, for
every world w in W, there is a star world (or mate world) called w*. These star
worlds are of serious import to the relevant logician. For instance, to validate
contraposition, the relation R is such that if Rww1w2,
then Rww2*w1*. In addition, it is
important that w** = w. This way, double negation is also preserved for the
relevant logician. Here now we turn to the primary concern: what are these star
worlds supposed to be? It’s certainly nice that they allow us to verify
particular logical validities that we appreciate having in classical
Using something called the ‘normality condition’
we can have R handle conditionals at all worlds, not just normal ones. See
Priest (2008) page 189.
, but we presumably decided that
classical logic was getting it wrong at some point; the worry may be that these
star worlds are too problematic if we want to be sure that relevant logic is
getting it right in the ways classical logic wasn’t.
classically, w* = w for all worlds.
These star worlds offer an
interesting turn on negation. Indeed, negation now becomes an intensional
operator, and behaves much differently from how we expect it to behave as it
does in the classical case. It does not on the face of it seem obvious as to why
one might require star worlds in their semantics if they have nonnormal worlds.
The primary issue one runs into with only including normal worlds in a relevant
logic is that with negation. If negation is defined intrinsically for the world
itself – that is to say, evaluated at the world in question – then the
purpose of a relevant logic is ignored. For instance, if negation is evaluable
at a world as being the complement of a statement’s truth value, A →
(B ∨ ¬B) is valid at that world.
This is pointed out by Priest (1992)
regarding Routley. The important point here is that a relevant logic requires
something important about implication, and negation will overwrite this
importance if not treated carefully.
In essence, if the
interpretation only considers negation intrinsically, the entire point of
introducing nonnormal worlds is lost. That is to say, LEM could become a logical
truth at even nonnormal worlds, but the entire purpose of nonnormal worlds was
to introduce possibilities where laws of logic may fail. Graham Priest offers an
explanation of the ∗ operator and why it belongs. He says that “∗
is, in fact, a device for ensuring that there are non-normal worlds that are
inconsistent and in-complete, and so where certain “laws of logic” may
Priest (1992), page
Thus, the ∗ operator provides a look into precisely the
kinds of impossible worlds we wish to include. While possible worlds are in
general not a problematic topic, these nonnormal and logically impossible worlds
are of a strikingly distinct character than the usual worlds. What we are left
with is the question of what to make of these nonnormal worlds, and determine
how precisely they might differ from the actual world and other possible worlds.
There are several different kinds of impossible worlds that have been proposed,
each deserving its own short treatment and explanation.
The first kind of impossible
Many thanks are due to Berto
(2009) for explicitly underlining these various definitions of impossible worlds
across different articles.
are those which are most analogous to
possible worlds. Take possible worlds to be representations of sentences or
propositions, or as states of affairs that could have been but are not. For
instance, it could perhaps be physically possible that I am six inches taller.
Thus, there is some world different from the actual world we inhabit in which I
am such a height and not much else is different. Likewise, for impossible
worlds, it might be that there are some things that simply cannot be the case.
Impossible worlds would be those where states of affairs that cannot happen,
happen. These things that can’t happen are impossibilities and, as such,
populate the set of impossible worlds; these worlds are things which we would
expect to be exactly not possible. The other three options for what an
impossible world is are relative to a logic or a logical law instead of being an
analogy to traditional possible worlds. For instance, an impossible world may be
one in which the laws of logic are simply different than they are in possible
worlds. That is to say, relative to some logic, an impossible world would be one
in which the collection of truths in that world would not be true under that
logic’s interpretations. For example, there may be some world at which A and
A → C are true. However, at such an impossible
world, it might not follow that C is true simply because modus ponens
Another case may be one in which impossible worlds are regarded as simply being
worlds in which classical interpretations are different. This then would be a
special case of the previous kind of impossible world, privileging the classical
perspective in classifying these worlds.
Finally, impossible worlds may just be worlds in which actual contradictions
exist. For instance, violations of the law of noncontradiction may be
common-place and accepted. These definitions are not necessarily exclusive nor
exhaustive ones and many can be taken simultaneously. The real question might be
why we should allow impossible worlds at all in the first place.
This way of thinking is perhaps similar to how other
logics operate. For instance, the paraconsistent logician already knows what it
is for modus ponens to fail. Why we should necessarily believe impossible worlds
to be paraconsistent worlds is perhaps a matter of personal preference on the
truth of paraconsistent logics.
See Priest (2008) sec. 9.7 for an elucidation on this
argue that there is no good reason to rule out
impossible worlds if one accepts possible worlds. If one is comfortable with the
idea of physically impossible worlds where things travel faster than the speed
of light, one should also consider those worlds in which logical laws are
See Priest (1992) for a
similar and more detailed description of these impossible worlds.
These raise the question, why must the whole world itself
be impossible? Why not just have parts of a world be impossible? This runs along
the idea of our worlds not being real, existing things but instead states of
information. As Restall puts it, “we can consider ways that parts of our
world could be, and ways that parts of our world couldn’t be. All of these
entities will be called ''states''”
Thus, our impossible worlds can be thought of as states
in which our information is incomplete or perhaps inconsistent. These states are
ways in which worlds could be, and again there is perhaps no good reason to
suspect that this could not be the case.
Restall (1999), page 4.
It is at this point that
we turn to the Law of Excluded Middle. LEM is not valid for any interpretations I that are structurally
similar to the one I have explained thus far.
As Priest might say, while there are no
states of information at which logical laws may not be the case, these are
precisely the kinds of states where we might expect them to be false; namely,
impossible ones. For a better understanding of impossible worlds, see Priest
What follows is
the tableau for LEM:
reference, all interpretations I that are similar will be such that I =
A ∨ ¬A,
A, -0 (2)
¬A, -0 (3)
A, +0# (5)
As you can see, we end up stuck at (5), being unable to infer anything about the
truth of A in 0.
The only move we can make between worlds
and their mate worlds is with negation itself. Without something like ¬A,
-0# (which would yield A, +0 and close our branch),
LEM is not a validity. The question thus becomes, what is missing from our
interpretation to keep LEM valid?
Here is where we introduce the so far overlooked relation of content inclusion.
Content inclusion is a reflexive and transitive binary relation on worlds in
confused, (4) is required by the normality condition (as 0 is normal) and (5)
follows from (3) and (4).
and is a
rather intuitive relation. We say that if w ⊑ w’, then everything that
is true at w is true at w’.
These facts are not necessary for
the demonstration of LEM. They are just useful to know.
As a result of this introduction, we have a new set of tableaux
rules and ways of evaluating validity in our language.
to Priest (2008), we see that ⊑ satisfies the following constraints:#
If vw(p) = 1 then vw’(p) = 1 # w’* ⊑ w* # if
Rw’w1w2 then (w ∈ N and w1
⊑ w2) or (w ∉ N and
Note that the identity relation is one which
satisfies these conditions, and so our previous interpretations which were
lacking content inclusion simply need identity and we have a similar structure.
The relation of
content inclusion allows us to make the crucial inference we needed before to
demonstrate LEM’s validity. For instance, we can introduce the following
constraint: if a is a normal world, then whatever is true in a’s star world
is also true in a itself.
See Priest (2008) and Read (1988) for an introduction and
an in-depth discussion of the intuitive reading.
∈N, a* ⊑
A ∨ ¬A,
A, -0 (7)
¬A, -0 (8)
A, +0# (10)
The crucial line for the branch to close is now
available in (11) and follows from the constraint imposed above. While this
preserves the validity of LEM and the tableaux rules make intuitive sense, we
find ourselves appealing to the notion of inclusion between normal worlds and
their counterparts. We must evaluate LEM as being valid by appealing to
impossible worlds. Is this a satisfying fact? I argue that this development is
unsettling and that accepting these costs for LEM is too large a price.
Content inclusion seems like an intuitive kind of
relation. Indeed, it might make a lot of intuitive sense that a world contains
the truths of the worlds that are included in it. If we consider possible
worlds, for instance, it seems that, at least in principle, we would be able to
organize these worlds in terms of proximity to the actual world, where the
nearest of possible worlds are the most like the actual world, such that for
almost every fact p both the actual world and the possible worlds agree. It
makes even more intuitive sense in the case where our worlds are simply states
of information or ways the world can be. Per Restall, “there is a
relationship of ''involvement'' x ≤ y between states. To say
that x ≤ y is just to say that
being y includes being x, or that y involves x”.
Another way of phrasing this is that if we have a certain amount of information
in one state, we do not lose that information in a later one.
Restall (1999) page 5. He offers a far more feature rich
account of inclusion than Priest (2008) and fleshes out the notion of
“states” much more clearly than I can do justice in the space I
From this position we may now
see where content inclusion and impossible worlds fit together. Indeed, Edwin
Mares says that “we must distinguish between the assertion of ¬A and
the denial of A… If ¬A obtains in [a situation] s, then it obtains in all
⊑–extensions of s”.
This discussion is similar to how an intuitionist
logician may read their account of validity. For the intuitionist account, see
Priest (2008). Involvement behaves similarly to the heredity rule in
intuitionist logic. The only strong reason I have for the intuitionist logician
not being a target here as well is related to the fact that the intuitionist
does not require nonnormal worlds, whereas the relevant logician (in at least
this case) demands them.
Looking at the constraint we introduce
for the content inclusion operation
Mares (2008). Mares has an excellent discussion on
situations and states and goes into great detail on their import to the
semantics of relevant logic.
, we discover a concerning fact. If our world w
is a normal world, then w’s star world, w*, is content included in w. As 0,
the actual world, is a normal world, then 0* ⊑ 0. As a result, those facts
which obtain in 0* obtain in its ⊑-extension, 0. What this all effectively
means, then, is that what is true in our logically impossible worlds has bearing
on what is true in the actual world. While this works out quite nicely for a
demonstration that LEM is a logical truth in the actual world, it leaves us with
some concerning metaphysical considerations. Most prominently, we must answer
the question of what these impossible worlds are for if content inclusion aligns
with its intuitive reading
a ∈N, a* ⊑ a, N the
set of normal worlds.
, then everything that is true at impossible worlds is true
in their ⊑-extensions. Thus, it must be considered how different definitions
of impossible worlds may impact the actual world. As described above, there are
at least four possible kinds of impossible worlds. If impossible worlds are ways
in which the world cannot be, there are seriously strong issues presented. For
if a logically impossible situation obtains in w*, then that logical
impossibility likewise obtains in w. But this seems paradoxical; the logical
impossibility in question occurs in w* precisely because it cannot obtain in w.
This seems to be the most problematic definition of an impossible world if
content inclusion is part of our interpretation. The other three considerations
of impossible worlds are likewise concerning. If the actual world is one in
which classical logic has the correct interpretation for logical laws and our
impossible worlds are ones in which classical laws do not obtain, then similarly
we would not expect them to obtain in content included extensions of those
impossible worlds. If an impossible world is one in which actual contradictions,
all the worse for the actual world, for then those contradictions exist there as
well. It appears then that the actual world is in fact some kind of impossible
world, and not very normal indeed. But if the actual world is not normal, then
it is not the case that the content inclusion restriction
that if w ⊑ w’, then everything that is true at w is true at
holds for our world, and LEM cannot be
verified. It might seem that we are therefore left only with logically
impossible worlds or information states such that some logical laws are false,
but nothing violates those laws. In the case of physical laws, it may perhaps be
the case that objects could travel beyond the speed of light, it just isn’t
the case that any object does or ever will, perhaps due to other facts in the
world. Analogously for logical laws, it perhaps is the case that LEM or modus
ponens are false, but if they are not logical laws nothing is in such a way in
the world to demonstrate their failure. We would therefore have no way of
determining if this is the case without some sort of prior metaphysical
commitment compelling us to think this is the case. As it were, there is nothing
in the language of the logic itself to suggest if our logical laws are
''actually'' valid or invalid, but rather only a determination of what is
possibly so. Our logic leaves it unclear what we should accept. The
consideration we are forced to make is a metaphysical one: what is the nature of
our worlds and are we at ease with the actual world being some kind of logical
As I have argued here, attempts by the relevant logician to save the Law of
Excluded Middle entail serious metaphysical issues about the nature of this
world with respect to how it relates to impossible worlds. The issues of the
semantics of R and ∗, while not being very intuitive, are not necessarily
the biggest issue facing particular relevant logics when it comes to
intelligibility. It remains unclear what it might mean for impossible worlds to
share their true content in the actual world. In order for the relevant logician
to recognize LEM as valid, they need some sort of way of relating the content of
a logically impossible world to the actual world. Whether these worlds are mere
possibilities or existing objects, or if they are simply ways worlds can be
taken as information states, the problem of content inclusion seems like it
renders our logic unusable for determining validity anyways. We need some kind
of deeper, more robust metaphysical or philosophical framework by which we can
determine what these kinds of worlds might look like, and it must be careful to
avoid being circular.
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Nolan, D. P. (2013). Impossible Worlds.
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Priest, G. (1992). What Is a Non-Normal World?
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Dilyn Corner (C) 2020-2022
a ∈N, a* ⊑ a, N the
set of normal worlds.