Impossible Worlds and Content Inclusion
In this paper, I will argue that the relevant logician[In
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
worldsI will use “worlds”
to include both senses of worlds outlined above, but will distinguish the two
when necessary.
See Priest, (2008), page
190-1.
See Restall, (1999) for a more
interesting discussion of states.
Restall,
(1999) page 7.
See Berto (2007) for a
more detailed discussion on R’s intuitive readings.
recall that our interpretation includes N, the set of all
normal worlds, a subset of W. Nonnormal worlds are the worlds in W –
N
Using something called the ‘normality condition’
we can have R handle conditionals at all worlds, not just normal ones. See
Priest (2008) page 189.
Note that,
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
fail”.
Priest (1992), page
298-9.
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
world
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
fails.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.
Some
authors
See Priest (2008) sec. 9.7 for an elucidation on this
point.
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
different.
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''”
(emphasis his).Restall (1999), page 4.
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
(1992).
For
reference, all interpretations I that are similar will be such that I =
A ∨ ¬A,
-0 (1)
A, -0 (2)
¬A, -0 (3)
r00#0#
(4)
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.For those
confused, (4) is required by the normality condition (as 0 is normal) and (5)
follows from (3) and (4).
These facts are not necessary for
the demonstration of LEM. They are just useful to know.
Due
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
Rww1w2)
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.
See Priest (2008) and Read (1988) for an introduction and
an in-depth discussion of the intuitive reading.
If a
∈N, a* ⊑
a.
A ∨ ¬A,
-0 (6)
A, -0 (7)
¬A, -0 (8)
r00#0#
(9)
A, +0# (10)
0# ≼
0
(11)
A, +0
(12)
x
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”.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
have.
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.
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.
a ∈N, a* ⊑ a, N the
set of normal worlds.
We say
that if w ⊑ w’, then everything that is true at w is true at
w’.
a ∈N, a* ⊑ a, N the
set of normal worlds.