One of the most popular and well known accounts of the
identity-conditions of facts and properties is the modal theory. According to this theory: i) two
facts are identical iff they are necessarily equivalent to each other;
and ii) two properties are identical iff they are necessarily
coextensive to each other. That is, the modal theory holds that: i) the
fact that = the fact that
iff, necessarily, ( iff ); and ii) the property of being
= the property of being iff, necessarily, for any , ( is iff is ). This theory is prima
facie attractive, since it is simple to formulate and provides an
account of the identity-conditions of facts and properties in terms of
(at least relatively) well understood notions. Everything else being
equal, the theory is also more parsimonious than rival theories that
reject it, since, everything else being equal, there are less facts if
the modal theory holds than if it fails to hold and there are distinct
facts that are necessarily equivalent to each other.
A prominent argument against the modal theory is the dual-detector
argument originally due to Elliot Sober.
Briefly, according to this argument, there could be a machine that, as a
result of containing detectors measuring different aspects of an input,
is causally sensitive to one fact without being causally sensitive to
another necessarily equivalent fact. Since, by Leibniz’s law, it follows
from this that, contra the modal theory, there are distinct facts that
are necessarily equivalent to each other, the argument concludes that
the modal theory is false. Despite this argument’s prominence,
discussions of the argument by both its proponents and opponents have
been brief and cursory. This paper will provide a more sustained
evaluation of the dual-detector argument and will argue that such an
evaluation shows that the argument is unsuccessful.
I will proceed as follows. In section 1, I
will formulate the dual-detector argument before then arguing in
section 2 that it is unsuccessful. In
section 3, I will then consider two variants
of the argument and I will argue that these variants, and more generally
that all variants, are also unsuccessful.
Before proceeding to section 1, it will be
useful to briefly discuss another common argument against the modal
theory—the constituency argument—in order to set it aside (see, for example, Audi 2016).
Suppose (1) and (2) are
true, where ‘’ refers to some
particular wire.
(1) is a closed straight-sided figure that
has three angles.
(2) is a closed straight-sided figure that
has three sides.
According to the constituency argument, since the fact expressed by
(1) has angularity as a constituent while the fact
expressed by (2) doesn’t have this property as a
constituent, the facts expressed by (1) and (2) are not identical to each other. Since the modal
theory entails that the facts expressed by (1) and
(2) are identical to each other (since they are
necessarily equivalent to each other), the constituency argument
concludes from this that the modal theory is false.
The constituency argument arguably begs the question against the
modal theory by in effect assuming the rival structured theory of facts.
According to this rival theory, facts are structured in the same kind of
way that sentences are structured. In particular, according to the
structured theory, facts are built up out of objects, properties,
relations, operators and quantifiers in the same way that sentences are
built up out of names, predicates, operator expressions and quantifier
expressions. If the structured theory holds so
that the facts expressed by (1) and (2) are built up out of objects, properties,
relations, operators and quantifiers in the same kind of way that
sentences are built up out of names, predicates and other expressions,
then it is plausible that the fact expressed by (1) has angularity as a constituent while the fact
expressed by (2) doesn’t have this constituent.
This is much less plausible, however, if the structured theory is false
and facts aren’t structured like sentences. For example, if facts are
instead structured like visual experiences or pictures, then, since it
is prima facie plausible to associate (1) and (2) with the same (type) of visual experience or
picture, it is prima facie plausible that (1) and
(2) express the same fact and hence prima facie
plausible that the facts expressed by (1) and (2) don’t differ in what constituents they have. (This
is because it is at least prima facie plausible that any picture that
represents as being a closed
straight-sided figure that has three angles also represents
as being a closed straight-sided
figure that has three sides, and vice versa.) Since the
argument from constituency provides no reason to think that facts are
structured in the way that the structured theory holds that they are
structured, rather than some other way, the argument therefore fails to
provide a good reason to think that (1) and (2) express distinct facts and hence fails to provide
a good reason to reject the modal theory.
It is important to appreciate that the structured theory is neither
self-evident nor prima facie highly plausible, and hence it cannot
simply be assumed to hold in the above argument from constituency
without begging the question against the modal theory. Three brief
reasons for this are the following: First, prior to investigation and
argument, the claim that facts are structured like sentences is no more
plausible than the claim that facts have some other type of structure,
such as that of visual experiences or pictures. Second, while (1) and (2) arguably differ in
their cognitive significance, since a linguistically competent person
arguably might endorse one of them while rejecting the other, such a
difference in cognitive significance is widely thought to be able to be
explained by a difference in what mode of presentation the facts
expressed by (1) and (2)
have when expressed by these sentences, where this explanation does not
require that the facts expressed by these sentences are non-identical
(see, for example, Braun 1998; McKay and
Nelson 2010). Third, the structured theory conflicts with
claims that are widely thought to be at least as prima facie plausible
as the structured theory itself, such as the claim made by (3).
(3) ‘ is self-identical’ expresses the same
fact as ‘ is identical to ’.
(3) conflicts with the structured theory,
since, if the structured theory is true, the fact expressed by ‘ is self-identical’ has the property of
being self-identical as a constituent while the fact expressed by ‘ is identical to ’ lacks this constituent and instead has
the property of being identical to as a constituent. Due
to the above difficulty with the constituency argument, and since we
cannot simply assume the structured theory in arguing against the modal
theory, I will assume in the following that the constituency argument
against the modal theory fails.
The Dual-Detector Argument
The dual-detector argument is not meant to rely on the cogency of the
constituency argument discussed above, nor is it meant to rely on the
truth of the structured theory of facts. Instead, the dual-detector
argument is meant to provide a separate reason for rejecting the modal
theory. The argument involves a machine that contains two detectors: a closed
straight-sided figure detector and a three-angle detector. These
detectors are linked in a series in , so that, if a wire (or several wires)
are inputted into , they are first
inputted into the closed straight-sided figure detector and then, if
they are outputted by this first detector, they are inputted into the
three-angle detector. If the wire (or wires) are then outputted by the
three-angle detector, they are then outputted by . Indeed, I will assume in the following
that what it is for something (or some things) to be outputted by is just for it (or them) to be
outputted by this second detector.
The closed straight-sided figure detector in works so that “when given a piece of
wire as input, it will output the piece of wire if and only if the wire
is a closed [(plane)] figure and all sides of the figure are straight”
(Sober 1982,
185). More explicitly, let us say that: i) when given a piece
of wire as input that is a closed figure all of whose sides are
straight, the closed straight-sided figure detector outputs the wire,
and it does this because the wire is a closed figure all of
whose sides are straight; whereas, ii) when given a piece of wire (or
several pieces of wire) as input that is not a single closed figure all
of whose sides are straight, the closed-straight-sided figure detector
does not output it (or them). The three-angle detector, on the other
hand, works so that “when given any number of straight pieces of wire,
it outputs them if and only if they have three angles” (Sober 1982, 185). More explicitly: i)
when given one or more pieces of wire with straight sides that
collectively have three angles, the three-angle detector outputs them
and it does this because the wire (or wires) collectively have
three angles; whereas, ii) when given one or more pieces of wire with
straight sides that don’t collectively have three angles, the
three-angle detector does not output them. The three-angle detector is
causally sensitive to whether the input has three angles, and not to
whether it has three sides, since, when given a four-sided open figure,
it will output the object (since it has three angles), and it will fail
to do this if the four-sided figure is closed. In addition, when the
three-angle detector is given three unconnected pieces of wire, each
containing exactly one angle, the detector will output them, even though
it is made up of six straight line segments.
Sober states the dual-detector argument as follows:
Now consider a particular object—a piece of wire—which is fed into
the machine, passes through both [detectors], and is then outputted by
the machine. What property of the object caused it to be
outputted? Given the mechanism at work here, I think that the cause was
the object’s having the property of being a closed straight-sided
figure having three angles (i.e., its being a triangle), and not
its being a closed straight-sided figure having three sides
(i.e., its being a trilateral). If this is right, and if a difference in
causal efficacy is enough to insure a difference in property, it follows
that being a triangle is not the same property as being a trilateral,
even though “triangle” and “trilateral” are logically (mathematically)
equivalent. (Sober 1982, 185,
author’s emphasis)
Let ‘[]’ abbreviate ‘the
fact that ’, and suppose that
is the piece of wire that is fed
into . Let us also suppose that
the above process of being fed
into and then being sequentially outputted by the two detectors has
occurred. Then, according to Sober’s dual-detector argument, (Angle)
is true while (Side) is false.
Angle. [
is a closed straight-sided figure that has three angles] causes
[ outputs ].
Side.
[ is a closed
straight-sided figure that has three sides] causes [ outputs ].
The dual-detector argument then employs Leibniz’s law to infer from
this that, since they differ in what they cause, [ is a closed straight-sided figure
having three angles] is not identical to the necessarily
equivalent [ is a closed
straight-sided figure having three sides]. The argument then
infers from (4) and the non-identity of these
facts that the property of being a closed straight-sided figure that has
three angles (or being triangular) is not identical to the
necessarily coextensive property of being a closed straight-sided figure
that has three sides (or being trilateral).
(4) For any , IF is ,
is , and the property of being
the property of being , THEN .
Since these facts and properties are respectively necessarily
equivalent to each other and necessarily coextensive with each other
(and hence are identical to each other according to the modal theory),
the dual-detector argument then concludes from the above results that
the modal theory is false.
Against the Dual-Detector
Argument
One initial problem with the dual-detector argument is that (Angle)
is not strictly speaking true, at least if we assume as we did above
that the above described process involving and has already occurred.
Angle. [
is a closed straight-sided figure that has three angles] causes
[ outputs ].
To see why this is the case, let us suppose that, after being fed
into and put inside the closed
straight-sided figure detector at , is outputted by the closed
straight-sided figure detector so that, at , is inside the three-angle detector. Let
us also suppose that being inside
the three-angle detector at
results in being outputted by the
three-angle detector at , and
hence results in being outputted
by at . Finally, let us also suppose that
the times , and are all past times. Then the fact
that is a closed straight-sided
figure that has three angles (either simpliciter or at the present time)
does not cause to do anything to
, since is no longer interacting with .
The above problem with the dual-detector argument shows that, as it
is most charitably understood, it is not (Angle) that is true
according to the argument, but is instead either (Angle) or (Angle).
Angle. [At , is a closed straight-sided figure that
has three angles] causes [ outputs at ].
Angle. [At , is a closed straight-sided figure that
has three angles] causes [ outputs at ].
As a result of this need to relativise to either time or time , we therefore have two versions of
the dual-detector argument. The first version—the -version—holds that (Angle) is true and (Side) is false, from which
it infers that, contra the modal theory, the necessarily equivalent
facts [at , is a closed straight-sided figure that
has three angles] and [at , is a closed straight-sided figure that
has three sides] are non-identical.
Side. [At , is a closed straight-sided figure that
has three sides] causes [
outputs at ].
The second version of the dual-detector argument—the -version—holds instead that (Angle) is true and (Side) is false, from which
it infers that, contra the modal theory, the necessarily equivalent
facts [at , is a closed straight-sided figure that
has three angles] and [at , is a closed straight-sided figure that
has three sides] are non-identical.
Side. [At , is a closed straight-sided figure that
has three sides] causes [
outputs at ].
As we will see, both these versions of the dual-detector argument
have serious problems.
The -version of the
dual-detector argument can be quickly seen to fail as follows: It is
[ has three angles at ] that causes to be outputted by the three-angle
detector at , rather than say
[at , has three angles and is blue]
that causes this fact (even supposing that is blue at ). This is intuitively because [at
, has three angles and is blue] goes
beyond what is causally relevant to whether is outputted by the three-angle
detector at . Similarly, it is
[ has three angles at ] that causes to be outputted by the three-angle
detector at rather than [at
, is a closed straight-sided
figure that has three angles] that causes this fact. This is
because the latter fact also goes beyond what is causally relevant to
whether gets outputted by the
three-angle detector at .
Since getting outputted by the
three-angle detector just is what it is for to be outputted by , it follows that (Angle) is false.
Angle. [At , is a closed straight-sided figure that
has three angles] causes [ outputs at ].
Since the falsity of (Angle) conflicts with the
-version of the dual-detector
argument, this version of the argument fails.
I will now argue that the -version of dual-detector argument
is also unsuccessful and hence that both versions of the dual-detector
argument fail. I will do this by first giving an argument from causal
exclusion that, contrary to the dual-detector argument, (Angle) is false. I will
then argue that, even if this causal exclusion argument is rejected, it
is not possible to justify both the truth of (Angle) and the falsity of
(Side), the justification
of both of which is required for the -version of the argument to be
successful. (Or at least, I will argue that one cannot justify the truth
of (Angle) and the falsity of
(Side) without appealing to
some other argument against the modal theory that, if successful, would
refute the modal theory by itself and hence would render the
dual-detector argument superfluous.)
To set up the needed background for the argument from causal
exclusion against (Angle), note that, in the
case of processing , [ is a closed straight-sided figure at
] causes to be outputted by the closed
straight-sided figure detector, and so causes to be in the three-angle detector at
. Hence we have (5)
(5) [ is a closed straight-sided figure at
] causes [ is in the three-angle detector at ].
Since [ is in the three-angle
detector at ] and [ has three angles at ] collectively cause to be outputted by the three-angle
detector at , which is what it
is to be outputted by at , we also have (6).
(6) [ is in the three-angle detector at ] and [ has three angles at ] collectively cause [ outputs at ].
Since plausibly one of the causes of having three angles at is that it had three angles at
previous times before , (7) plausibly also holds.
(7) [ has three angles at ] causes [ has three angles at ].
Assuming, as is plausible, that the causal transitivity principle (T)
holds in this causal situation, (5–7) then entail (Angle).
T.
IF the members of
collectively cause , the
members of collectively
cause and , collectively cause ; THEN the members of
collectively cause .
Angle. [
is a closed straight-sided figure at ] and [ has three angles at ] collectively cause [ outputs at ].
With the above background in place, it might seem like it should now
be easy to derive (Angle) from (Angle), and hence
establish that (Angle) holds.
Angle. [At , is a closed straight-sided figure that
has three angles] causes [ outputs at ].
However using the above background, we can now give the following
argument from causal exclusion that (Angle) is instead false:
Just as [at , is a closed straight-sided figure that
has three angles] fails to cause the closed straight-sided figure
detector to output at (since the former fact goes beyond
what is causally relevant), [at , is a closed straight-sided figure that
has three angles] fails to cause the closed straight-sided figure
detector to output (since this
fact also goes beyond what is causally relevant) and hence this fact
fails to cause [ is in the
three-angle detector at ].
Hence we have (8).
(8) [at , is a closed straight-sided figure that
has three angles] does not cause [
is in the three angle detector at ].
Similarly, while [ has three
angles at ] is a cause of
[ has three angles at ], it is not the case that [at , is a closed straight-sided figure that
has three angles] causes this fact, since it goes beyond what is
causally relevant. Hence we have (9).
(9) [at , is a closed straight-sided figure that
has three angles] does not cause [
has three angles at ].
Since [at , is a closed straight-sided figure that
has three angles] is also not caused by either [ is in the three-angle detector at ] or [ has three angles at ], and the causal chain that leads
up to [ outputs at ] goes through [ is in the three-angle detector at ] and [ has three angles at ], it therefore follows from (8) and (9) that [at , is a closed straight-sided figure that
has three angles] isn’t part of the causal chain that leads to [ outputs at ] and hence does not cause it. Hence
(Angle) is false.
Angle. [At , is a closed straight-sided figure that
has three angles] causes [ outputs at ].
A more rigorous version of the above argument against (Angle) can be given by
appealing to the version of the principle of causal exclusion given by
(PCE).
PCE.
In cases where there is no genuine causal overdetermination, if
is a set of facts that occur at a
time whose members collectively
completely cause , then is the unique set of facts that occur
at and collectively completely
cause .
In (PCE), a fact is said to occur at a certain
time iff the fact only concerns how things are at that time. Genuine
causal overdeterminism, on the other hand, occurs when two independent
causal processes converge on the same effect, such as when a house burns
down because a lit match starts a fire in the garbage at the same time
as lightning strikes the house.
Since there is no genuine causal overdetermination in the case of
being outputted by , (PCE) can be used to argue that (Angle) is false as follows:
Suppose, for reductio, that (Angle) is true. Then [ is a closed straight-sided figure that
has three angles at ] together
with the members of some possibly empty set completely cause [ outputs at ]. Since (Angle) holds, it is
also true that [ is a closed
straight-sided figure at ],
[ has three angles at ] together with the members of some
possibly empty set
collectively completely cause [
outputs at ]. Since the relevant facts occur at
the same time, these two consequences together with (PCE) then entail (10).
(10) [ is a closed straight-sided figure at
],[ has three angles at ], [ is a closed straight-sided figure that
has three angles] and the members of some possibly empty set collectively completely cause [ outputs at ].
If (10) is true, then [ is a closed straight-sided figure at
], [ has three angles at ] and the members of by themselves collectively
completely cause [ outputs at ], since [at , is closed straight-sided figure that
has three angles] is superfluous given the presence of [ is a closed straight-sided figure at
] and [ has three angles at ]. Given (PCE), however, this
consequence conflicts with (10). Hence, the
reductio assumption (Angle) is false.
The above argument shows that (Angle) fails to hold if (PCE)
holds. Not all philosophers, however, accept (PCE), and these
philosophers will not be convinced by the above argument from causal
exclusion that the dual-detector argument fails. For example, some
philosophers reject (PCE) on the grounds that it conflicts with
the popular counterfactual dependency thesis (Dep).
Dep.
Suppose that and obtain, and that, had failed to obtain, it would have been
that failed to obtain. Then,
causes .
Other philosophers reject (PCE) because they hold that, in cases where
there is no genuine causal overdetermination of a fact, there can still
be multiple complete causal chains that converge on that fact, provided
these chains are systematically related to each other in the right way.
In particular, some philosophers hold that there can be multiple such
causal chains provided that, for each such chain, either that chain
generates all the other chains, or that chain is generated by at least
one other such chain. Someone who endorses this view, for example, might
endorse (Conj).
Conj.
If and together with the members of a set
collectively completely cause
, then the conjunction of and together with the members of collectively completely cause .
It follows from (Conj) that, contra (PCE), if there is one
causal chain leading to that
contains the facts and occurring at a time , then there is a further causal chain
which is systematically related to it by virtue of containing the
conjunction of and instead of and themselves. Given (Conj),
it is natural to hold that this further causal chain containing the
conjunction of and is generated by the former chain
containing its conjuncts.
In light of the above views, the argument from causal exclusion does
not by itself decisively refute the -version of the dual-detector
argument. In addition to facing the argument from causal exclusion,
however, the -version of the
dual-detector argument faces the problem that, even if the caual
exclusion argument fails, it doesn’t appear possible to justify the
truth of (Angle) while also
justifying the falsity of (Side). (Or at least, it
doesn’t seem possible to do this without relying on some other argument
against the modal theory which, if successful, would by itself refute
the modal theory. I will discuss two attempts to give such a
justification, and I will argue that both these attempts fail. The
failure of these two attempts will give us reason to think that no such
justification is possible, and hence reason to think that, even if (PCE) and
the argument from causal exclusion fail, the -version of the dual-detector
argument is still unsuccessful.
The first attempt to justify the truth of (Angle) (while also
justifying the falsehood of (Side) appeals to (Conj)
above. This first attempt accepts (Angle) on the basis of
the transitivity reasoning given for it above. It then infers from (Angle) and (Conj)
that the conjunction of [ is a
closed straight-sided figure at ] and [ has three angles at ] collectively (partially) cause
to output at . Assuming (as I will from now on)
that this conjunction is the fact [at , is a closed straight-sided figure that
has three angles], it follows from this that (Angle) is true.
Angle. [At , is a closed straight-sided figure that
has three angles] causes [ outputs at ].
Let us assume that the above justification of (Angle) is successful. The
question that now needs to be addressed is whether we can go on to
justify the falsehood of (Side).
Side. [At , is a closed straight-sided figure that
has three sides] causes [
outputs at ].
One argument that tries to justify the falsehood of (Side) is the following:
Unlike (Angle), (Side) cannot be generated
from the causal facts given to us in the description of processing given in the dual-detector argument
using causal generational principles such as (T) and (Conj). As a result, the
truth of (Side) would require some
additional primitive causal fact to hold in the case of processing , which would be unparsimonious.
Moreover, since any such additional primitive causal fact would only
contingently hold, the possibility of such a fact holding can be removed
by simply stipulating that no such additional primitive causal fact
holds in the possible case of
processing that we are concerned
with. Hence, according to this argument, the truth of (Side) can be ruled out
either on parsimony grounds or by stipulation.
The problem with this argument for the falsity of (Side) is that it begs the
question against the modal theory. It does this because, if the modal
theory is true, then, contra the above argument, (Side) can be generated
from the causal facts given to us in the description of the case of
processing in the dual-detector argument and the
generational principles (T) and (Conj) in the same way
that (Angle) can be so generated.
This is because, if the modal theory is true, then [at , is a closed straight-sided figure that
has three sides] is the conjunction of [ is a closed straight-sided figure at
] and [ has three angles at ], just as much as [at , is a closed straight-sided figure that
has three angles] is. Hence, if the modal theory is true, then
(Side) can be derived from
(Angle) and (Conj) in
the same way that (Angle) can.
An alternative way of trying to justify the falsehood of (Side) appeals to (Conj).
Conj. If the conjunction of and partially causes , then and collectively partially cause .
We can give the same kind of argument from parsimony and contingency
for the falsity of (Side) as was given
above for the falsity of (Side), with the difference
that this argument for the falsity of (Side), unlike the
argument for the falsity of (Side), does not beg the
question against the modal theory.
Side. [
is a closed straight-sided figure at ] and [ has three sides at ] collectively cause [ outputs at ].
Indeed, plausibly both opponents and proponents of the modal theory
should reject (Side). Given the
falsity of (Side), however, the
falsity of (Side) follows from (Conj).
If we are justified in endorsing (Conj), then, we can use it to
justify the falsehood of (Side).
One problem with (Conj) is that the principle
directly conflicts with the modal theory. This is because, if the modal
theory holds, then (Conj) has the absurd
consequence that, if partially
causes , then any fact that is necessitated by also causes . (This is because, according to the
modal theory, if a fact
necessitates a fact , then is the conjunction of and .) If [Suzy throws a rock] causes [the
window breaks], for example, then, if the modal theory holds, (Conj) entails that [Suzy
throws a rock or Suzy does not throw a rock] (which is necessitated by
[Suzy throws a rock]) also causes [the window breaks], which is absurd.
In light of this, one problem with (Conj) is that, if it is
accepted, then we don’t need the dual-detector argument to refute the
modal theory, since (Conj) by itself achieves this
task. If the dual-detector argument needs to rely on (Conj) in order to be
successful, then, the argument is superfluous.
A second (more serious) problem with (Conj) is that it is not clear
why we should believe it. A proponent of (Conj) might attempt to justify
the principle by arguing that, in ordinary language, sentences of the
form (11) are equivalent to sentences of the
form (12).
(11) because and .
(12) because and because .
Such a proponent might then argue that (on its relevant causal use)
(11) is equivalent to (11)
and (12) is equivalent to (12).
(11) [ and ] causes [].
(12) [ and ] collectively cause [].
Assuming that these equivalences all hold, it follows that (11)
entails (12), from which it follows that (Conj) holds.
A problem with this attempted justification for (Conj) is that (12) is plausibly ambiguous between a conjunctive
reading and a non-conjunctive reading, just like (13) is (cf. Marshall 2021, 8035).
(13) Jane wants to go swimming and
go hiking.
(13) has a non-conjunctive reading on
which the proposition Jane is described as desiring is the proposition
that Jane goes swimming and hiking. On this reading, (13) is true iff (13) is
true.
(13) Jane wants to go (swimming and
hiking).
(13) also has a conjunctive reading on
which (13) is true iff (13) is
true.
(13) Jane wants to go swimming and
Jane wants to go hiking.
(11) is plausibly similarly ambiguous
between a non-conjunctive reading on which it is equivalent to (11) and a
conjunctive reading on which it is equivalent to (11).
(11) because ( and ).
(11) ( because ) and ( because ).
On its conjunctive reading, while (11) is
equivalent to ((12) (on its causal use), there
is no reason to think that (on its causal use) (11) is equivalent to (11)
(or at least no such reason has yet been provided).
On its non-conjunctive reading, on the other hand, there is no reason to
think that (11) is equivalent to (12). As a result, appealing to natural language
does not appear to help a proponent of the dual-detector argument
justify (Conj). In light of this, it is
not clear how (Conj) might be justified. As a result, it does not appear
possible to justify the truth of (Angle) by appealing to (Conj)
while also justifying the falsity of (Side).
I will discuss one further attempt to justify both the truth of (Angle) and the falsity of
(Side). Instead of
appealing to (Conj), this second attempt appeals to the
popular counterfactual dependency thesis (Dep) stated above.
Dep.
Suppose that and obtain, and that, had failed to obtain, it would have been
that failed to obtain. Then,
causes .
Assuming that (Dep) holds, we can derive (Angle) as follows: In the
case of outputting , had it not been that, at , was a closed straight-sided figure that
had three angles, then either: i) would not have been a closed
straight-sided figure at ; or
ii) would not have had three
angles at , in which case
would also not have had three
angles at . If had failed to be a closed
straight-sided figure at ,
would not have been outputted by
the closed straight-sided figure detector at , and hence would not have been outputted by at . On the other hand, if had failed to have three angles at
, it would not have been
outputted by the three-angle detector at , and hence would also not have been
outputted by at . Hence, had it not been that, at
, was a closed straight-sided figure that
had three angles, would
not have outputted at . It therefore follows from (Dep)
that (Angle) is true.
Angle. [At , is a closed straight-sided figure that
has three angles] causes [ outputs at ].
Assuming that (Dep) holds, then, a proponent of the
dual-detector argument can use (Dep) to justify (Angle). Unfortunately for
proponents of the dual-detector argument, however, if (Dep)
holds it can also be used to justify the truth of (Side). To see why, note
that, had it not been that, at , was a closed straight-sided figure that
had three sides, then
would also either: i) not have been a closed straight-sided figure at
or ii) not have had three
angles at , in which case it
would not have had three angles at . Hence, had it not been that, at
, was a closed straight-sided figure
having three sides, at least one of the detectors would not
have outputted , and so would not have outputted at . Hence, it also follows from (Dep)
that (Side) is true. Hence, a
proponent of the dual-detector argument cannot use (Dep) to justify the
combination of (Angle) being true and (Side) being false. This
second attempt at justifying the truth of (Angle) and the falsehood of
(Side) therefore fails.
I have now discussed two attempts to justify the truth of (Angle) and the falsity of
(Side), and I have argued
that both of these attempts fail. As far as I can see, other attempts to
do this are equally unsuccessful. If this is the case, then both the
-version and the -version of the dual-detector
argument fail.
Variants of the Dual-Detector
Argument
In the face of the failure of the original version of Sober’s
dual-detector argument, it might be thought that the argument can be
modified so that it evades the problems discussed in section 2. In particular, it might be thought that these
problems can be evaded by replacing the necessarily equivalent facts
expressed by (1) and (2) with some other necessarily
equivalent facts and describing a machine that is causally sensitive to
one of these facts but not the other.
(1) is a closed straight-sided figure that
has three angles at .
(2) is a closed straight-sided figure that
has three sides at .
As far as I can see, however, this cannot be done.
To illustrate the difficulty involved in successfully modifying the
dual-detector argument in the above manner, I will briefly consider two
attempts to do this that replace the facts expressed by (1) and (2) with
the facts expressed by (14) and (15), where is a circular wire and where the
facts expressed by (14) and (15) are both necessarily equivalent to the fact
that is a circle.
(14) is a closed (plane) figure all
of whose points are equidistant from a point.
(15) is a closed (plane) figure of
constant curvature.
For the first attempt, consider a machine that, when given a closed
(plane) figure as an input, scans that figure by having a distinct
curvature detector for each point of the figure. Suppose that each of
these detectors measures the curvature of their associated point in the
figure and sends the result of this measurement in the form of a signal
to the CPU of .
Further, suppose that, if all the signals the CPU receives are of the
same value, then the fact that the signals it receives have the same
value causes the figure to be outputted by . Finally, suppose that the
circular wire is inputted
into this machine , is
scanned by it, and is then outputted by it. It might then be claimed
that, in this case, (Curv) is true while (Dist) is false, and
that, due to Leibniz’s law, this difference in truth-value entails that
the modal theory is false.
Curv.
[ is a closed figure
with constant curvature] causes [ outputs ].
Dist.
[ is a closed figure
all of whose points are equidistant from a point] causes [ outputs ].
A problem with this first attempt at finding a successful variant of
the dual-detector argument is that it is no more obvious that (Curv)
holds than it is that (Angle) holds in Sober’s
original case.
Angle. [At , is a closed straight-sided figure that
has three angles] causes [ outputs at ].
Instead, using transitivity reasoning, what can be uncontroversially
established in the variant case of machine is a claim along the lines
of (Curv), just as what can be
uncontroversially established using such reasoning in Sober’s original
case of machine is (Angle).
Curv. [Point of has curvature ], [point of has curvature ] collectively cause [ outputs ].
Angle. [
is a closed straight-sided figure at ] and [ has three angles at ] collectively cause [ outputs at ].
Moreover, an opponent of the modal theory who wishes to show that (Curv)
and (Dist) differ in their truth-value faces the
same challenges that a proponent of Sober’s original version of the
dual-detector argument faces in showing that (Angle) and (Side) differ in their
truth-value. First, they need to resist an argument from causal
exclusion that (Curv) entails the falsehood of
(Curv). And second, they need to find some
way of justifying the truth of (Curv) while also justifying the falsehood
of (Dist), a task that appears to be just as
difficult as finding a way of justifying the truth of (Angle) while also
justifying the falsehood of (Side). Hence, this first
attempt at describing a machine that is differentially sensitive to the
facts expressed by (14) and (15) results in a variant of the dual-detector
argument that is no more successful than Sober’s original argument.
For a second attempt to show that there could be a machine that is
causally sensitive to one of the facts expressed by (14) and (15) but not
the other, consider a machine that contains an extendable
straight rod that rotates around one of its endpoints. When given a
closed figure as input, works by placing this rod
inside the inputted closed figure, fixing the location of one of the
rod’s endpoints, extending the length of the rod until its other
endpoint touches the inputted figure, and then rotating the rod around
its fixed endpoint while keeping the length of the rod fixed. If the rod
does a full rotation without moving the inputted figure or losing touch
with it, then the fact that it does this causes to output the figure.
Suppose now that the circular wire is inputted into and that the rod of is placed inside of and does a full rotation meeting
the above conditions, so that gets outputted by . It might then be claimed
that, in this case, (Dist) is true and (Curv) is false, and
hence that the modal theory is false.
The problem with this second variant of Sober’s version of the
dual-detector argument is that, if is a circle that is inputted
into and then outputted by , then there is no reason to
think that (Curv) and (Dist) differ in their
truth-value. In particular, if is so inputted and outputted, it
is equally plausible to say that the machine measures the curvature of
the points of as it is to
say that it measures the equidistance of those points from a common
point. After all, the rod would fail to do its full rotation (while
touching but not moving )
if the points of didn’t
have constant curvature, just as it would fail to do this if the points
of weren’t equally distant
from some common point. There is therefore no grounds for thinking that
being outputted by is due to one of these facts
rather than the other. Hence, also fails to be a
demonstrable case of a machine that is causally sensitive to one of the
facts expressed by (14) and (15) and not the other.
Other variations of Sober’s original version of the dual-detector
argument face similar problems to those described above. Indeed, the
above two attempts to construct a successful variant of Sober’s original
version of the argument arguably illustrate a dilemma facing any such
attempt. This dilemma is the following: Suppose we have a machine whose
output is intended to be caused by the fact and not by the necessarily
equivalent fact . Then the
machine will either contain multiple detectors that differ in what
aspects of the input they measure (as in the cases of and ), or the machine will only
contain detectors (or a single detector) that don’t so differ (as in the
case of ). If the
machine contains multiple detectors that differ in what aspects of the
input they measure, then the argument against the modal theory based on
this machine will arguably face the same challenges facing Sober’s
original argument and the first variant of it discussed above. In
particular, the argument will need to resist an argument from causal
exclusion and will face the same difficulties in justifying the claim
that the input being outputted is caused by and not by that Sober’s original dual-detector
argument faces in justifying the truth of (Angle) and the falsity of
(Side). On the other hand,
if the machine contains only a single detector (or multiple detectors
that don’t differ in what aspects of the input they measure), then it
will arguably fail to be even initially plausible that and differ in whether they cause the
input to be outputted just as there is no even initial plausibility for
thinking that the facts expressed by (14) and
(15) differ in whether they cause to be outputted by machine . Hence, whether or not we
have a machine that contains detectors that differ in what aspects they
measure, the argument against the modal theory based on this machine
will arguably fail. In light of this, it is reasonable to conclude that,
not only does Sober’s original version of the dual-detector argument
fail, but it is not possible to modify the argument so that it is
successful. If this is correct, then all variants of the dual-detector
argument fail and some other kind of argument will be needed if we are
to have reason to reject the modal theory of facts and properties.
Acknowledgements
Research in this paper was supported by an Early Career Scheme grant
from the Research Grants Council of Hong Kong SAR,China (LU23607616).
Thanks to Andrew Brenner, Daniel Waxman and three anonymous referees for
their valuable comments on this paper.