# Falsification and consciousness

Johannes Kleiner<EMAIL_ADDRESS>Munich Center for Mathematical
Philosophy, Ludwig Maximilian University of Munich  Erik Hoel
<EMAIL_ADDRESS>Allen Discovery Center, Tufts University, Medford, MA, USA

###### Abstract

The search for a scientific theory of consciousness should result in theories
that are falsifiable. However, here we show that falsification is especially
problematic for theories of consciousness. We formally describe the standard
experimental setup for testing these theories. Based on a theory’s application
to some physical system, such as the brain, testing requires comparing a
theory’s predicted experience (given some internal observables of the system
like brain imaging data) with an inferred experience (using report or
behavior). If there is a mismatch between inference and prediction, a theory
is falsified. We show that if inference and prediction are independent, it
follows that any minimally informative theory of consciousness is
automatically falsified. This is deeply problematic since the field’s reliance
on report or behavior to infer conscious experiences implies such
independence, so this fragility affects many contemporary theories of
consciousness. Furthermore, we show that if inference and prediction are
strictly dependent, it follows that a theory is unfalsifiable. This affects
theories which claim consciousness to be determined by report or behavior.
Finally, we explore possible ways out of this dilemma.

##  1 Introduction

Successful scientific fields move from exploratory studies and observations to
the point where theories are proposed that can offer precise predictions.
Within neuroscience the attempt to understand consciousness has moved out of
the exploratory stage and there are now a number of theories of consciousness
capable of predictions that have been advanced by various authors (Koch et
al., 2016).

At this point in the field’s development falsification has become relevant. In
general, scientific theories should strive to make testable predictions
(Popper, 1959). In the search for a scientific theory of consciousness,
falsifiability must be considered explicitly as it is commonly assumed that
consciousness itself cannot be directly observed, instead it can only be
inferred based off of report or behavior.

Contemporary neuroscientific theories of consciousness first began to be
proposed in the early 1990s (Crick, 1994). Some have been based directly on
neurophysiological correlates, such as proposing that consciousness is
associated with neurons firing at a particular frequency (Crick and Koch,
1990) or activity in some particular area of the brain like the claustrum
(Crick and Koch, 2005). Other theories have focused more on the dynamics of
neural processing, such as the degree of recurrent neural connectivity (Lamme,
2006). Others yet have focused on the “global workspace” of the brain, based
on how signals are propagated across different brain regions (Baars, 1997).
Specifically, Global Neuronal Workspace theory claims that consciousness is
the result of an “avalanche” or “ignition” of widespread neural activity
created by an interconnected but dispersed network of neurons with long-range
connections (Sergent and Dehaene, 2004).

Another avenue of research strives to derive a theory of consciousness from
analysis of phenomenal experience. The most promising example thereof is
Integrated Information Theory (Tononi, 2004, 2008; Oizumi et al., 2014).
Historically, Integrated Information Theory is the first well-formalized
theory of consciousness. It was the first (and arguably may still be the lone)
theory that makes precise quantitative predictions about both the contents and
level of consciousness (Tononi, 2004). Specifically, the theory takes the form
of a function, the input of which is data derived from some physical system’s
internal observables, while the output of this function are predictions about
the contents of consciousness (represented mathematically as an element of an
experience space) and the level of consciousness (represented by a scalar
value $\Phi$).

Both Global Neuronal Workspace (GNW) and Integrated Information Theory (IIT)
have gained widespread popularity, sparked general interest in consciousness,
and have led to dozens if not hundreds of new empirical studies (Massimini et
al., 2005; Del Cul et al., 2007; Dehaene and Changeux, 2011; Gosseries et al.,
2014; Wenzel et al., 2019). Indeed, there are already significant resources
being spent attempting to falsify either GNW or IIT in the form of a global
effort pre-registering predictions from the two theories so that testing can
be conducted in controlled circumstances by researchers across the world
(Ball., 2019; Reardon, 2019). We therefore often refer to both GNW and IIT as
exemplar theories within consciousness research and show how our results apply
to both. However, our results and reasoning apply to most contemporary
theories, e.g. (Lau and Rosenthal, 2011; Chang et al., 2019), particularly in
their ideal forms. Note that we refer to both “theories” of consciousness and
also “models” of consciousness, and use these interchangeably (Seth, 2007).

Due to IIT’s level of formalization as a theory, it has triggered the most in-
depth responses, expansions, and criticisms (Cerullo, 2015; Bayne, 2018;
Mediano et al., 2019; Kleiner and Tull, 2020) since well-formalized theories
are much easier to criticize than non-formalized theories. Recently one
criticism levied against IIT was based on how the theory predicts feedfoward
neural networks have zero $\Phi$ and recurrent neural networks have non-zero
$\Phi$. Since a given recurrent neural network can be “unfolded” into a
feedfoward one while preserving its output function, this has been argued to
render IIT outside the realm of science (Doerig et al., 2019). Replies have
criticised the assumptions which underlie the derivation of this argument
(Kleiner, 2020; Tsuchiya et al., 2019).

Here we frame and expand concerns around testing and falsification of theories
by examining a more general question: what are the conditions under which
theories of consciousness (beyond IIT alone) can be falsified? We outline a
parsimonious description of theory testing with minimal assumptions based on
first principles. In this agnostic setup falsifying a theory of consciousness
is the result of finding a mismatch between the inferred contents of
consciousness (usually based on report or behavior) and the contents of
consciousness as predicted by the theory (based on the internal observables of
the system under question).

This mismatch between prediction and inference is critical for an empirically
meaningful scientific agenda, because a theory’s prediction of the state and
content of consciousness on its own cannot be assessed. For instance, imagine
a theory that predicts (based on internal observables like brain dynamics)
that a subject is seeing an image of a cat. Without any reference to report or
outside information, there can be no falsification of this theory, since it
cannot be assessed whether the subject was actually seeing a “dog” rather than
“cat.” Falsifying a theory of consciousness is based on finding such
mismatches between reported experiences and predictions.

In the following work, we formalize this by describing the prototypical
experimental setup for testing a theory of consciousness. We come to a
surprising conclusion: a widespread experimental assumption implies that most
contemporary theories of consciousness are already falsified.

The assumption in question is the independence of an experimenter’s inferences
about consciousness from a theory’s predictions. To demonstrate the problems
this independence creates for contemporary theories, we introduce a
“substitution argument.” This argument is based on the fact that many systems
are equivalent in their reports (e.g., their outputs are identical for the
same inputs) and yet their internal observables may differ greatly. This
argument constitutes both a generalization and correction of the “unfolding
argument” against IIT presented in Doerig et al. (2019). Examples of such
substitutions may involve substituting a brain with a Turing machine or a
cellular automaton since both types of systems are capable of universal
computation (Turing, 1937; Wolfram, 1984) and hence may emulate the brain’s
responses, or replacing a deep neural network with a single-layer neural
network, since both types of networks can approximate any given function
(Hornik et al., 1989; Schäfer and Zimmermann, 2006).

Crucially, our results do not imply that falsifications are impossible.
Rather, they show that the independence assumption implies that whenever there
is an experiment where a theory’s predictions based on internal observables
and a system’s reports agree, there exists also an actual physical system that
falsifies the theory. One consequence is that the “unfolding argument”
concerning IIT (Doerig et al., 2019) is merely a small subset of a much larger
issue that affects all contemporary theories which seek to make predictions
about experience off of internal observables. Our conclusion shows that if
independence holds, all such theories come “pre-falsified.” Thus, instead of
putting the blame of this problem on individual theories of consciousness, we
show that it is due to issues of falsification in the scientific study of
consciousness, particularly the field’s contemporary useage of report or
behavior to infer conscious experiences.

A simple response to avoid this problem is to claim that report and inference
are not independent. This is the case, e.g., in behaviorist theories of
consciousness, but arguably also in Global Workspace Theory (Baars, 2005), the
“attention schema” theory of consciousness (Graziano and Webb, 2015) or “fame
in the brain” (Dennett, 1993) proposals. We study this answer in detail and
find that making a theory’s predictions and an experimenter’s inferences
strictly dependent leads to pathological unfalsifiability.

Our results show that if independence of prediction and inference holds true,
as in contemporary cases where report about experiences is relied upon, it is
likely that no current theory of consciousness is correct. Alternatively, if
the assumption of independence is rejected, theories rapidly become
unfalsifiable. While this dilemma may seem like a highly negative conclusion,
we take it to show that our understanding of testing theories of consciousness
may need to change to deal with these issues.

##  2 Formal description of testing theories

Here we provide a formal framework for experimentally testing a particular
class of theories of consciousness. The class we consider makes predictions
about the conscious experience of physical systems based on observations or
measurements. This class describes many contemporary theories, including
leading theories such as Integrated Information Theory (Oizumi et al., 2014),
Global Neuronal Workspace Theory (Dehaene and Changeux, 2004), Predictive
Processing (when applied to account for conscious experience (Dolega and
Dewhurst, 2020; Clark, 2019; Seth, 2014; Hobson et al., 2014; Hohwy, 2012)) or
Higher Order Thought Theory (Rosenthal, 2002). In some cases, contemporary
theories in this class may lack the specificity to actually make precise
predictions in their current form. Therefore, the formalisms we introduce may
sometimes describe a more advanced form of a theory, one that can actually
make predictions.

In the following section, we introduce the necessary terms to define how to
falsify this class of theories: how measurement of a physical system’s
observables results in datasets (Section 2.1), how a theory makes use of those
datasets to offer predictions about consciousness (Section 2.2), how an
experimenter makes inferences about a physical system’s experiences (Section
2.3), and finally how falsification of a theory occurs when there is a
mismatch between a theory’s prediction and an experimenter’s inference
(Section 2.4). In Section 2.5 we give a summary of the introduced terms. In
subsequent sections we explore the consequences of this setup, such as how all
contemporary theories are already falsified if the data used by inferences and
predictions are independent, and also how theories are unfalsifiable if this
is changed to a strict form of dependency.

###  2.1 Experiments

All experimental attempts to either falsify or confirm a member of the class
of theories we consider begin by examining some particular physical system
which has some specific physical configuration, state, or dynamics, $p$. For
example, in IIT, the class of systems $P$ may be some Markov chains, set of
logic gates, or neurons in the brain, and every $p\in P$ denotes that system
being in a particular state at some time $t$. On the other hand, for Global
Neuronal Workspace, $P$ might comprise the set of long-range cortical
connections that make up the global workspace of the brain, with $p$ being the
activity of that global workspace at that time.

Testing a physical system necessitates experiments or observations. For
instance, neuroimaging tools like fMRI or EEG have to be used in order to
obtain information about the brain. This information is used to create
datasets such as functional networks, wiring diagrams, models, or transition
probability matrices. To formalize this process, we denote by ${\mathscr{O}}$
all possible datasets that can result from observations of $P$. Each
$o\in{\mathscr{O}}$ is one particular dataset, the result of carrying out some
set of measurements on $p$. We denote the datasets that can result from
measurements on $p$ as $obs(p)$. Formally:

$obs:P\twoheadrightarrow{\mathscr{O}}\>,$ (1)

where $obs$ is a _correspondence_ , which is a “generalized function” that
allows more than one element in the image $obs(p)$ (functions are a special
case of correspondences). A correspondence is necessary because, for a given
$p$, various possible datasets may arise, e.g., due to different measurement
techniques such as fMRI vs. EEG, or due to the stochastic behaviour of the
system, or due to varying experimental parameters. In the real world, data
obtained from experiments may be incomplete or noisy, or neuroscientific
findings difficult to reproduce (Gilmore et al., 2017). Thus for every $p\in
P$, there is a whole class of datasets which can result from the experiment.

Note that $obs$ describes the experiment, the choice of observables, and all
conditions during an experiment that generates the dataset $o$ necessary to
apply the theory, which may differ from theory to theory, such as
interventions in the case of IIT. In all realistic cases, the correspondence
$obs$ is likely quite complicated since it describes the whole experimental
setup. For our argument it simply suffices that this mapping exists, even if
it is not known in detail.

It is also worth noting here that all leading neuroscientific theories of
consciousness, from IIT to GNW, assume that experiences are not observable or
directly measurable when applying the theory to physical systems. That is,
experiences themselves are never identified or used in $obs$, but are rather
inferred based on some dataset $o$ that contains report or other behavioural
indicators.

Next we explore how the datasets in ${\mathscr{O}}$ are used to make
predictions about the experience of a physical system.

###  2.2 Predictions

A theory of consciousness makes predictions about the experience of some
physical system in some configuration, state, or dynamics, $p$, based on some
dataset $o$. To this end, a theory carries within its definition a set or
space $E$ whose elements correspond to various different _conscious
experiences_ a system could have. The interpretation of this set varies from
theory to theory, ranging from descriptions of the level of conscious
experience in early versions of IIT, descriptions of the level and content of
conscious experience in contemporary IIT (Kleiner and Tull, 2020), or the
description only of whether a presented stimuli is experienced in GNW or HOT.
We sometimes refer to elements $e$ of $E$ simply as _experiences_.

Formally, this means that a prediction considers an experimental dataset $o\in
O$ (determined by $obs$) and specifies an element of the experience space $E$.
We denote this as $pred$, for “prediction,” which is a map from
${\mathscr{O}}$ to $E$. The details of how individual datasets are being used
to make predictions again do not matter for the sake of our investigation.
What matters is that a procedure exists, and this is captured by $pred$.
However, we have to take into account that a single dataset $o\in O$ may not
predict only one single experience. In general, $pred$ may only allow an
experimenter to constrain experience of the system in that it only specifies a
subset of all experiences a theory models. We denote this subset by $pred(o)$.
Thus, $pred$ is also a correspondence

$pred:{\mathscr{O}}\twoheadrightarrow E\>.$

Shown in Figure 1 are the full set of terms needed to formally define how most
contemporary theories of consciousness make predictions about experience. So
far, what we have said is very general. Indeed, the force and generalizability
of our argument comes from the fact that we do not have to define $pred$
explicitly for the various models we consider. It suffices that it exists, in
some form or the other, for the models under consideration.

It is crucial to note that predicting states of consciousness alone does not
suffice to test a model of consciousness. Some have previously criticized
theories of consciousness, IIT in particular, just based off of their counter-
intuitive predictions. An example is the criticism that relatively simply
grid-like networks have high $\Phi$ (Aaronson, 2014; Tononi, 2014). However,
debates about counter-intuitive predictions are not meaningful by themselves,
since $pred$ alone does not contain enough information to say whether a theory
is true or false. The most a theory could be criticized for is either not
fitting our own phenomenology or not being parsimonious enough, neither of
which are necessarily violated by counter-intuitive predictions. For example,
it may actually be parsimonious to assume that many physical systems have
consciousness (Goff, 2017). That is, speculation about acceptable predictions
by theories of consciousness must implicitly rely on a comparative reference
to be meaningful, and speculations that are not explicit about their reference
are uninformative.

$P$
---
${\mathscr{O}}$
---
$E$
---
$obs$
---
$pred$
---
Figure 1: We assume that an experimental setup apt for a particular model of
consciousness has been chosen for some class of physical systems $P$, wherein
$p\in P$ represents the dynamics or configurations of a particular physical
system. ${\mathscr{O}}$ then denotes all datasets that can arise from
observations or measurements on $P$. Measuring the observables of $p$ maps to
datasets $o\in{\mathscr{O}}$, which is denoted by the $obs$ correspondence.
$E$ represents the mathematical description of experience given by the theory
or model of consciousness under consideration. In the simplest case, this is
just a set whose elements indicate whether a stimulus has been perceived
consciously or not, but far more complicated structures can arise (e.g., in
IIT). The correspondence $pred$ describes the process of prediction as a map
from ${\mathscr{O}}$ to $E$.

###  2.3 Inferences

As discussed in the previous section, a theory is unfalsifiable given just
predictions alone, and so $pred$ must be compared to something else. Ideally
this would be the actual conscious experience of the system under
investigation. However, as noted previously, the class of theories we focus on
here assumes that experience itself is not part of the observables. For this
reason, the experience of a system must be inferred separately from a theory’s
prediction to create a basis of comparison. Most commonly, such inferences are
based on _reports_. For instance, an inference might be based on an
experimental participant reporting on the switching of some perceptually
bistable image (Blake et al., 2014) or on reports about seen vs. unseen images
in masking paradigms (Alais et al., 2010).

It has been pointed out that report in a trial may interfere with the actual
isolation of consciousness, and there has recently been the introduction of
so-called “no-report paradigms” (Tsuchiya et al., 2015). In these cases,
report is first correlated to some autonomous phenomenon like optokinetic
nystagmus (stereotyped eye movement), and then the experimenter can use this
instead of the subject’s direct reports to infer their experiences. Indeed,
there can even be simpler cases where report is merely assumed: e.g., that in
showing a red square a participant will experience a red square without
necessarily asking the participant, since previously that participant has
proved compos mentis. Similarly, in cases of non-humans incapable of verbal
report, “report” can be broadly construed as behavior or output.

All these cases can be broadly described as being a case of inference off of
some data. This data might be actual reports (like a participant’s button
pushes) or may be based off of physiological reactions (like no-report
paradigms) or may be the outputs of a neural network or set of logic gates,
such as the results of an image classification task (LeCun et al., 2015).
Therefore, the inference can be represented as a function, $inf(o)$, between a
dataset $o$ generated by observation or measurement of the physical system,
and the set of postulated experiences in the model of consciousness, $E$:

$inf:{\mathscr{O}}\rightarrow E\>.$

Defining $inf$ as a function means that we assume that for every experimental
dataset $o$, one single experience in $E$ is inferred during the experiment.
Here we use a function instead of a correspondence for technical and formal
ease, which does affect our results (if two correspondences to the same space
are given, one of them can be turned into a function using quotient spaces).
The $inf$ function is flexible enough to encompass both direct report, no-
report, input/output analysis, and also assumed-report cases. It is a mapping
that describes the process of inferring the conscious experience of a system
from data recorded in the experiments. Both $inf$ and $pred$ are depicted in
Figure 2.

It is worth noting that we have used here the same class ${\mathscr{O}}$ as in
the definition of the prediction mapping $pred$ above. This makes sense
because the inference process also uses data obtained in experimental trials,
such as reports by a subject. So both $pred$ and $inf$ can be described to
operate on the same total dataset measured, even though they usually use
different parts of this dataset (cf. below).

$P$
---
${\mathscr{O}}$
---
$E$
---
$obs$
---
$pred$
---
$inf$
---
Figure 2: Two maps are necessary for a full experimental setup, one that
describes a theory’s predictions about experience ($pred$), another that
describes the experimenter’s inference about it ($inf$). Both map from a
dataset $o\in O$ collected in an experimental trail to some subset of
experiences described by the model, $E$.

###  2.4 Falsification

We have now introduced all elements which are necessary to formally say what a
falsification of a theory of consciousness is. To falsify a theory of
consciousness requires mismatch between an experimenter’s inference (generally
based on report) and the predicted consciousness of the subject. In order to
describe this, we consider some particular experimental trial, as well as
$inf$ and $pred$.

######  Definition 2.1.

There is a _falsification_ at $o\in{\mathscr{O}}$ if we have

$inf(o)\not\in pred(o)\>.$ (2)

This definition can be spelled out in terms of individual components of $E$.
To this end, for any given dataset $o\in{\mathscr{O}}$, let $e_{r}:=inf(o)$
denote the experience that is being inferred, and let $e_{p}\in obs(o)$ be one
of the experiences that is predicted based off of some dataset. Then (2)
simply states that we have $e_{p}\neq e_{r}$ for all possible predictions
$e_{p}$. None of the predicted states of experience is equal to the inferred
experience.

What does Equation (2) mean? There are two cases which are possible. Either,
the prediction based on the theory of consciousness is correct and the
inferred experience is wrong. Or the prediction is wrong, so that in this case
the model would be falsified. In short: Either the prediction process or the
inference process is wrong.

We remark that if there is a dataset $o$ on which the inference procedure
$inf$ or the prediction procedure $pred$ cannot be used, then this dataset
cannot be used in falsifying a model of consciousness. Thus, when it comes to
falsifications, we can restrict to datasets $o$ for which both procedures are
defined.

In order to understand in more detail what is going on if (2) holds, we have
to look into a single dataset $o\in{\mathscr{O}}$. This will be of use later.

Generally, $inf$ and $obs$ will make use of different part of the data
obtained in an experimental trial. E.g., in the context of IIT or GNW, data
about the internal structure and state of the brain will be used for the
prediction. This data can be obtained from an fMRI scan or EEG measurement.
The state of consciousness on the other hand can be inferred from verbal
reports. Pictorially, we may represent this as in Figure 3. We use the
following notation:

  * $o_{i}$

For a chosen dataset $o\in O$, we denote the part of the dataset which is used
for the prediction process by $o_{i}$ (for ‘internal’ data). This can be
thought of as data about the internal workings of the system. We call $o_{i}$
the _prediction data_ in $o$.

  * $o_{r}$

For a chosen dataset $o\in O$, we denote the part of the dataset which is used
for inferring the state of experience by $o_{r}$ (for ‘report’ data). We call
it the _inference data_ in $o$.

Note that in both cases, the subscript can be read similarly as the notation
for restricting a set.

$p$
---
$o\in{\mathscr{O}}$
---
$pred(o)$
---
$o_{r}$
---
$o_{i}$
---
$inf(o)$
---
$e_{r}$
---
$pred(o)$
---
$e_{p}$
---
$E$
---
Figure 3: This figure represents the same setup as Figure 2. The left circle
depicts one single dataset $o$. $o_{i}$ (orange) is the part of the dataset
used for prediction. $o_{r}$ (green) is the part of the dataset used for
inferring the state of experience. Usually the green area comprises verbal
reports or button presses, whereas the orange area comprises the data obtained
from brain scans. The right circle depicts the experience space $E$ of a
theory under consideration. $e_{p}$ denotes a predicted experience while
$e_{r}$ denotes the inferred experience. Therefore, in total, to represent
some specific experimental trial we use $p\in P$, $o\in{\mathscr{O}}$,
$e_{r}\in E$ and $e_{p}\in E$, where $e_{p}\in pred(o)$.

###  2.5 Summary

In summary, for testing of a theory of consciousness we have introduced the
following notion:

  * $P$

denotes a class of physical systems, each in various different configurations.
In most cases, every $p\in P$ thus describes a physical system in a particular
state, dynamical trajectory, or configuration.

  * $obs$

is a correspondence which contains all details on how the measurements are set
up and what is measured. It describes how measurement results (datasets) are
determined by a system configuration under investigation. This correspondence
is given, though usually not explicitly known, once a choice of measurement
scheme has been made.

  * ${\mathscr{O}}$

is the class of all possible datasets that can result from observations or
measurements of the systems in the class $P$. Any single experimental trail
results in a single dataset $o\in O$, whose data is used for making
predictions based on the theory of consciousness and for inference purposes.

  * $pred$

describes the process of making predictions by applying some theory of
consciousness to a dataset $o$. It is therefore a mapping from ${\mathscr{O}}$
to $E$.

  * $E$

denotes the space of possible experiences specified by the theory under
consideration. The result of the prediction is a subset of this space, denoted
as $pred(o)$. Elements of this subset are denoted by $e_{i}$ and describe
predicted experiences.

  * $inf$

describes the process of inferring a state of experience from some observed
data, e.g. verbal reports, button presses or using no-report paradigms.
Inferred experiences are denoted by $e_{r}$.

##  3 The substitution argument

Substitutions are changes of physical systems (i.e., the substitution of one
for another) that leave the inference data invariant, but may change the
result of the prediction process. A specific case of substitution, the
unfolding of a reentrant neural network to a feed-forward one, was recently
applied to IIT to argue that IIT cannot explain consciousness (Doerig et al.,
2019).

Here we show that, in general, the contemporary notion of falsification in the
science of consciousness exhibits this fundamental flaw for almost all
contemporary theories, rather than being a problem for a particular theory.
This flaw is based on the independence between the data used for inferences
about consciousness (like reports) and the data used to make predictions about
consciousness. We discuss various responses to this flaw in Section
LABEL:sec:responses.

We begin by defining what a substitution is in Section 3.1, show that it
implies falsifications in Section 3.2, and analyze the particularly
problematic case of universal substitutions in Section 3.3. In Section 3.4, we
prove that universal substitutions exist if prediction and inference data are
independent and give some examples of already-known cases.

###  3.1 Substitutions

In order to define formally what a substitution is, we work with the inference
content $o_{r}$ of a dataset $o$ as introduced in Section 2.4. We first denote
the class of all physical configurations which could have produced the
inference content $o_{r}$ upon measurement by $P_{o_{r}}$. Using the
correspondence $obs$ which describes the relation between physical systems and
measurement results, this can be defined as

$P_{o_{r}}:=\\{\,p\in P\,|\,o_{r}\in obs(p)\,\\}\>,$ (3)

where $obs(p)$ denotes all possible datasets that can be measured if the
system $p$ is under investigation and where $o_{r}\in obs(p)$ is a shorthand
for $o\in obs(p)$ with inference content $o_{r}$.

Any map of the form $S:P_{o_{r}}\rightarrow P_{o_{r}}$ takes a system
configuration $p$ which can produce inference content $o_{r}$ to another
system’s configuration $S(p)$ which can produce the same inference content.
This allows us to define what a substitution is formally. In what follows, the
$\circ$ indicates the composition of the correspondences $obs$ and $pred$ to
give a correspondence from $P$ to $E$, which could also be denoted as
$pred(obs(p))$, and $\cap$ denotes the intersection of sets.

######  Definition 3.1.

There is a _$o_{r}$ -substitution_ if there is a transformation
$S:P_{o_{r}}\rightarrow P_{o_{r}}$ such that at least for one $p\in P_{o_{r}}$

$pred\circ obs(p)\,\cap\,pred\circ obs(S(p))=\emptyset\>.$ (4)

In words, a substitution requires there to be a transformation $S$ which keeps
the inference data constant but changes the prediction of the system. So much
in fact that the prediction of the original configuration $p$ and of the
transformed configuration $S(p)$ are fully incompatible, i.e. there is no
single experience $e$ which is contained in both predictions. Given some
inference data $o_{r}$, an $o_{r}$-substitution then requires this to be the
case for at least one system configuration $p$ that gives this inference data.
In other words, the transformation $S$ is such that for at least one $p$, the
_predictions change completely_ , while the inference content $o_{r}$ is
preserved.

A pictorial definition of substitutions is given in Figure 4. We remark that
if $obs$ were a function, so that $obs(p)$ only contained one element,
Equation (4) would be equivalent to $pred(obs(p))\neq pred(obs(S(p)))$.

We will find below that the really problematic case arises if there is an
$o_{r}$-substitution for every possible inference content $o_{r}$. We call
this case a universal substitution:

######  Definition 3.2.

There is a _universal substitution_ if there is an $o_{r}$-substitution for
every $o_{r}$.

We recall that according to the notation introduced in Section 2.4, the
inference content of any dataset $o\in{\mathscr{O}}$ is denoted by $o_{r}$
(adding the subscript $r$). Thus the requirement is that there is an
$o_{r}$-substitution for every inference data that can pertain in the
experiment under consideration (for every inference data that is listed in
${\mathscr{O}}$).

$T(p)$
---
$o$ resp. $o^{\prime}$
---
$o_{r}=o_{r}^{\prime}$
---
$o_{i}$
---
$p$
---
$o_{i}^{\prime}$
---
$pred(o)$
---
$inf(o)$
---
$e_{r}=e_{r}^{\prime}$
---
$pred(o)$
---
$e_{p}$
---
$E$
---
$pred(o^{\prime})$
---
$e_{p}^{\prime}$
---
$pred(o^{\prime})$
---
Figure 4:  This picture illustrates substitutions. Assume that some dataset
$o$ with inference content $o_{r}$ is given. A substitution is a
transformation $S$ of physical systems which leaves the inference content
$o_{r}$ invariant but which changes the result of the prediction process. Thus
whereas $p$ and $S(p)$ have the same inference content $o_{r}$, the prediction
content of experimental datasets is different; different in fact to such an
extend that the predictions of consciousness based on these datasets are
incompatible (illustrated by the non-overlapping gray circles on the right).
Here we have used that by definition of $P_{o_{r}}$, every $\tilde{p}\in
P_{o_{r}}$ yields at least one dataset $o^{\prime}$ with the same inference
content as $o$ and have identified $o$ and $o^{\prime}$ in the drawing.

###  3.2 Substitutions imply falsifications

The force of our argument comes from the fact that if there are substitutions,
then this necessarily leads to mismatches between inferences and predictions.
This is shown by the following lemma.

######  Lemma 3.3.

If there is a $o_{r}$-substitution, there is a falsification at some
$o\in{\mathscr{O}}$.

###### Proof.

Let $p$ be the physical system in Definition 3.1 and define $p^{\prime}=S(p)$.
Let $o\in obs(p)$ be a dataset of $p$ which has inference content $o_{r}$ and
let $o^{\prime}$ be a dataset of $p^{\prime}$ which has the same inference
content $o_{r}$, guaranteed to exist by the definition of $P_{o_{r}}$ in (3).
Equation (4) implies that

$pred(o)\cap pred(o^{\prime})=\emptyset\>.$ (5)

Since, however, $o_{r}=o_{r}^{\prime}$, we have $inf(o)=inf(o^{\prime})$. Thus
we have either $inf(o)\not\in pred(o)$ or $inf(o^{\prime})\not\in
pred(o^{\prime})$, or both. Thus there is either a falsification at $o$, a
falsification at $o^{\prime}$, or both. ∎

The last lemma shows that if there are substitutions, then there are
necessarily falsifications. This might, however, not be considered too
problematic, since it could always be the case that the model is right whereas
the inferred experience is wrong. Inaccessible predictions are not unusual in
science. A fully problematic case only pertains for universal substitutions,
i.e., if there is an $o_{r}$-substitution for every inference content $o_{r}$
that can arise in an experiment under consideration.

###  3.3 Universal substitutions imply complete falsification

In Section 2.4, we have defined falsifications for individual datasets
$o\in{\mathscr{O}}$. Using the ‘insight view’ of single datasets, we can
refine this definition somewhat by relating it to the inference content only.

######  Definition 3.4.

There is an $o_{r}$-falsification if there is a falsification for some
$o\in{\mathscr{O}}$ which has inference content $o_{r}$.

This definition is weaker than the original definition, because among all
datasets which have inference content $o_{r}$, only one needs to exhibit a
falsification. Using this notion, the next lemma specifies the exact relation
between substitutions and falsifications.

######  Lemma 3.5.

If there is an $o_{r}$-substitution, there is an $o_{r}$-falsification.

###### Proof.

This lemma follows directly from the proof of Lemma 3.3 because the datasets
$o$ and $o^{\prime}$ used in that proof both have inference content $o_{r}$. ∎

This finally allows us to show our first main result. It shows that if a
universal substitution exists, the theory of consciousness under consideration
is falsified. We explain the meaning of this proposition after the proof.

######  Proposition 3.6.

If there is a universal substitution, there is an $o_{r}$-falsification _for
all possible_ inference contents $o_{r}$.

###### Proof.

By definition of universal substitution, there is an $o_{r}$-substitution for
every $o_{r}$. Thus the claim follows directly from Lemma 3.5. ∎

In combination with Definition 3.4, this proposition states that for every
possible report (or any other type of inference procedure, cf. our use of
terminology in Section 2.4), there is a dataset $o$ which contains the
report’s data and for which we have

$inf(o_{r})\notin pred(o)\>,$ (6)

where we have slightly abused notation in writing $inf(o_{r})$ instead of
$inf(o)$ for clarity. This implies that one of two cases needs to pertain:
Either at least one of the inferred experiences $inf(o_{r})$ is correct, in
which case the corresponding prediction is wrong and the theory needs to be
considered falsified. The only other option is that for _all_ inference
contents $o_{r}$, the prediction $pred(o)$ is correct, which qua (6) implies
that no single inference $inf(o_{r})$ points at the correct experience, so
that the inference procedure is completely wrong. This shows that Proposition
3.6 can equivalently be stated as follows.

######  Proposition 3.7.

If there is a universal substitution, either every single inference operation
is wrong or the theory under consideration is already falsified.

Next, we discuss under which circumstances a universal substitution exists.

###  3.4 When does a universal substitution exist?

In the last section, we have seen that if a universal substitution exists,
this has strong consequences. In this section, we discuss under what
conditions universal substitutions exist.

####  3.4.1 Theories need to be minimally informative

We have taken great care above to make sure that our notion of prediction is
compatible with incomplete or noisy datasets. This is the reason why $pred$ is
a correspondence, the most general object one could consider. For the purpose
of this section, we add a gentle assumption which restricts $pred$ slightly:
we assume that every prediction carries at least a minimal amount of
information. In our case, this means that for every prediction $pred(o)$,
there is at least one other prediction $pred(o^{\prime})$ which is different
from $pred(o)$. Put in simple terms, this means that we don’t consider
theories of consciousness which have only a single prediction.

In order to take this into account, for every $o\in{\mathscr{O}}$, we define
$\bar{o}:=obs(obs^{-1}(o))$, which comprises exactly all those datasets which
can be generated by physical systems $p$ that also generate $o$. When applying
our previous definitions, this can be fleshed out as

$\bar{o}=\\{\,o^{\prime}\,|\,\exists\,p\textrm{ such that }o\in obs(p)\textrm{
and }o^{\prime}\in obs(p)\,\\}\>.$ (7)

Using this, we can state our _minimal information assumption_ in a way that is
compatible with the general setup displayed in Figure 2:

We assume that the theories of consciousness under consideration are
_minimally informative_ in that for every $o\in{\mathscr{O}}$, there exists an
$o^{\prime}\in{\mathscr{O}}$ such that

$pred(\bar{o})\cap pred(\bar{o}^{\prime})=\emptyset\>.$ (8)

####  3.4.2 Inference and prediction data are independent

We have already noted, that in most experiments, the prediction content
$o_{i}$ and inference content $o_{r}$ consist of different parts of a dataset.
What is more, they are usually assumed to be independent, in the sense that
changes in $o_{i}$ are possible while keeping $o_{r}$ constant. This is
captured by the next definition.

######  Definition 3.8.

_Inference and prediction data are independent_ if for any $o_{i}$,
$o_{i}^{\prime}$ and $o_{r}$, there is a variation

$\nu:P\rightarrow P$ (9)

such that $o_{i}\in obs(p)$, $o_{i}^{\prime}\in obs(\nu(p))$ but $o_{r}\in
obs(p)$ and $o_{r}\in obs(\nu(p))$ for some $p\in P$.

Here, we use the same shorthand as in (3). For example, the requirement
$o_{i}\in obs(p)$ is a shorthand for there being an $o\in obs(p)$ which has
prediction content $o_{i}$. Note that if inference and prediction data are not
independent, e.g. because they have a common cause, problems of tautologies
loom large, cf. Section LABEL:sec:responses. Throughout the text we often
refer to Definition 3.8 simply as “independence”.

####  3.4.3 Universal substitutions exist

Combining the last two sections, we can now prove that universal substitutions
exist.

######  Proposition 3.9.

If inference and prediction data are independent, universal substitutions
exist.

###### Proof.

To show that a universal substitution exists, we need to show that for every
$o\in{\mathscr{O}}$, an $o_{r}$-substitution exists (Definition 3.1). Thus
assume that an arbitrary $o\in{\mathscr{O}}$ is given. The minimal information
assumption guarantees that there is an $o^{\prime}$ such that Equation (8)
holds. As before, we denote the prediction content of $o$ and $o^{\prime}$ by
$o_{i}$ and $o_{i}^{\prime}$, respectively, and the inference content of $o$
by $o_{r}$.

Since inference and prediction data are independent, there exists a $p\in P$
as well as a $\nu:P\rightarrow P$ such that $o_{i}\in obs(p)$,
$o_{i}^{\prime}\in obs(\nu(p))$, $o_{r}\in obs(p)$ and $o_{r}\in obs(\nu(p))$.
By Definition (7), the first two of these four conditions imply that
$obs(p)\subset\bar{o}$ and $obs(\nu(p))\subset\bar{o}^{\prime}$. Thus Equation
(8) applies and allows us to conclude that

$pred(obs(p))\cap pred(obs(\nu(p))=\emptyset\>.$

Via Equation (3), the latter two of the four conditions imply that $p\in
P_{o_{r}}$ and $\nu(p)\in P_{o_{r}}$. Thus we may restrict $\nu$ to
$P_{o_{r}}$ to obtain a map

$S:P_{o_{r}}\rightarrow P_{o_{r}}\>,$

which in light of the first part of this proof exhibits at least one $p\in
P_{o_{r}}$ which satisfies (4). Thus we have shown that an
$o_{r}$-substitution exists. Since $o$ was arbitrary, it follows that a
universal substitution exists. ∎

The intuition behind this proof is very simple. In virtue of our assumption
that theories of consciousness need to be minimally informative, for any
dataset $o$, there is another dataset $o^{\prime}$ which makes a non-
overlapping prediction. But in virtue of inference and prediction data being
independent, we can find a variation that changes the prediction content as
prescribed by $o$ and $o^{\prime}$, but keeps the inference content constant.
This suffices to show that there exists a transformation $S$ as required by
the definition of a substitution.

Combining this result with Proposition 3.7, we finally can state our main
theorem.

######  Theorem 3.10.

If inference and prediction data are independent, either every single
inference operation is wrong or the theory under consideration is already
falsified.

###### Proof.

The theorem follows by combining Proposition 3.9 and Proposition 3.7. ∎

In the next section, we give several examples of universal substitutions,
before discussing various possible responses to our result in Section
LABEL:sec:responses.

####  3.4.4 Examples of data independence

Our main theorem shows that testing a theory of consciousness will necessarily
lead to its falsification if inference and prediction data are independent
(Definition 3.8), and if at least one single inference can be trusted (Theorem
3.10). In this section, we give several examples that illustrate the
independence of inference and prediction data. We take report to mean output,
behavior, or verbal report itself and assume that prediction data derives from
internal measurements.

Artificial neural networks. ANNs, particularly those trained using deep
learning, have grown increasingly powerful and capable of human-like
performance (LeCun et al., 2015; Bojarski et al., 2016). For any ANN, report
(output) is a function of node states. Crucially, this function is non-
injective, i.e. some nodes are not part of the output. E.g., in deep learning,
the report is typically taken to consist of the last layer of the ANN, while
the hidden layers are not taken to be part of the output. Correspondingly, for
any given inference data, one can construct a ANN with arbitrary prediction
data by adding nodes, changing connections and changing those nodes which are
not part of the output. Put differently, one can always substitute a given ANN
with another with different internal observables but identical or near-
identical reports. From a mathematical perspective it is well-known that both
feed-forward ANNs and recurrent ANNs can approximate any given function
(Hornik et al., 1989; Schäfer and Zimmermann, 2006). Since reports are just
some function, it follows that there are viable universal substitutions.

A special case thereof is the unfolding transformation considered in Doerig et
al. (2019) in the context of IIT. The arguments in this paper constitute a
proof of the fact that for ANNs, inference and prediction data are independent
(Definition 3.8). Crucially, our main theorem shows that this has implications
for all minimally informative theories of consciousness. A similar result
(using a different characterization of theories of consciousness than
minimally informative) has been shown in (Kleiner, 2020).

Universal computers. Turing machines are extremely different in architecture
than ANNs. Since they are capable of universal computation (Turing, 1937) they
should provide an ideal candidate for a universal substitution. Indeed, this
is exactly the reasoning behind the Turing test of conversational artificial
intelligence (Turing, 2009). Therefore, if one believes it is possible for a
sufficiently fast Turing machine to pass the Turing test, one needs to accept
that substitutions exist. Notably, Turing machines are just one example of
universal computation, and there are other instances of different parameter
spaces or physical systems that are capable thereof, such as cellular automata
(Wolfram, 1984).

Universal intelligences. There are models of universal intelligence that allow
for maximally intelligent behavior across any set of tasks (Hutter, 2003). For
instance, consider the AIXI model, the gold-standard for universal
intelligence, which operates via Solomonoff induction (Solomonoff, 1964;
Hutter, 2004). The AIXI model generates an optimal decision making over some
class of problems, and methods linked to it have already been been applied to
a range of behaviors, such as creating “AI physicists” (Wu and Tegmark, 2019).
Its universality indicates it is a prime candidate for universal
substitutions. Notably, unlike a Turing machine, it avoids issues of precisely
how it is accomplishing universal substitution of report, since the algorithm
that governs the AIXI model behavior is well-described and “relatively”
simple.

The above are all real and viable classes of systems that are used everyday in
science and engineering which all provide different viable universal
substitutions if inferences are based on reports or outputs. They show that in
normal experimental setups such as the ones commonly used in neuroscientific
research into consciousness (Frith et al., 1999), inference and prediction
data are indeed independent, and dependency is not investigated nor properly
considered. It is always possible to substitute the physical system under
consideration with another that has different internal observables, and
therefore different predictions, but similar or identical reports. Indeed,
recent research in using the work introduced in this work shows that even
different spatiotemporal models of a system can be substituted for one
another, leading to falsification (Hanson and Walker, 2020). We have not
considered possible but less reasonable examples of universal substitutions,
like astronomically-large look-up ledgers of reports.

As an example of our Main Theorem 3.10, we consider the case of IIT. Since the
theory is normally applied in Boolean networks, logic gates, or artificial
neural networks, one usually takes report to mean “output.” In this case, it
has already been proven that systems with different internal structures and
hence different predicted experiences, can have identical input/output (and
therefore identical reports or inferences about report) (Albantakis and
Tononi, 2019). To take another case: within IIT it has already been
acknowledged that a Turing machine may have a wildly different predicted
contents of consciousness for the same behavior or reports (Koch, 2019).
Therefore, data independence during testing has already been shown to apply to
IIT under its normal assumptions.

##  4 Inference and prediction data are strictly dependent

An immediate response to our main result would be to claim that it offers
support of theories which define conscious experience in terms of what is
accessible to report. This is the case, e.g., for behaviourist theories of
consciousness but might arguably also be the case for some interpretations of
global workspace theory or fame in the brain proposals. In this section, we
show that this response is not valid, as theories of this kind, where
inference and prediction data are strictly dependent, are empirically
unfalsifiable.

In order to analyse this case, we first need to say something about
unfalsifiable theories of consciousness. Clearly, the goal of the scientific
study as a whole is to find, eventually, a theory of consciousness that cannot
be falsified in experimental situations, because it is compatible with all
experimental evidence. This is a necessary condition (though not sufficient)
any purportedly “true” theory of consciousness needs to satisfy. Therefore,
unfalsifiability per se is not a bad thing. However, unfasifiability can also
be due to pathological assumptions that underlie a theory of consciousness,
assumptions which render an experimental investigation meaningless.
Specifically, a pathological dependence between inferences and predictions
leads to theories which are empirically unfalsifiable.

Such empirically unfalsifiable theories can be identified neatly in our
formalism. To see how, recall that ${\mathscr{O}}$ denotes the class of all
datasets that can result from an experiment under investigation. Put
differently, it contains all datasets that can appear in nature when probed in
the experiment. This is _not_ the class of all possible datasets of type
${\mathscr{O}}$ one can think of. Many datasets which are of the same form as
elements of ${\mathscr{O}}$ might simply not arise in the experiment under
consideration. We denote the class of all possible datasets as:

$\overline{\mathscr{O}}:\textrm{ All possible datasets of type
}{\mathscr{O}}\>.$

By construction, ${\mathscr{O}}$ is a subset of $\overline{\mathscr{O}}$,
which describes which among the possible datasets actually arises across
experimental trails. Hence, ${\mathscr{O}}$ also determines which theory of
consciousness is compatible with (i.e. not falsified by) the experimental
investigation.

Introduction of $\overline{\mathscr{O}}$ allows us to distinguish the
pathological and non-pathological cases of unfalsifiability mentioned above.
Whereas any purportedly true theory should only be unfalsifiable with respect
to the experimental data ${\mathscr{O}}$, a pathological unfalsifiability
pertains if a theory cannot be falsified at all, i.e. over
$\overline{\mathscr{O}}$. This is captured by the following definition.

######  Definition 4.1.

A theory of consciousness which does not have a falsification over
$\overline{\mathscr{O}}$ is _empirically unfalsifiable_.

Here, we use the term ‘empirically unfalsifiable’ to highlight and refer to
the pathological notion of unfalsifiability. Intuitively speaking, a theory
which satisfies this definition appears to be true independently of any
experimental investigation, and without the need for any such investigation.
Using $\overline{\mathscr{O}}$, we can also define the notion of strict
dependence in a useful way.

######  Definition 4.2.

Inference and prediction data are _strictly dependent_ if there is a function
$f$ such that for any $o\in\overline{\mathscr{O}}$, we have $o_{i}=f(o_{r})$.

This definition says that there exists a function $f$ which for every possible
inference data $o_{r}$ allows to deduce the prediction data $o_{i}$. We remark
that the definition refers to $\overline{\mathscr{O}}$ and not
${\mathscr{O}}$, as the dependence of inference and prediction considered here
holds by assumption and is not simply asserting a contingency in nature.

The definition is satisfied, for example, if inference data is equal to
prediction data, i.e. if $o_{i}=o_{r}$, where $f$ is simply the identity. This
is the case, e.g., for behaviourist theories (Skinner, 1938) of consciousness,
where consciousness is equated directly with report or behavior, or for
precursors of functionalist theories of consciousness that are based on
behavior or input/output (Putnam, 1960). The definition is also satisfied in
the case where prediction data is always a subset of the inference data:

$o_{i}\subseteq o_{r}\>.$ (10)

Here, $f$ is simply the restriction function. This arguably applies to global
workspace theory (Baars, 2005), the “attention schema” theory of consciousness
(Graziano and Webb, 2015) or “fame in the brain” (Dennett, 1993) proposals.

In all these cases, consciousness is generated by – and hence needs to be
predicted via – what is accessible to report or output. In terms of Block’s
distinction between phenomenal consciousness and access consciousness (Block,
1996), Equation (10) holds true whenever a theory of consciousness is under
investigation where access consciousness determines phenomenal consciousness.

Our second main theorem is the following.

######  Theorem 4.3.

If a theory of consciousness implies that inference and prediction data are
strictly dependent, then it is either already falsified or empirically
unfalsifiable.

###### Proof.

To prove the theorem, it is useful to consider the inference and prediction
content of datasets explicitly. The possible pairings that can occur in an
experiment are given by

${\mathscr{O}}_{\textrm{exp}}:=\\{\,(o_{i},o_{r})\ |\
o\in{\mathscr{O}}\,\\}\>,$ (11)

where we have again used our notation that $o_{i}$ denotes the prediction data
of $o$, and similar for $o_{r}$. To define the possible pairings that can
occur in $\overline{\mathscr{O}}$, we let ${\mathscr{O}}_{i}$ denote the class
of all prediction contents that arise in ${\mathscr{O}}$, and
${\mathscr{O}}_{r}$ denote the class of all inference contents that arise in
${\mathscr{O}}$. The set of all conceivable pairings is then given by

$\displaystyle{\mathscr{O}}_{\textrm{all}}:=$
$\displaystyle\\{\,(o_{i},o_{r}^{\prime})\ |\ o\in{\mathscr{O}},\
o^{\prime}\in{\mathscr{O}}\\}$ (12) $\displaystyle=$
$\displaystyle\\{\,(o_{i},o_{r}^{\prime})\ |\ o_{i}\in{\mathscr{O}}_{i},\
o_{r}^{\prime}\in{\mathscr{O}}_{r}\,\\}\>.$ (13)

Crucially, here, $o_{i}$ and $o_{r}^{\prime}$ do not have to be part of the
same dataset $o$. Combined with Definition 2.1, we conclude that there is a
falsification over $\overline{\mathscr{O}}$ if for some
$(o_{i},o_{r}^{\prime})\in{\mathscr{O}}_{\textrm{all}}$, we have $inf(o)\notin
pred(o^{\prime})$, and there is a falsification over ${\mathscr{O}}$ if for
some $(o_{i},o_{r})\in{\mathscr{O}}_{\textrm{exp}}$, we have $inf(o)\notin
pred(o)$.

Next we show that if inference and prediction data are strictly dependent,
then ${\mathscr{O}}_{\textrm{all}}={\mathscr{O}}_{\textrm{exp}}$ holds. We
start with the set ${\mathscr{O}}_{\textrm{all}}$ as defined in (12).
Expanding this definition in words, it reads

${\mathscr{O}}_{\textrm{all}}=\\{\,(d_{i},d_{r})\ |\
\exists\,o\in{\mathscr{O}}\textrm{ such that }d_{r}=o_{r}\textrm{ and
}\exists\,\tilde{o}\in{\mathscr{O}}\textrm{ such that
}d_{i}=\tilde{o}_{i}\,\\}\>,$ (14)

where we have symbols $d_{i}$ and $d_{r}$ to denote prediction and inference
data independently of any dataset $o$.

Assume that the first condition in this expression, $d_{r}=o_{r}$ holds for
some $o\in{\mathscr{O}}$. Since inference and prediction data are strictly
dependent, we have $d_{i}=f(d_{r})$. Furthermore, for the same reason, the
prediction content $o_{i}$ of the dataset $o$ satisfies $o_{i}=f(o_{r})$.
Applying the function $f$ to both sides of the first condition gives
$f(d_{r})=f(o_{r})$, which thus in turn implies $o_{i}=d_{i}$. This means that
the $o$ that satisfies the first condition in (14) automatically also
satisfies the second condition. Therefore, due to inference and prediction
data being strictly dependent, (14) is equivalent to

${\mathscr{O}}_{\textrm{all}}=\\{\,(d_{i},d_{r})\ |\
\exists\,o\in{\mathscr{O}}\textrm{ such that }d_{r}=o_{r}\textrm{ and
}d_{i}=o_{i}\,\\}\>.$ (15)

This, however, is exactly ${\mathscr{O}}_{\textrm{exp}}$ as defined in (11).
Thus we conclude that if inference and prediction data are strictly dependent,
${\mathscr{O}}_{\textrm{all}}={\mathscr{O}}_{\textrm{exp}}$ necessarily holds.

Returning to the characterisation of falsification in terms of
${\mathscr{O}}_{\textrm{exp}}$ and ${\mathscr{O}}_{\textrm{all}}$ above, what
we have just found implies that there is a falsification over ${\mathscr{O}}$
if and only if there is a falsification over $\overline{\mathscr{O}}$. Thus
either there is a falsification over ${\mathscr{O}}$, in which case the theory
is already falsified or there is no falsification over
$\overline{\mathscr{O}}$, in which case the theory under consideration is
empirically unfalsifiable. ∎

The gist of this proof is that if inference and prediction data are strictly
dependent, then as far as the inference and prediction contents go,
${\mathscr{O}}$ and $\overline{\mathscr{O}}$ are the same. I.e, the experiment
does not add anything to the evaluation of the theory. It is sufficient to
know only all possible datasets to decide whether there is a falsification. In
practise, this would mean that knowledge of the experimental design (which
reports are to be collected, on the one hand, which possible data a
measurement device can produce, one the other) is sufficient to evaluate the
theory, which is clearly at odds with the role of empirical evidence required
in any scientific investigation. Thus such theories are empirically
unfalsifiable.

To give an intuitive example of the theorem, let us examine a theory that uses
the information accessible to report in a system to predict conscious
experience (perhaps this information is “famous” in the brain or is within
some accessible global workspace). In terms of our notation, we can assume
that $o_{r}$ denotes everything that is accessible to report, and $o_{i}$
denotes that part which is used by the theory to predict conscious experience.
Thus in this case we have $o_{i}\subseteq o_{r}$. Since the predicted contents
are always part of what can be reported, there can never be any mismatch
between reports and predictions. However, this is not only the case for
${\mathscr{O}}_{\textrm{exp}}$ but also, in virtue of the theory’s definition,
for all possible datasets, i.e., ${\mathscr{O}}_{\textrm{all}}$. Therefore
such theories are empirically unfalsifiable. Experiments add no information to
whether the theory is true or not, and such theories are empirically
uninformative or tautological.

##  5 Objections

In this section, we discuss a number of possible objections to our results.

###  5.1 Restricting inferences to humans only

The examples given in Section 3.4.4 show that data independence holds during
the usual testing setups. This is because prima facie it seems reasonable to
base inferences either on report capability or intelligent behavior in a
manner agnostic of the actual physical makeup of the system. Yet this entails
independence, so in these cases our conclusions apply.

One response to our results might be to restrict all testing of theories of
consciousness solely to humans. In our formalisms this is equivalent to making
the strength of inferences based not on reports themselves but on an
underlying biological homology. Such an $inf$ function may still pick out
specific experiences via reports, but the weight of the inference is carried
by homology rather than report or behavior. This would mean that the
substitution argument does not significantly affect consciousness research, as
reports of non-human systems would simply not count. Theories of
consciousness, so this idea goes, would be supported by abductive reasoning
from testing in humans alone.

Overall there are strong reasons to reject this restriction of inferences. One
significant issue is that this objection is equivalent to saying that reports
or behavior in non-humans carry no information about consciousness. Even if
non-humans contradicted a theory (like an organism acting in pain while a
theory predicted a lack of pain) the theory would be presumed to be correct
above any behavior or report. This is highly problematic for consciousness
research which often uses non-human animal models (Boly et al., 2013). For
instance, cephalopods are among the most intelligent animals yet are quite
distant on the tree of life from humans and have a distinct neuroanatomy, and
still are used for consciousness research (Mather, 2008). Even in artificial
intelligence research, there is increasing evidence that deep neural networks
produced brain-like structures such as grid cells, shape tuning, and visual
illusions, and many others (Richards et al., 2019). Given these similarities,
it becomes questionable why the strength of inferences should be based on
homology instead of capability of report, intelligence, or even functionally
brain-like structures.

What is more, restricting inferences to humans alone might not be sufficient
to avoid our results. Depending on the theory under consideration, data
independence might exist just in human brains alone. That is, it is probable
that there are transformations (as in Equation (9)) available within the brain
wherein $o_{r}$ is fixed but $o_{i}$ varies. This is particularly true once
one allows for interventions on the human brain by experimenters, such as
perturbations like transcranial magnetic stimulation, which is already used in
consciousness research (Rounis et al., 2010; Napolitani et al., 2014).

For these reasons this objection does not appear viable. At minimum it is
clear that if the objection were taken seriously, it would imply significant
changes to consciousness research which would make the field extremely
restricted with strong a priori assumptions.

###  5.2 Reductio ad absurdum

Another hypothetical objection to our results is to argue that they could just
as well be applied to scientific theories in other fields. If this turned out
to be true this wouldn’t imply our argument is necessarily incorrect. But the
fact that other scientific theories do not seem especially problematic with
regard to falsification would generate the question of whether some assumption
is illegitimately strong. In order to address this, we explain which of our
assumptions is specific to theories of consciousness and wouldn’t hold when
applied to other scientific theories. Subsequently, we give an example to
illustrate this point.

The assumption in question is that ${\mathscr{O}}$, the class of all datasets
that can result from observations or measurements of a system, is determined
by the physical configurations in $P$ alone. I.e., every single dataset $o$,
including both its prediction content $o_{i}$ and its inference content
$o_{r}$, is determined by $p$, and _not_ by a conscious experience in $E$. In
Figure 2, this is reflected in the fact that there is an arrow from $P$ to
${\mathscr{O}}$, but no arrow from $E$ to ${\mathscr{O}}$.

This assumption expresses the standard paradigm of testing theories of
consciousness in neuroscience, according to which both the data used to
predict a state of consciousness and the reports of a system are determined by
its physical configuration alone. This, in turn, may be traced back to
consciousness’ assumed subjective and private nature, which implies that any
empirical access to states of consciousness in scientific investigations is
necessarily mediated by a subject’s reports, and to general physicalist
assumptions.

This is different from experiments in other natural sciences. If there are two
quantities of interest whose relation is to be modelled by a scientific
theory, then in all reasonable cases there are two independent means of
collecting information relevant to a test of the theory, one providing a
dataset that is determined by the first quantity, and one providing a dataset
that is determined by the second quantity.

Consider, as an example, the case of temperature $T$ and its relation to
microphysical states. To apply our argument, the temperature $T$ would replace
the experience space $E$ and $p$ would denote a microphyiscal configuration.
In order to test any particular theory about how temperature is determined by
microphysical states, one would make use of two different measurements. The
first measurement would access the microphysical states and would allow
measurement of, say, the mean kinetic energy (if that’s what the theory under
consideration utilizes). This first measurement would provide a dataset
$o_{m}$ that replaces the prediction data $o_{i}$ above. For the second
measurement, one would use a thermometer or some other measuring device to
obtain a dataset $o_{t}$ that replaces our inference data $o_{r}$ above.
Comparison of the inferred temperature with the temperature that is predicted
based on $o_{m}$ would allow testing of the theory under consideration. These
independent means provide independent access to each of the two datasets in
question. Combining $o_{m}$ and $o_{t}$ in one dataset $o$, the diagrammatic
representation is

$P\longrightarrow{\mathscr{O}}\longleftarrow T\>,$

which differs from the case of theories of consciousness considered here,
wherein the physical system determines both datasets.

###  5.3 Theories could be based on phenomenology

Another response to the issue of independence/dependence identified here is to
propose that a theory of consciousness may not have to be falsified but can be
judged by other characteristics. This is reminiscent of ideas put forward in
connection with String Theory, which some have argued can be judged by
elegance or parsimony alone (Carroll, 2018).

In addition to elegance and parsimony, in consciousness science, one could in
particular consider a theory’s fit with phenomenology, i.e. how well a theory
describes the general structure of conscious experience. Examples of theories
that are constructed based on a fit with phenomenology are recent versions of
IIT (Oizumi et al., 2014) or any view that proposes developing theories based
on isomorphisms between the structure of experiences and the structure of
physical systems or processes (Tsuchiya et al., 2019).

It might be suggested that phenomenological theories might be immune to
aspects of the issues we outline in our results (Negro, 2020). We emphasize
that in order to avoid our results, and indeed the need for any experimental
testing at all, a theory constructed from phenomenology has to be _uniquely
derivable_ from conscious experience. However, to date, no such derivation
exists, as phenomenology seems to generally undermine the postulates of IIT
(Bayne, 2018; Barrett and Mediano, 2019), and because it is unknown what the
scope and nature of non-human experience is. Therefore theories based on
phenomenology can only confidently identify systems with human-like conscious
experiences and cannot currently do so uniquely. Thus they cannot avoid the
need for testing.

As long as no unique and correct derivation exists across the space of
possible conscious experiences, the use of experimental tests to assess
theories of consciousness, and hence our results, cannot be avoided.

###  5.4 Rejecting falsifiability

Another response to our findings might be to deny the importance of
falsifications within the scientific methodology. Such responses may reference
a Lakatosian conception of science, according to which science does not
proceed by discarding theories immediately upon falsification, but instead
consists of _research programs_ built around a family of theories (Lakatos,
1980). These research programs have a _protective belt_ which consists of non-
essential assumptions that are required to make predictions, and which can
easily be modified in response to falsifications, as well as a _hard core_
that is immune to falsifications. Within the Lakatosian conception of science
research programs are either progressive or degenerating based on whether they
can “anticipate theoretically novel facts in its growth” or not (Lakatos,
1980).

It is important to note, however, that Lakatos does not actually break with
falsificationism. This is why Lakatos description of science is often called
“refined falsificationism” in philosophy of science (Radnitzky, 1991). Thus
cases of testing theories’ predictions remain relevant in a Lakatosian view in
order to distinguish between progressive and degenerating research programs.
Therefore our results generally translate into this view of scientific
progress. In particular, Theorem 3.10 shows that for every single inference
procedure that is taken to be valid, there exists a system for which the
theory makes a wrong prediction. This implies necessarily that a research
program is degenerating. That is, independence implies that there is always an
available substitution that can falsify any particular prediction coming from
the research program.

##  6 Conclusion

In this paper, we have subjected the usual scheme for testing theories of
consciousness to a thorough formal analysis. We have shown that there appear
to be deep problems inherent in this scheme which need to be addressed.

Crucially, in contrast to other similar results (Doerig et al., 2019), we do
not put the blame on individual theories of consciousness, but rather show
that a key assumption that is usually being made is responsible for the
problems: an experimenter’s inference about consciousness and a theory’s
predictions are generally implicitly assumed to be independent during testing
across contemporary theories. As we formally prove, if this independence
holds, substitutions or changes to physical systems are possible that falsify
any given contemporary theory. Whenever there is an experimental test of a
theory of consciousness on some physical system which does not lead to a
falsification, there necessary exists another physical system which, if it had
been tested, would have produced a falsification of that theory. We emphasize
that this problem does not only affect one particular type of theory, for
example those based on causal interactions like IIT; theorems apply to all
contemporary neuroscientific theories of consciousness if independence holds.

In the second part of our results, we examine the case where independence
doesn’t hold. We show that if an experimenter’s inferences about consciousness
and a theory’s predictions are instead considered to be strictly dependent,
empirical unfalsifiability follows, which renders any type of experiment to
test a theory uninformative. This affects all theories wherein consciousness
is predicted off of reports or behavior (such as behaviorism), theories based
off of input/output functions, and also theories that equate consciousness
with on accessible or reportable information.

Thus theories of consciousness seem caught between between Scylla and
Charybdis, requiring delicate navigation. In our opinion there may only be two
possible paths forward to avoid these dilemmas, which we briefly outline
below. Each requires a revision of the current scheme of testing or developing
theories of consciousness.

##### Lenient dependency.

When combined, our main theorems show that both independence and strict
dependence of inference and prediction data are problematic and thus neither
can be assumed in an experimental investigation. This raises the question of
whether there are reasonable cases where inference and prediction are
dependent, but not strictly dependent.

A priori, in the space of possible relationships between inference and
prediction data, there seems to be room for relationships that are neither
independent (Section 3) nor strictly dependent (Section 4). We define this
relationships of this kind as cases of lenient dependency. No current theory
or testing paradigm that we know of satisfies this definition. Yet cases of
lenient dependency cannot be excluded to exist. Such cases would technically
not be beholden to either Theorem 3.10 or Theorem 4.3.

There seems to be two general possibilities of how lenient dependencies could
be built. On the one hand, one could hope to find novel forms of inference
that allow to surpass the problems we have identified here. This would likely
constitute a major change in the methodologies of experimental testing of
theories of consciousness. On the other hand, another possibility to attain
lenient dependence would be to construct theories of consciousness which yield
prediction functions that are designed to explicitly have a leniently
dependent link to inference functions. This would likely constitute a major
change in constructing theories of consciousness.

##### Physics is not causally closed.

Another way to avoid our conclusion is to only consider theories of
consciousness which do not describe the physical as causally closed (Kim,
1998). That is, the presence or absence of a particular experience itself
would have to make a difference to the configuration, dynamics, or states of
physical systems above and beyond what would be predicted with just
information about the physical system itself. If a theory of consciousness
does not describe the physical as closed, a whole other range of predictions
are possible: predictions which concern the physical domain itself, e.g.,
changes in the dynamics of the system which depend on the dynamics of
conscious experience. These predictions are not considered in our setup and
may serve to test a theory of consciousness without the problems we have
explored here.

##  7 Acknowledgments

We would like to thank David Chalmers, Ned Block, and the participants of the
NYU philosophy of mind discussion group for valuable comments and discussion.
Thanks also to Stephan Sellmaier, Timo Freiesleben, Mark Wulff Carstensen and
Sofiia Rappe for feedback on early versions of the manuscript.

Author contributions: J.K.and E.H. conceived the project and wrote the
article. Competing interests: The authors declare no competing interests.

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