Falsification and consciousness
FFalsification and consciousness
Johannes Kleiner ∗ and Erik Hoel † Munich Center for Mathematical Philosophy, Ludwig Maximilian University of Munich Allen Discovery Center, Tufts University, Medford, MA, USA
April 8, 2020
Abstract
The search for a scientific theory of consciousness should result in theories that are falsifiable. How-ever, here we show that falsification is especially problematic for theories of consciousness. We formallydescribe the standard experimental setup for testing these theories. Based on a theory’s applicationto 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 (usingreport or behavior). If there is a mismatch between inference and prediction, a theory is falsified. Weshow that if inference and prediction are independent, it follows that any minimally informative theoryof consciousness is automatically falsified. This is deeply problematic since the field’s reliance on reportor behavior to infer conscious experiences implies such independence, so this fragility affects many con-temporary theories of consciousness. Furthermore, we show that if inference and prediction are strictlydependent, it follows that a theory is unfalsifiable. This affects theories which claim consciousness to bedetermined by report or behavior. Finally, we explore possible ways out of this dilemma.
All successful scientific fields grow in complexity over time, moving from exploratory studies and observationsto the point where theories are proposed that can offer precise predictions. Within neuroscience the attemptto understand consciousness has moved out of the exploratory stage and there are now a number of theoriesof consciousness that have been advanced by different authors, some quite formal and complicated (Kochet al., 2016).At this point in the field’s development falsification has become an issue. In general, scientific theoriesshould strive to be falsifiable (Popper, 1959). Yet in some fields, particularly theoretical physics, issues offalsifiability have become a point of major contention as some proposed and even popular physical theoriesmay be unfalsifiable (Ellis and Silk, 2014; Woit, 2006). In the search for a scientific theory of consciousness,falsifiability must be considered explicitly to avoid a similar circumstance. This is especially importantbecause consciousness research is unlike other fields in that it is commonly assumed that consciousness itselfcannot be directly observed, instead it can only be inferred based off of brain states, which are physicalphenomena.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 thatconsciousness is associated with neurons firing at a particular frequency (Crick and Koch, 1990) or activity insome particular area of the brain like the claustrum (Crick and Koch, 2005). Other theories have focused moreon the properties of 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 arepropagated widely across different brain regions (Baars, 1997). Specifically, a global neuronal workspacetheory has been proposed in which consciousness is the result of an “avalanche” or “ignition” of widespread ∗ [email protected] † [email protected] a r X i v : . [ q - b i o . N C ] A p r eural activity created by an interconnected but dispersed network of neurons with long-range connections(Sergent and Dehaene, 2004).Another avenue of research has been associating the complexity of brain activity with consciousness(Tononi and Edelman, 1998) which evolved into its most specific instantiation: Integrated InformationTheory (Tononi, 2004, 2008; Oizumi et al., 2014). Historically, Integrated Information Theory is the firstwell-formalized theory of consciousness. It was the first (and arguably may still be the lone) theory thatmakes 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 physicalsystem’s internal observables, while the output of this function are predictions about the contents of con-sciousness (represented mathematically as an element of an experience space) and the level of consciousness(represented by a scalar value Φ).Both Global Neuronal Workspace (GNW) and Integrated Information Theory (IIT) have gained widespreadpopularity, sparked general interest in consciousness, and lead to dozens if not hundreds of new empiricalstudies (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 eitherGNW or IIT in the form of a global effort that is pre-registering predictions from the two theories so thattesting can be conducted in controlled circumstances by researchers across the world (Reardon, 2019). Wetherefore often refer to both GNW and IIT as exemplar theories within consciousness research and showhow our results apply to both. While there are too many other relevant theories, e.g., (Lau and Rosenthal,2011; Chang et al., 2019), to fully discuss, our results and reasoning apply to most if not all contemporarytheories, particularly in their ideal and precise forms. Note that we refer to both “theories” of consciousnessand 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-formalizedtheories are much easier to criticize than non-formalized theories (a positive attribute). Recently one criticismlevied against IIT was based on the fact that the theory predicts feedfoward neural networks have zero Φand recurrent neural networks have non-zero Φ. Since it is known that a given recurrent neural network canbe “unfolded” into a feedfoward one while preserving its output function, this is argued to render IIT wrongor outside the realm of science (Doerig et al., 2019). Replies have criticised the assumptions which underliethe derivation of this argument (Kleiner, 2019; Tsuchiya et al., 2019).Here we frame and expand concerns around testing and falsification of theories by examining a muchmore general question: what are the conditions under which theories of consciousness (beyond IIT alone) canbe falsified? We outline a parsimonious description of theory testing with minimal background assumptionsbased on first principles. In this agnostic setup falsifying a theory of consciousness is the result of findinga mismatch between the inferred contents of consciousness (based usually on report or behavior) and thecontents of consciousness as predicted by the theory given the internal observables of the system underquestion.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. Instead,it requires some comparison, that is, an inference of an agent’s conscious experience, e.g., using report orbehavior. For instance, imagine a theory that predicts (based on such internal observables as brain dynamics)that a subject is seeing an image of a cat. Without any reference to report or outside information, therecan 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 needs to be based on finding such mismatchesbetween reported experience and prediction.In the following work, we formalize this by describing the ideal experimental setup for testing a theoryof consciousness based on a physical system, its observables, predictions of conscious content from thoseobservables, and inferences by the experimenter based on report or behavior.We come to a surprising conclusion: it is likely that all contemporary theories of consciousness arealready falsified. This is because contemporary testing rests on an assumption of independence whereinan experimenter’s inferences about consciousness are independent from a theory’s predictions, meaning thatinferences can be kept steady while predictions vary. To demonstrate the problems this independence createsfor most contemporary theories, we introduce a “substitution argument.” This argument is based on thefact that many systems are equivalent in their reports (e.g., their outputs are identical for the same inputs)2nd yet their internal observables may differ greatly, and constitutes both a generalization and correctionof the unfolding argument presented in Doerig et al. (2019). Examples of such substitutions may involvesubstituting a brain with a Turing machine or a cellular automaton since both types of systems are capableof universal computation (Turing, 1937; Wolfram, 1984), or replacing a given neural network with a differentrecurrent neural network or a feedfoward neural network, since both types of networks can approximate anygiven function (Hornik et al., 1989; Sch¨afer and Zimmermann, 2006), or any of these with some schema foruniversal intelligence, like the AIXI model, which is based on Solomonoff induction (Hutter, 2004).While for some particular theory of consciousness there may be a system where there is no significantmismatch between the theory’s predictions based on internal observables and that system’s reports, if infer-ences and predictions are independent, there exists some different system with different internal observablesbut equivalent inferences to the original system. This second system can be “substituted” in for the originalone in the testing of the theory. Since internal observables following this substitution are then by defini-tion changed, this will lead to differing predictions of consciousness but with the same inferences by theexperimenter about consciousness. In turn this triggers a mismatch between predictions and inferences, thusfalsifying the theory. One consequence is that 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 tomake predictions about experience off of internal observables. We identify the root cause of this issue asthe assumption of independence between inferences and predictions. Thus, instead of putting the blame ofthis problem on individual theories of consciousness, we show that it is due to issues of falsification in thescientific study of consciousness, particularly the field’s reliance on using report or behavior to infer consciousexperiences.A simple response to avoid this problem is to claim that report and inference are not independent. Thisis, e.g., the case in behaviorist theories of consciousness, but arguably also in in Global Workspace Theory(Baars, 2005), the “attention schema” theory of consciousness (Graziano and Webb, 2015) or “fame in thebrain” (Dennett, 1993) proposals. We thus also subject this response to a detailed formal analysis. Wefind that making a theory’s predictions and an experimenter’s inferences dependent leads to unfalsifiability.Correspondingly, this assumption is not compatible with a scientific approach to consciousness either.Our results show that if independence of prediction and inference holds true, as in cases where reportabout experiences is relied upon, it is likely that no contemporary theory of consciousness is correct. Each is“pre-falsified,” so to speak. Alternatively, if the reasonable assumption of independence is rejected, theoriesrapidly become unfalsifiable. While this twin dilemma may seem a highly negative set of conclusions, thepositive side of our work is that it constrains the future set of viable theories of consciousness in a usefulway. We discuss various paths forward for researchers who seek to develop and test such theories. Here we provide a formal framework for experimentally testing a particular class of theories of consciousness.The class of theories we consider make predictions about the conscious experience of physical systems basedon observations or measurements . This class includes many contemporary theories, including leading theoriessuch as Integrated Information Theory (Oizumi et al., 2014), Global Neuronal Workspace Theory (Dehaeneand Changeux, 2004), Predictive Processing (when applied to account for conscious experience (Dolega andDewhurst, 2020; Clark, 2019; Seth, 2014; Hobson et al., 2014; Hohwy, 2012)) or Higher Order ThoughtTheory, HOT (Rosenthal, 2002). In some cases, contemporary theories in this class may lack the specificityto actually make predictions in their current form. Therefore, the formalisms we introduce may sometimesdescribe a more advanced or more developed form of a theory, one that can actually be applied to makepredictions.In the following section, we introduce the necessary background and terms to define how to falsify thisclass of theories: how measurement of a physical system’s observables results in data sets (section 2.1),how a theory makes use of those data sets to offer a predictions about consciousness (section 2.2), howan experimenter makes an inference about a physical system’s experiences (section 2.3), and finally howfalsification of a theory occurs when there is a mismatch between a theory’s prediction and an experimenter’sinference about experiences (section 2.4). In section 2.5 we give a summary of the introduced terms. Insubsequent sections we explore the consequences of this setup, such as how all contemporary theories are3lready falsified if the data used by inferences and predictions are independent, and also how theories areunfalsifiable if this is changed to dependency. All experimental attempts to either falsify or confirm a theory of consciousness (in the class 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 ∈ P denotes that system being in a particular state at some time t . Onthe other hand, for Global Neuronal Workspace, P might denote the set of long-range cortical connectionsthat make up the global workspace of the brain, with p being the activity of that global workspace at thattime.Testing a physical system necessitates experiments or observations on the system under consideration.For instance, measurements based on neuroimaging tools like fMRI or EEG have to be carried out in orderto obtain information about the brain. This information might be used to create detailed data sets suchas functional networks, wiring diagrams, models, or transition probability matrices. To formalize this, wedenote by O all possible data sets that can result from observations of P . Each o ∈ O is one particular dataset, and is the result of carrying out some set of measurements on p . We denote the process of measurementof the observables of p as obs ( p ): obs : P (cid:16) O , (1)where obs is a correspondence , which is a “generalized function” that allows for there to be more than oneelement in the image obs ( p ). 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 stochasticbehaviour of the system under measurements, 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 ∈ P , there is a whole class of data sets which can result from theexperiment.Note that obs describes the experiment, the choice of observables, and all conditions during an experimentthat generates the data set o necessary to apply the theory, which may differ from theory to theory. Inall realistic cases, the correspondence obs is likely quite complicated. It depends on what measurements areperformed and might be difficult to state simply. But for our argument, it 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 physicalsystems. That is, experiences themselves are never identified or used in obs , but are rather inferred basedon some data set o , and o does not contain the experiences themselves. In the subsequent sections, we explore how the data sets in O are used to make predictions about theexperience of a physical system. A theory of consciousness makes predictions about the experience of some physical system in some configu-ration, state, or dynamics, p , based on some data set o . To this end, a theory carries within its definitiona 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 consciousexperience in early versions of IIT, descriptions of the content of conscious experience in contemporary IIT, In the case of a function f : A → B , for every a ∈ A there is exactly one element f ( a ) ∈ B . A correspondence g : A (cid:16) B canspecify more than one element for every a ∈ A : The image g ( a ) is a subset of B . Functions are special cases of correspondences. For example, in cases where a theory necessitates perturbations, knock-outs, or noise injections, such as IIT, this is includedin obs , since such interventions always manipulate observables then measure the consequence on other observables. That is, obs describes the whole experimental setup. Even though phenomenology, i.e., the structure of experience, is used to derive the axioms of IIT, the applications of thepostulates never include experience themselves, rather, calculating the integrated information is done using some third-persondata set o . O Eobs pred
Figure 1:
We assume that an experimental setup apt for a particular model of consciousness has been chosen forsome class of physical systems P , wherein p ∈ P represents the dynamics or configurations of a physical system. O then denotes all data sets that can arise from observations or measurements on P . Measuring the observables of p maps to data sets o ∈ O , which is denoted by the obs correspondence. E represents the mathematical descriptionof experience given the theory or model of consciousness under consideration. In the simplest case, this is just a setwhose element indicate whether a stimulus has been perceived consciously or not, but far more complicated structurescan arise (e.g., in IIT). The correspondence pred describes the process of prediction as a map from O to E . or the description only of whether a presented stimuli is experienced in GNW or HOT. We sometimes referto elements e of E as states of experience or simply experiences .Formally, this means that a prediction considers an experimental dataset o ∈ O (determined by obs )and specifies an element of the experience space E . We denote this as pred , for “prediction,” which is amap from O to E . The details of how individual data sets are being used to make predictions again do notmatter 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 data set o ∈ O may not predict only one singleexperience. In general, pred may only allow an experimenter to constrain experience of the system in thatit only specifies a subset of all experiences a theory models. We denote this subset pred ( o ). Thus, pred isalso a correspondence pred : O (cid:16) E .
Shown in Figure 1 are the full set of terms needed to formally define how most contemporary theoriesof consciousness make predictions about experience. So far, what we have said is very general. Indeed, theforce and generalizability of our argument comes from the fact that we do not have to define pred explicitlyfor the various models we consider. It suffices that it exists, in some form or the other, for the models underconsideration.It is crucial to note that predicting states of consciousness alone does not suffice to test a model ofconsciousness. Some have previously criticized theories of consciousness, IIT in particular, just based off oftheir counter-intuitive predictions. An example is the criticism that relatively simply grid-like networks havehigh Φ (Aaronson, 2014; Tononi, 2014). However, debates about counter-intuitive predictions of such a theoryare not meaningful by themselves, since pred alone does not contain enough information to say whether atheory is true or false. The most a theory could be criticized for is either not fitting our own phenomenologyor 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 implicitlyrely on a comparative reference to be meaningful, and speculations that are not explicit about their referenceare uninformative.
As discussed in the previous section, a theory is unfalsifiable given just predictions alone. Falsifications requirethat the results of pred must be compared to something else. Ideally this would be the actual consciousexperience of the system under investigation. However, as noted previously, the class of theories that wefocus on here assume that experience itself is not part of the observables. For this reason, the experienceof a system must be inferred separately from a theory’s prediction to create a basis of comparison. Mostcommonly, such inferences are based on reports . For instance, an inference might be based on an experimentalparticipant reporting on the switching of some perceptually bistable image (Blake et al., 2014) or based onreports about seen vs. unseen images in masking paradigms (Alais et al., 2010). In such common cases thereport would be some key press indicating a certain visual experience.It has been pointed out that such a 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). Insuch cases, report is first correlated to some autonomous phenomenon like optokinetic nystagmus (stereotypedeye movement), and then the experimenter can use this instead of the subject’s direct reports to infer their5 O Eobs predinf
Figure 2:
Two maps are necessary for a full experimental setup, one that describes a theory’s predictions aboutexperience ( pred ), another that describes the experimenter’s inference about it ( inf ). Both map from a data set o ∈ O collected in an experimental trail to some subset of experiences described by the model, E . experiences. Indeed, there can even be simpler cases where report is merely assumed: e.g., that in showinga red square a participant will experience a red square without necessarily asking the participant, sincepreviously that participant has proved compos mentis. Similarly, in cases of non-humans incapable of verbalreport, “report” can be broadly construed as behavior or output.Despite the number of ways of inferring experiences, all these cases can be broadly described as beinga case of inference off of some data. This data might be actual verbal reports (like a participant’s buttonpushes) or may be based off of physiological reactions (like no-report paradigms) or may be the outputsof a neural network of 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 data set o generated byobservation or measurement of the physical system, and the set of postulated experiences in the model ofconsciousness, E : inf : O → E .
Defining inf as a function means that we assume that for every experimental data set o , one single experiencein E is inferred during the experiment. Here we use a function instead of a correspondence for technical andformal ease, which does affect our results. The inf function is flexible enough to encompass both directreport, no-report, input/output analysis, and also assumed-report cases. It is a mapping that describes theprocess 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 O as in the definition of the prediction mapping pred above. This makes sense because the inference process also uses data obtained in experimental trial,such as reports by a subject. Note that this fully compatible with use of data that has been collected inadvance. For example, in cases where inference does not actually draw from the observables during theexperiment (like an assumption of compos mentis), it must draw from some other observables (how composmentis was decided), which can be formally included in our setup if necessary. So both pred and inf can bedescribed to operate on the same total data set measured, even though they usually use different parts ofthis data set (cf. below). We have now introduced all elements which are necessary to formally say what a falsification of a theory ofconsciousness 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, weconsider some particular experimental trial, as well as inf and pred just introduced.
Definition 2.1.
There is a falsification at o ∈ O if we have inf ( o ) (cid:54)∈ pred ( o ) . (2)This definition can be spelled out in terms of individual components of E . To this end, for any givendata set o ∈ O , let e r := inf ( o ) denote the experience that is being inferred, and let e p ∈ obs ( o ) be one of This assumption is not a restriction of our formalism, because if two correspondences to the same space are given, onecan turn of of them into a function by coarse-graining. This makes the definitions involving experimental falsification, such asDefinition 2, conceptually and mathematically simple. o ∈ 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 data set o . o i (orange) is the part of the dataset used for prediction. o r (green) is the part of the data set usedfor inferring the state of experience. Usually the green area comprises verbal reports or button presses(typically only one), whereas the orange area comprises the data obtained from brain scans. The right circledepicts the experience space E of a theory under consideration. e p denotes a predicted experience while e r denotes the inferred experience inferred (generally from reports). Therefore, in total to represent somespecific experimental trial we use p ∈ P , o ∈ O , e r ∈ E and e p ∈ E , where e p ∈ pred ( o ).the experiences that is predicted based off of some data set. Then (2) simply states that we have e p (cid:54) = e r for all possible predictions e p . None of the predicted states of experience is, in other words, equal to theinferred experience.What does Equation (2) mean? There are two cases which are possible. Either the prediction based onthe model of consciousness is correct, so that according to this equation the inferred experience is wrongand therefore the particular prediction at o cannot be tested. Alternatively, the prediction could be wrong,so that in this case the model is falsified. In short: Either the prediction process or the inference process iswrong.We remark that if there is a data set 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 itcomes to falsifications, we can restrict to data sets o for which both procedures are defined, i.e. for whichboth a prediction and a falsification is possible.In order to understand in more detail what is going on if (2) holds, we have to look into a single dataset o ∈ 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 theprediction. This data can be obtained from an fMRI scan or EEG measurement. The state of consciousnesson the other hand can be inferred from verbal reports. Pictorially, we may represent this as in Figure 3. Weuse the following notation: o i For a chosen data set o ∈ O , we denote the part of the data set which is used for the prediction processby 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 data set o ∈ O , we denote the part of the data set which is used for inferring the stateof experience by o r (adding the subscript r , similar to how restrictions are denoted in sets). We callit the inference data in o . In summary, the formalism we have introduced so far describes an experimental test of a theory of conscious-ness. We have introduced the following notion: 7 denotes a class of physical systems, each in various different configurations. In most cases, every p ∈ 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 ismeasured. It describes how measurement results (data sets) are determined by a system configurationunder investigation. This correspondence is given, though usually not explicitly known, once a choiceof measurement scheme has been made. O is the class of all possible data sets that can result from observations or measurements of the systemsin the class P . Any single experimental trail results in a single data set o ∈ O , whose data is used formaking 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 data set o .It is therefore a mapping from O to E . E denotes the space of possible experiences specified by the theory under consideration. The result ofthe 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 . In this section, we subject the notion of falsification as used in contemporary experiments to a thoroughformal analysis. We do this by considering changes of physical systems we call substitutions.Substitutions are changes of physical systems (i.e., the substitution of one for another) that leave theinference data invariant, but may change the result of the prediction process. Note that a specific case ofsuch a substitution, the unfolding of a reentrant neural network to a feed-forward one, was recently appliedto 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 consciousnessexhibits a fundamental flaw for almost all contemporary theories, rather than being a problem for a particulartheory. 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 tothis flaw in Section 4.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, weprove that universal substitutions exist if prediction and inference data is independent and give some examplesof already-known cases.
Substitutions are changes of physical systems that leave the inference data invariant. In order to defineformally what a substitution is, we work with the inference content o r of a data set o as introduced inSection 2.4. We first denote the class of all physical configurations which could have produced the inferencecontent o r upon measurement by P o r . Using the correlation obs which describes the relation between physicalsystems and measurement results, this can be defined as P o r := { p ∈ P | o r ∈ obs ( p ) } , (3)where obs ( p ) denotes all possible data sets that can be measured if the system p is under investigation andwhere o r ∈ obs ( p ) is a shorthand for o ∈ obs ( p ) with inference content o r .Any map of the form S : P o r → 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 definewhat a substitution is formally. 8 ( p ) o resp. o (cid:48) o r = o (cid:48) r o i p o (cid:48) i pred ( o ) inf ( o ) e r = e (cid:48) r pred ( o ) e p E pred ( o (cid:48) ) e (cid:48) p pred ( o (cid:48) ) Figure 4: This picture illustrates substitutions. Assume that some data set 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 sameinference content o r , the prediction content of experimental data sets is different. Different in fact to suchan extend that the predictions of consciousness based on these datasets are incompatible (illustrated by thenon-overlapping circles on the right). Here we have used that by definition of P o r , every ˜ p ∈ P o r yields atleast one data set o (cid:48) with the same inference content as o and have identified o and o (cid:48) in the drawing. Definition 3.1.
There is a o r -substitution if there is a transformation S : P o r → P o r such that at least forone p ∈ P o r pred ◦ obs ( p ) ∩ pred ◦ obs ( S ( p )) = ∅ . (4)In words, this definition requires there to be a transformation S which keeps the inference data constantbut 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 iscontained in both predictions. Given some inference data o r , an o r -substitution then requires this to be thecase 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 )) (cid:54) = pred ( obs ( S ( p ))).We will find below that the really problematic case arises if there is an o r -substitution for every possibleinference 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 . The force of our argument comes from the fact that if there are substitutions (transformation S which satisfy(4)), then this necessarily leads to mismatches between inferences and predictions. This is shown by thefollowing lemma. Lemma 3.3.
If there is a o r -substitution, there is a falsification at some o ∈ O .Proof. Let p be the physical system in Definition 3.1 and define p (cid:48) = S ( p ). Let o ∈ obs ( p ) be a datasetof p which has inference content o r and let o (cid:48) be a dataset of p (cid:48) 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 ) ∩ pred ( o (cid:48) ) = ∅ . (5) Here, the ◦ indicates the composition of the correspondences obs and pred to give a correspondence from P to E , whichcould also be denoted as pred ( obs ( p )), and ∩ denotes the intersection of sets. We recall that according to the notation introduced in Section 2.4, the inference content of any data set o ∈ 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 pertainin the experiment under consideration (for every inference data that is listed in O ). o r = o (cid:48) r , we have inf ( o ) = inf ( o (cid:48) ). Thus we have either inf ( o ) (cid:54)∈ pred ( o ) or inf ( o (cid:48) ) (cid:54)∈ pred ( o (cid:48) ), or both. Thus there is either a falsification at o , a falsification at o (cid:48) , 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 whereasthe inferred experience is wrong. Inaccessible predictions are not unusual in science. A fully problematiccase 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. In Section 2.4, we have defined falsifications for individual data sets o ∈ O . Using the ‘insight view’ of singledata sets, 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 ∈ O which has inferencecontent o r . This definition is weaker than the original definition, because among all data sets which have inferencecontent o r , only one needs to exhibit a falsification. Using this notion, the next lemma specifies the exactrelation 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 data sets o and o (cid:48) used in thatproof both have inference content o r .This finally allows us to show our main result. It shows that if a universal substitution exists, the model ofconsciousness under consideration indeed is highly problematic. We explain the meaning of this propositionafter the proof. Proposition 3.6.
If there is a universal substitution, there is an o r -falsification for all possible inferencecontents o r .Proof. By definition of universal substitution, there is an o r -substitution for every o r . Thus the claim followsdirectly from Lemma 3.5.What is the meaning of Proposition 3.6? In combination with Definition 3.4, this proposition states thatfor every possible report (or any other type of inference procedure, cf. our use of terminology in Section 2.4),there is a data set o which contains the report’s data and for which we have inf ( o r ) / ∈ pred ( o ) , (6)where we have slightly abused notation in writing inf ( o r ) instead of inf ( o ) for clarity. This implies that oneof two cases needs to pertain: Either at least one of the inferred experiences inf ( o r ) is correct, in which casethe corresponding prediction is wrong and the model needs to be considered falsified. The only other optionis that for all inference contents o r , the prediction pred ( o ) is correct, which qua (6) implies that no singleinference inf ( o r ) points at the correct experience, so that the inference procedure is completely rubbish.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 thetheory under consideration is already falsified.
In Section 4, we interpret this result. But first, we discuss under which circumstances a universalsubstitution exists.
In the last section, we have seen that if a universal substitution exists, this has strong consequences for atheory under consideration. In this section, we discuss whether a universal substitution exists.10 .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 incompleteor noisy data sets. This is the reason why pred is a correspondence, the most general object one couldconsider. For the purpose of this section, we add an gentle assumption which restricts pred slightly: Weassume that every prediction carries at least a minimal amount of information. In our case, this means thatfor every prediction pred ( o ), there is at least one other prediction pred ( o (cid:48) ) which is different from pred ( o ).Put in simple terms, this means that we don’t consider theories of consciousness which have only a singleprediction.A simple way of formalizing this assumption would be to assume that for every o ∈ O , there is a o (cid:48) ∈ O with pred ( o ) ∩ pred ( o (cid:48) ) = ∅ . This ensures that pred can make multiple predictions: that is, that for every pred ( o ) there is some pred ( o (cid:48) ) that gives a different prediction. Indeed, if pred ( o ) were all of E , that wouldbe a meaningless prediction.However, this formalization is to naive, as it ignores the fact that data sets come from physical systems p when measured in experimental trails. In order to take this into account, for every o ∈ O , we define¯ o := obs ( obs − ( o )), which comprises exactly all those data sets which can be generated by physical systems p that also generate o . When applying our previous definitions, this can be fleshed out as¯ o = { o (cid:48) | ∃ p such that o ∈ obs ( p ) and o (cid:48) ∈ obs ( p ) } . (7)Using this, we can state our minimal information assumption in a way that is compatible with the generalsetup displayed in Figure 2:We assume that the theories of consciousness under consideration are minimally informative in that forevery o ∈ O , there exists an o (cid:48) ∈ O such that pred (¯ o ) ∩ pred (¯ o (cid:48) ) = ∅ . (8) We have already noted, that in most experiments, the prediction content o i and inference content o r consistof different parts of a data set. What is more, they are independent, in the sense changes in o i are possiblewhile 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 (cid:48) i and o r , there is a variation ν : P → P (9) such that o i ∈ obs ( p ) , o (cid:48) i ∈ obs ( ν ( p )) but o r ∈ obs ( p ) and o r ∈ obs ( ν ( p )) for some p ∈ P . 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 4.
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 ∈ O , a o r -substitutionexists (Definition 3.1). Thus assume that an arbitrary o ∈ O is given. The minimal information assumptionguarantees that there is an o (cid:48) such that Equation (8) holds. As before, we denote the prediction content of o and o (cid:48) by o i and o (cid:48) i , respectively, and the inference content of o by o r .Since inference and prediction data are independent, there exists a p ∈ P as well as a ν : P → P suchthat o i ∈ obs ( p ), o (cid:48) i ∈ obs ( ν ( p )), o r ∈ obs ( p ) and o r ∈ obs ( ν ( p )). By Definition (7), the first two of these four We remark that here, we use the same shorthand as in (3). For example, the requirement o i ∈ obs ( p ) is a shorthand forthere being an o ∈ obs ( p ) which has prediction content o i . obs ( p ) ⊂ ¯ o and obs ( ν ( p )) ⊂ ¯ o (cid:48) . Thus Equation (8) applies and allows us to concludethat pred ( obs ( p )) ∩ pred ( obs ( ν ( p )) = ∅ . Via Equation (3), the latter two of the four conditions imply that p ∈ P o r and ν ( p ) ∈ P o r . Thus we mayrestrict ν to P o r to obtain a map S : P o r → P o r , which in light of the first part of this proof exhibits at least one p ∈ P o r which satisfies (4). Thus we haveshown 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 consciousnessneed to be minimally informative, for any data set o , there is another data set o (cid:48) which makes a non-overlapping prediction. But in virtue of inference and prediction data being independent, we can find avariation that changes the prediction content as prescribed by o and o (cid:48) , but keeps the inference contentconstant. This suffices to show that there exists a transformation S as required by the definition of asubstitution.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 iswrong or the theory under consideration is already falsified.Proof.
The theorem follows by combining Proposition 3.9 and Proposition 3.7.This concludes the statement of our main claim. In the next section, we give several examples of universalsubstitutions, before discussing various possible responses to our result in Section 4.
Our main theorem shows that testing a theory of consciousness will necessarily lead to its falsificationif inference and prediction data are independent (Definition 3.8), and if at least one single inference canbe trusted (Theorem 3.10). In this section, we give several examples that illustrate the independence ofinferences that rely on report and prediction data. We take report to mean output, behavior, or verbalreport itself.
Artificial neural networks . ANNs, particularly those trained using deep learning, have grown increasinglypowerful 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 notpart 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 changingthose nodes which are not part of the output. Put differently, one can always substitute a given ANN withanother with different internal observables but identical or near-identical reports. From a mathematicalperspective it is well-known that both single-layer ANNs and recurrent ANNs can approximate any givenfunction (Hornik et al., 1989; Sch¨afer and Zimmermann, 2006). Since reports are just some function, thereare viable universal substitutions that provably exist.A special case thereof is the unfolding transformation considered in Doerig et al. (2019) in the contextof IIT. The arguments in this paper constitute a proof of the fact that for ANNs, inference and predictiondata are independent (Definition 3.8). Crucially, our main theorem shows that this has implications forall minimally informative theories of consciousness. A similar results (using a different characterization oftheories of consciousness than minimally informative) has been shown in Kleiner (2019).
Universal computers . Turing machines are extremely different architecture than ANNs. Since they arecapable of universal computation (Turing, 1937) they should provide an ideal candidate for a universal sub-stitution. 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 Turing machine (or some sped-up variant thereof)to pass the Turing test, one should accept the substitution argument. Notably, Turing machines are just oneexample of universal computation, and there are other instances of different parameter spaces or physicalsystems that are capable, such as cellular automata (Wolfram, 1984).12 niversal intelligences . There are models of universal intelligence that allow for maximally intelligentbehavior across any set of tasks (Hutter, 2003). For instance, consider the AIXI model, the gold-standard foruniversal intelligence, which operates via Solomonoff induction (Solomonoff, 1964; Hutter, 2004). The AIXImodel generates via an algorithm an optimal decision making over some class of problems, and methodslinked to it have already been been applied to a range of behaviors, such as creating ”AI physicists” (Wuand Tegmark, 2019). Its universality indicates it is a prime candidate for universal substitution of reports.Notably, unlike a Turing Machine, it avoids issues of precisely how it is accomplishing universal substitutionof report, since the algorithm that governs the AIXI model behavior is well-described and ”relatively” simple(compared to, say, the human brain).These are all real and viable classes systems that are used everyday in science and engineering whichall provide different viable universal substitutions if inferences are based on reports or outputs. We haveavoided possible but less reasonable examples of universal substitutions, like astronomically-large look-upledgers of reports.The above examples show that in normal experimental setups such as the ones commonly used in neurosci-entific research into consciousness (Frith et al., 1999), inference and prediction data are indeed independent,and dependency is not investigated nor properly considered. So it is always possible to substitute the phys-ical system under consideration with another that has different internal observables, and therefore differentpredictions, but similar or identical reports (since inferences are generally based solely on reports).As an example of our Main Theorem 3.10, we consider the case of IIT. For this theory it has already beendirectly proven, with multiple examples, that inferences and predictions are independent. Since the theoryis normally applied in Boolean networks, logic gates, or artificial neural networks, we take report to mean“output.” It has already been proven that systems with different observables and corresponding differentpredictions from IIT, can have identical input/output (and therefore reports or inferences about report)(Albantakis and Tononi, 2019). Similar research has shown that Φ is not invariant under isomorphismsof labeling schemes, despite those schemes having identical input/output (Hanson and Walker, 2019). Totake another case: within IIT it has already been acknowledged that a Turing machine may have a wildlydifferent 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. Note that this has been showneven in the strictest sense of universality wherein the reports of the system are not only highly similar, butcompletely identical (even the dynamics of the internal observables, but not their causal relationships, areidentical in some already demonstrated cases).One immediate response may be to argue that universal substitution, in the strictest sense of universality(for each and every possible inference content), is impossible despite the examples we have shown. Thatis, even a computer capable of passing the Turing test might not have, for instance, a sense of smell, andtherefore would not count as truly universal. Alternatively, it might be argued that perhaps there are someclasses of problems that a Turing machine cannot solve but the brain can (Penrose, 1994).However, the issues from independence do not actually require universal substitutions. Instead, it requiresonly that there be independence between o i and o r . So even if o and o (cid:48) do not fully overlap there can stillbe mismatch, and therefore falsification, which can be created via partial substitutions. Therefore, there isstrong evidence to believe that data independence given the normal assumptions behind testing scientifictheories of consciousness while relying on report, behavior, or output. It may be argued that the examples we have given here do show that data independence holds, but only when inf is based on reports (or outputs or behavior more generally). This is true. Yet report being the basisof inferences about consciousness is an assumption shared widely, perhaps universally, within contemporaryconsciousness research.The reasons for this is an attempt to be as agnostic as possible, since it is unknown a priori (without atheory) which systems are conscious, without looking at behavior, output, or report. In a maximally agnosticview, inf should solely depend on the richness of the available report and/or the intelligence of the systemunder question. That is, an experimenter’s confidence in inferences about experiences should grow withrichness of report, intelligence of behavior, or some combination of the two. This is quite reasonable. Yetrelying on solely on reports entails data independence and therefore creates fragility in theories and allows13or falsifications via substitutions.One response to the availability of substitutions might be to restrict all testing of theories of consciousnesssolely to humans. If testers simply refuse to accept reports about experiences from non-humans, testing canproceed apace. There are some significant downsides to this strategy, since this is equivalent to saying that allinferences in non-humans are wrong. First, many theories of consciousness clearly wish to answer questionsabout non-human organisms and things like artificial intelligence. The consequence of such a restriction isthat such predictions about consciousness outside of humans are no longer falsifiable. This drastically cutsdown the reach of theories of consciousness. Second, and more fundamentally, there are likely variations(as in 3.8) available within the brain wherein o r is fixed but o i varies, meaning that there is still dataindependence even just in brains alone.Another reasonable response to these issues is to explore setups and theories wherein there is data dependency , such as when predictions are based on reports or alternatively accessibility to report. As wewill next show, theories that imply data dependency have their own issues and flaws, most notably that oftriviality or unfalsifiability. Therefore, consciousness research is caught between a rock and a hard place.Clearly, however, our results show that inferences based on reports, behavior, or output alone cannot beused to test theories of consciousness. In this section, we list several possible responses to the previous results.
The simplest response to our result is to claim that inference and prediction data are not independent. Toshow that this claim has important negative consequences, we consider first the complete opposite case toindependence: the case in which inference and prediction data are strictly dependent.
In order to analyse the case where inference and prediction data are dependent, we first need to say some-thing about unfalsifiable theories of consciousness. Clearly, the goal of the scientific study as a whole isto find, eventually, a theory of consciousness that cannot be falsified in experimental situations, becauseit is compatible with all experimental evidence. This is a necessary condition (though not sufficient) anypurportedly “true” theory of consciousness needs to satisfy. Therefore, unfalsifiability per se is not a badthing. However, unfasifiability can also be due to pathological assumptions that underlie a theory of con-sciousness, assumptions which render an experimental investigation meaningless. Specifically, a pathologicaldependence 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 O denotes the class of all data sets that can result from an experiment under investigation. Put differently, itcontains all data sets that can appear in nature when probed in the experiment. This is not the class of allpossible data sets of type O one can think of. Many data sets which are of the same form as elements of O might simply not arise in the experiment under consideration. We denote the class of all possible data setsas: O : All possible data sets of type O . By construction, O is a subset of O , which describes which among the possible data sets actually arises acrossexperimental trails. Hence, O also determines which theory of consciousness is compatible with (i.e. notfalsified by) the experimental investigation.Introduction of O allows us to distinguish the pathological and non-pathological cases of unfalsifiabilitymentioned above. Whereas any purportedly true theory should only be unfalsifiable with respect to theexperimental data O , a pathological unfalsifiability pertains if a theory cannot be falsified at all, i.e. over O .This is captured by the following definition. Definition 4.1.
A theory of consciousness which does not have a falsification over O is empirically unfal-sifiable . O , we can also define thenotion 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 forany o ∈ O , we have o i = f ( o r ) . This definition should be understood as placing a condition on O : that there is a function f as described.If inference and prediction are assumed to be dependent, the class of possible data sets is smaller than if therewere no dependence. The definition refers to O and not O , as the dependence of inference and predictionconsidered here holds by assumption and is not simply asserting a contingency in nature.This definition of strict dependence 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, for example, the case for behaviourist theories (Skinner,1938) of consciousness, where consciousness is equated directly with report or behavior. It is also the casefor early functionalist theories of consciousness based on behavior or input/output (Putnam, 1960).Another example of strict dependence is the case where prediction data is always a subset of the inferencedata: o i ⊆ 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 – some form ofinference on the system about what is accessible to report or output. In terms of Block’s distinction betweenphenomenal consciousness and access consciousness (Block, 1996), Equation (10) holds true whenever atheory of consciousness is under investigation which models access consciousness alone.Our second main theorem proves this formally. 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 proof the theorem, it is useful to consider the inference and prediction content of data sets explicitly.The possible pairings that can occur in an experiment are given by O exp := { ( o i , o r ) | o ∈ O } , (11)where we have again used our notation that o i denotes the prediction data of o , and similar for o r . To definethe possible pairings that can occur in O , we let O i denote the class of all prediction contents that arise in O ,and O r denote the class of all inference contents that arise in O . The set of all conceivable pairings is thengiven by O all := { ( o i , o (cid:48) r ) | o ∈ O , o (cid:48) ∈ O } (12)= { ( o i , o (cid:48) r ) | o i ∈ O i , o (cid:48) r ∈ O r } . (13)Crucially, here, o i and o (cid:48) r do not have to be part of the same data set o . Combined with Definition 2.1, weconclude that there is a falsification over O if for some ( o i , o (cid:48) r ) ∈ O all , we have inf ( o ) / ∈ pred ( o (cid:48) ), and thereis a falsification over O if for some ( o i , o r ) ∈ O exp , we have inf ( o ) / ∈ pred ( o ).Next we show that if inference and prediction data are strictly dependent, then O all = O exp holds. Westart with the set O all as defined in (12). Expanding this definition in words, it reads O all = { ( d i , d r ) | ∃ o ∈ O such that d r = o r and ∃ ˜ o ∈ O such that d i = ˜ o i } , (14) We remark that this formalization of dependency is more fundamental than the one used in Definition 3.8, in that any ifinference and prediction data are independent as defined there, they are not dependent as defined here. These slightly differentways of formalizing are necessary to prove our results given the parsimonious formalization we apply throughout the paper.More detailed comparison would require introduction of more formalism, which we would like to avoid. We also remark thatboth Definition 3.8 and Definition 4.1 could be stated in a more symmetric way. Again, this is unnecessary and would onlymake matters more complicated. In fact, there exists an inverse substitution argument, which we do not discuss here, withsubscripts i and r replaced everywhere. d i and d r to denote prediction and inference data independently of any data set o .Assume that the first condition in this expression, d r = o r holds for some o ∈ O . Since inference andprediction data are strictly dependent, we have d i = f ( d r ). Furthermore, for the same reason, the predictioncontent o i of the data set o satisfies o i = f ( o r ). Applying the function f to both sides of the first conditiongives f ( d r ) = f ( o r ), which thus in turn implies o i = d i . This means that the o that satisfies the first conditionin (14) automatically also satisfies the second condition. Therefore, due to inference and prediction databeing strictly dependent, (14) is equivalent to O all = { ( d i , d r ) | ∃ o ∈ O such that d r = o r and d i = o i } . (15)This, however, is exactly O exp as defined in (11). Thus we conclude that if inference and prediction data arestrictly dependent, O all = O exp necessarily holds.Returning to the characterisation of falsification in terms of O exp and O all above, what we have just foundimplies that there is a falsification over O if and only if there is a falsification over O . Thus either there is afalsification over O , in which case the theory is already falsified or there is no falsification over O , in whichcase the theory under consideration is empirically unfalsifiable.The gist of this proof is that if inference and prediction are strictly dependent, then as far as the inferenceand prediction contents go, O and O are the same. I.e, the experiment does not add anything to the evaluationof the theory. It is sufficient to know only all possible data sets 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 toevaluate the theory, which is clearly ad odds with the role of empirical evidence required in any scientificinvestigation. Thus such theories are empirically unfalsifiable.To give an example of the theorem, let us examine a theory that uses the information accessible toreport in a system to predict conscious experience (perhaps this information is “famous” in the brain or iswithin some accessible global workspace). I.e., the prediction gives the same experience as the one inferredfrom what is accessible to report. Therefore, in terms of our notation, we can assume that o r denoteseverything that is accessible to report, and o i denotes that part which is used by the theory to predictconscious experience. Thus in this case we have o i ⊆ o r . Unless something goes wrong, the participant inthe experiment should be able to deterministically report what is accessible to report, by definition of thatinformation as accessible . Since the predicted contents are always precisely what can be reported, there cannever be any mismatch between reports and predictions. However, this is not only the case for O exp but also,in virtue of the theory’s definition, for all possible data sets, i.e., O all . Therefore such theories are empiricallyunfalsifiable in that experiments add no information to whether the theory is true or not, and therefore suchtheories are empirically uninformative or tautological.This concludes our treatment of the case where inference and prediction data are strictly dependent.Next, we consider other cases of dependencies. When combined, our main theorems show that both independence and strict dependence of inference andprediction data are problematic and thus must not be assumed in an experimental investigation. This raisesthe question of what happens in the case that inference and prediction are dependent, but not strictlydependent. Is there some middle ground which may work?This issue is worth future investigation, and a full formal analysis goes beyond the scope of this paper.However, there are some things we can point out which are not encouraging.First, it is important to note that even if a weaker form of independence holds than the one we haveworked with in Definition 3.8, there will still be available substitutions. The difference is only that thesemight not be universal, i.e., might not imply the worst possible scenario (where one single inference that isto be trusted implies that a theory is wrong). But in general they will still cause problems of theories ofconsciousness, and if inference procedures have any credibility, even non-universal substitutions will implythe theory under consideration is wrong. We expect that the same holds true for strict dependence, so thattheories which imply inference and prediction to be close to strict dependence will have to be dismissed.16he only remaining case that has a chance to avoid these problems is a case where inference is fine-tunedto match the prediction procedure in just the right way so as to avoid falsification but also unfalsifiability.Indeed, the definition of a inf that does not falsify a pred for any data set in O exp but does falsify a pred for some data sets in O all would render a theory of consciousness “true” under our formalisms. It may bepossible to construct such a theory. However, there are already notable problems, for this seems to requirean inf that is equally as complex as pred , in that a theorist is constructing inf as much as pred . It isusually taken to be important that inf has independent justification from pred , which seems difficult toachieve in the case of fine-tuning of inf . That is, it would no longer be the case that inferences rely simplyon verbal report or output, but something much more complex, and therefore difficult to justify. With thisindependent justification missing, the complexity of a fine-tuned inf implies that there is no longer any biasas to whether pred or inf is wrong during a mismatch. Thus even if a falsification occurs, it may be easyto blame this on inf and proceed with a modified version thereof. Indeed, if we allow for any inf to beconstructed of arbitrary complexity, then it is likely that for any pred an inf can be constructed that makesthose predictions “true.”Thus, while according to our present results, there does seem to be room for saving the usual falsificationsscheme, it is clear that to do so, inferences about consciousness must be complexified far beyond report,behavior, or output. This may prove its own challenge as theories of inference must be equally as complexas theories of consciousness themselves. One response to the issue of data independence/dependence identified here is to accept that a theory ofconsciousness may be unfalsifiable but can be judged by other characteristics. This is similar to argumentsaround certain physical theories, like String Theory. Some have argued that falsifiability is not necessary forsome physical theories but that they can be judged by elegance or parsimony instead (Carroll, 2018). Othershave argued that this makes such theories beyond science (Ellis and Silk, 2014; Woit, 2006).In addition to elegance and parsimony, in consciousness science, one could also consider a theory’s fitwith the phenomenology of experience. Examples that pursue this fit with phenomenology are IIT (Tononi,2008) and the isomorphism view (Tsuchiya et al., 2019). The underlying idea here is that the developmentof a theory of consciousness should aim to establish an isomorphism between phenomenal consciousnessand physical substrate (in IIT, this takes the form of axioms about phenomenology and their translation tomathematical postulates).Any proposal to negate falsifications in the scientific study of consciousness imply a substantial change ofscientific methodology, moving away the ideas that have enabled much progress in other disciplines as theybecame more advanced. Additionally, such proposals face a fundamental inverse problem when dealing onlywith parsimony/elegance and fit with phenomenology, as alone the degree of introspective detail we haveabout our own phenomenology may not provide enough information to fit a unique pred function.
A theory of consciousness might not need to be falsifiable in the sense we have laid out here if it solves theHard Problem of Consciousness (Chalmers, 1996), i.e. if it provides an a priori logical reduction of consciousexperience to the physical domain. This would take the form of a pred where it is necessary that a physicalsystem in a certain configuration, state, or trajectory, is associated with certain contents consciousness, andthere is no way such a thing could not be associated with these contents. In such a case, any inf functioncould be considered useless, since even in the cases of mismatch there would be no doubt to believe pred over inf , due to the necessity of the predictions.
One easy way to avoid our conclusions is to claim that experience is an observable when testing theories ofconsciousness. In terms of our notation (Figure 2), this would mean that obs would include measuring andidentifying experiences themselves, and therefore be a map from P and E to O . Equivalently this amounts17o assuming that in an experiment, the “actual” experience e of a system is accessible, next to e r and e p .This is a case to which our results do not apply.While promising at first sight, deeper inspection of this idea raises several severe issues. Most notably,it is unclear of how experience could be represented in experimental data. Science is an intersubjectiveendeavour (Nagel, 1989), and so in virtue of data being of use for science, it seems to always be associatedwith a third-person perspective, and hence cannot “contain” experience.Another option compatible with this line of response is to consider “direct observation” of experience,meaning that predictions should refer to the experience of the experimenter. Predictions of this sort wouldconstitute a significant change to scientific methodology, whose credibility needs to be evaluated in detail.More reasonably, consciousness may be indirectly observable in an experiment. This would require thatthe presence or absence of experience itself changes the configuration, trajectory, or states of physical systems,in other words that the physical is not (causally) closed (Kim, 1998). If a theory of consciousness does notdescribe the physical as closed, a whole other range of predictions are possible: predictions which concernthe observable, physical domain itself, e.g. changes in the dynamics of the system which depend on thedynamics of conscious experience. These predictions are not considered in our setup and may serve to testa theory of consciousness without the problems we have explored here. In this paper, we have subjected the usual scheme for testing theories of consciousness to a thorough formalanalysis. We have shown that deep problems are inherent in this scheme which need to be addressed beforedesigning any experimental investigation.Crucially, in contrast to other similar results (Doerig et al., 2019), we do not put the blame on individualtheories of consciousness, but rather show that a number of assumptions that are usually being made, whentaken together, are responsible for this problem. Most notably, an experimenter’s inference about conscious-ness and a theory’s predictions are generally implicitly assumed to be independent during testing. This isbecause normally the science of consciousness relies on reports, behavior, or output of systems to provideinferences about consciousness, which however implies independence between inferences and predictions. Aswe formally prove, if this independence holds and reports are relied upon, substitutions or changes to physi-cal systems are possible that falsify any given contemporary theory, under minimum assumptions. However,if an experimenter’s inferences about consciousness and a theory’s predictions are instead considered to bestrictly dependent, unfalsifiability follows. Theories of consciousness seem caught between between Scyllaand Charybdis.There may be several paths forward that avoid these dilemmas. We have discussed the various possibleresponses to our results in Section 4. The dilemmas outlined here usefully constrain the set of possibletestable theories of consciousness, helping to draw the negative space around a viable scientific theory.
Funding:
This research was supported by the Allen Discovery Center program through The Paul G. AllenFrontiers Group (12171).
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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