Does the ‘missing fundamental’ require an inferentialist explanation?

J. Judge

Forthcoming in Topoi, special issue on ‘Perception without Representation’ [do not cite this draft]

1 INTRODUCTION: REPRESENTATIONAL THEORIES, RELATIONAL

THEORIES AND PERCEPTUAL INFERENCE

Representationalists think that one’s perceptual experience involves one’s representing

the environment to be a certain way. This representation may not be an accurate

reflection of the way that the environment, in fact, is. Relational theorists, by contrast,

think that one’s perceptual experience is a matter of standing in a certain relation to the

environment. The mind-independent objects and properties that one perceives are

constitutive parts of the corresponding perceptual experience, on the relational view.

Relational theorists acknowledge that appearances can diverge from reality, but they

argue that one does not have to appeal to representational content in order to

accommodate this fact1.

In arbitrating between representational and relational theories of perception, then, it

seems as though perceptual illusions—cases in which a subject’s perceptual

experience diverges from the way the world really is—constitute an important

battleground. These debates, however, have mostly played out in relation to visual

1 Varying suggestions have been made, regarding approaches that can be taken on this score, including, for instance, the proposal that the errors in questions are errors of judgment, and not errors of perception itself. Such doxastic accounts have been advanced by, for instance, Armstrong (1968) and Glüer (2009); see Genone (2014) for an overview.

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experience. What of other modalities? In attempting to identify the parameters of the

representational versus relational debate with respect to audition, it is appropriate to

start by considering what is thought to be a key case of auditory illusion. In what follows,

I consider the phenomenon of the ‘missing fundamental’, as well as examining a notion

that is often deployed by representationalists to explain it—namely, perceptual

inference.

Some representationalists claim that perceptual content is delivered by means of an

unconscious, perceptual inference. This ‘inferentialist’ view was inaugurated by von

Helmholtz at the end of the 19th century, and subsequently developed by scientists such

as Richard Gregory, Irvin Rock, David Marr and Stephen Palmer. According to

inferentialists, perceivers do not have direct, unmediated access to the world. Rather,

the world appears as it does because our perceptual systems construct it, by means of

perceptual inference, to appear that way. On the other side of the debate, ‘direct’

theorists, who subscribe to the relational view of perception, argue that no such

inference occurs: the world appears as it does primarily because of how it really is, not

because of how we construct it to appear.

Though it is frequently deployed as an explanatory concept by inferentialists, the

notion of perceptual inference is somewhat opaque. It is commonly given a negative

description: it is merely an inference that is unconscious, as distinct from person-level,

conscious inferences. The positive features of unconscious inference are rarely spelled

out. In the next section, I examine the notion of inference, in order to clarify what an

appeal to subpersonal inference might entail; I formulate a ‘job description’ for

perceptual inference on this basis. I identify two sets of cases that commonly prompt the

invocation of perceptual inference: namely, cases of perceptual illusion, and cases of

veridical perception where the perceptual content outstrips the information present in

the stimulus. I then pose the following question: do we need to invoke perceptual

inference, as many authors do, in order to explain the phenomenon of the missing

fundamental? I argue that such an appeal is unnecessary, for two reasons. Firstly, the

missing fundamental is not, after all, a clear candidate for the ascription of inferential

capacity to the auditory system: it is neither an illusion, nor is it the case that the

stimulus is crucially impoverished. That is, it submits to a ‘direct’ explanation. Secondly,

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given the adequacy of a simpler explanation, I argue that we should avoid appealing to

perceptual inference. I close by considering objections, and offering replies.

This paper, then, has two central aims. The first is to examine a concept used to

explain perceptual illusions, which are themselves central to the dispute between

representational and relational theories of perception. Secondly, it attempts to extend

this debate, hitherto dominated by vision, towards other sensory modalities. Before I

turn to the missing fundamental itself, however, a preliminary investigation of the nature

of inference is required. What is it? And why is it commonly thought to be needed, in

order to explain aspects of how perceptual systems work?

2 WHAT IS INFERENCE? THE ‘JOB DESCRIPTION’ CHALLENGE

In a recent paper, Paul Boghossian (2014) notes that the nature of inference itself is

‘surprisingly understudied’ (ibid., p. 1) by philosophers, despite its being considered as a

cornerstone of the capacity for rational thought. He gives an account of what he thinks

inference consists in, at the personal level, explicitly declining to discuss subpersonal,

unconscious, automatic reasoning. Considering that the focus of this paper is on such

subpersonal ‘reasoning’, it might be objected that Boghossian’s discussion of inference

has no place here. However, as Boghossian observes, we have a surprisingly loose

grasp on what counts as person-level inference; I suggest that we have even less of a

grasp on what counts as subpersonal inference. To my knowledge, there are no

worked-out accounts of what the hallmarks and characteristics of subpersonal inference

are. For this reason, I suggest that we should begin with a paradigm case of inference,

so that we may begin to get a feel for the kind of process it is, before figuring out what

might constitute subpersonal inference.

My strategy, here, is similar to that employed by Ramsey (2007). If representation is

to be used as a posit in cognitive science, argues Ramsey, then this posit is going to

have to share some features with our commonsense, person-level notions of

representation, in order to be identifiable as representation in the first instance, and to

have the same sort of explanatory value that ordinary representation has. Ramsey

argues that we need to describe the role, or set of causal relations, that warrants the

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assigning of representational function to a structure or state. This description should

allow us to distinguish between representational and non-representational states. He

refers to the task of delineating this role as the ‘job description challenge’. I suggest that

the same applies to inference. I propose, then, to proceed by analysing person-level

inference, in order to find a characteristic feature, or set of features, that a given

subpersonal process must possess in order to count as a case of inference. Following

Boghossian, I argue that rule-following, with the associated representation of the rule,

counts as such a feature.

Boghossian (2014) canvasses various potential approaches, including a doxastic and

a counterfactual approach, and ultimately argues that an appeal to rule-following, which

entails the representation of the rule, is the best way of characterizing inference.

Boghossian begins by inviting us to consider an episode of thinking, which he calls

(Rain). Suppose I wake up one morning, and I recall that (1) it rained last night. This, in

combination with my knowledge that (2) if it rained last night, the streets are wet, leads

me to conclude that (3) the streets are wet. I select appropriate footwear as a result of

this episode of thinking, without having to look outside to verify that the streets are

indeed wet.

The question is, what is it for me—the conscious agent—to infer (3) from (1) and (2)?

Boghossian notes that it cannot be merely that I happen to judge (3) after judging (1)

and (2). Many thoughts can occur in succession without being related by inference. It

also cannot be the case that my judging (1) and (2) caused me to judge (3). Suppose

that I happen to see Ben. My seeing of Ben causes me to drop the coffee I am holding,

which stains my shirt. I form the belief that my shirt is stained as a result of having had

the belief that I see Ben; but we would stop short of saying that I inferred that my shirt

was stained on the basis that I saw Ben. So, the conclusion must be caused by the

premises ‘in the right way’, if it is to count as a case of inference. But what is this ‘right

way’?

Boghossian argues that a transition from beliefs to a conclusion only counts as an

inference when the thinker takes the condition to be supported by the presumed truth of

the premises in question, and draws the conclusion because of that fact. But what

governs this ‘taking’? Boghossian thinks that rule-following is the best way to

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characterize the move from premises to conclusion in an episode of inference. It has

two main strengths. Firstly, the rule-following account can incorporate the ‘taking

condition’ easily. Consider the following example:

(Email rule): Answer any email that requires an answer immediately upon receipt.

Anybody who answers an email immediately upon receipt, if they are following this rule,

would be treating the receipt of an email as a reason for responding immediately.

Secondly, the rule-following account accommodates the intuition that inferential capacity

is general and productive. Not only can I perform the (Rain) inference, but I can also

perform many other ones that are of a similar logical form. Consider the following

example from Boghossian:

(Space-time): If X is a Malament-Hogarth space-time, then it has no Cauchy

surface. X is a Malament-Hogarth space-time. Therefore, X does not have a Cauchy

surface.

Note that the subject doesn’t need to know anything about the terms in (Space-time):

the subject may have no idea what either a Malament-Hogarth space-time or a Cauchy

surface is. Nonetheless, the subject can be confident in the inference.

Boghossian also thinks that there must be a rule-encoding intentional state involved

in any episode of inference. Let us suppose that this requirement were dropped. Instead

of encoding the rule, a subject might just be disposed to act in accordance with it, under

appropriate conditions. But Kripke (1982) argues, along with Fodor (2008) and

Boghossian (Boghossian and Velleman 1989, Boghossian 2008), that this is not an

explication of rule-following. Rather, it is an abandonment of it. Why? Firstly, our

dispositions to behave are finite, but the rules that we follow are infinitary—that is, they

are operations on an infinite set. Secondly, the rules that we follow are meant to guide

our behaviour. My following the modus ponens rule explains why I am disposed to

perform the rain inference. The dispositional view says that I follow the modus ponens

rule by virtue of being disposed to act in accordance with it. But now, I am disposed to

perform the rain inference because I am disposed to act in accordance with the modus

ponens rule; and this is unsatisfactory. Any candidate for an inferential process, then,

must incorporate the representation of the rule, which the agent is following.

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Let us return to the task at hand. Recall that we are trying to formulate a ‘job

description’ for perceptual, unconscious inference, in order to understand when an

inferential explanation of a perceptual process is warranted. Boghossian proposes that

the hallmark of person-level inference is rule-following, where the rule is represented by

the agent; abandoning the ‘representation of the rule’ condition leads to an

abandonment of legitimate rule-following, and the preservation only of its appearance. I

submit, then, that rule-following should figure in our account of unconscious,

subpersonal inference, if subpersonal inference is to sufficiently resemble inference as

we know it, and hence have explanatory value in the theories in which it is posited.

Thus, I propose that perceptual inference ought to be considered as the unconscious

following of a rule, which is represented within the system.

It is important to emphasise that a system may be merely appearing to follow a rule

without actually doing so. We need a way of distinguishing between these cases.

Boghossian argues, as we saw above, that if a subject—or, in our case, a system—

does not internally represent the rule it appears to follow, and instead is merely

‘disposed’ to act in accordance with that rule, then it does not count as rule-following; it

is, in fact, an abandonment of it. And if it is an abandonment of rule-following, it is too

much of a departure from inference to count as a valid case of it. Crucial to the rule-

following account, then, is the internal representation of the rule. This is what enables

us to distinguish between systems that are legitimately rule-following from ones whose

states are merely causally co-varying with an external regularity.

How are we to know what the prima facie candidates for the ascription of inferential

capacity to perceptual systems are, then? In the (Rain) example, a subject reaches a

verdict about the world, in the absence of sufficient sensory information, by the

performance of an inference. Therefore, we should expect that perceptual inference

might be deployed to explain cases where the perceptual system manages to deliver a

verdict in the absence of sufficient information being present at its receptors: cases, in

other words, where there is a seeming discrepancy between experience and world.

There are two important kinds of cases that involve this kind of discrepancy. The first is

a case of perceptual illusion. The second is a case where the perceptual experience,

though veridical, is underdetermined by the stimulus present at the sensory receptors.

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Therefore, our two candidate cases for the ascription of inferential capacity to a sensory

system ought to be perceptual illusions, on one hand, and cases where the stimulus is

crucially impoverished, on the other. As we shall see in the next section, it is precisely

cases such as these that motivate the inferentialist view. When we turn to the ‘missing

fundamental’, in section 4, we will consider whether it counts as a case of perceptual

illusion, or a case of stimulus impoverishment. I will argue that it is neither, and hence,

that it is not, despite the treatment it receives in the literature, a clear candidate for an

inferentialist explanation. Moreover, given the difficulty in determining whether a

perceptual system is following an internally-represented rule, or merely causally

covarying with an environmental regularity—merely disposed to act in accordance with

an environmental regularity, that is—we should take a conservative approach and opt

for the simpler explanation. But first, a closer look at the motivations for ascribing

inferential capacity to perceptual systems.

3 WHY ARE PERCEPTUAL SYSTEMS THOUGHT TO PERFORM

INFERENCES?

Inferentialist explanations of visual perception are motivated by the observation that the

sensory stimulation at the retina is not sufficiently detailed to guarantee an

unambiguous percept. The retina only receives two-dimensional patterns of light; and

yet, somehow, the visual system delivers a stable percept of three-dimensional objects.

Palmer (1999) says that we ought to consider vision as ‘an interpretive process that

somehow transforms complex, moving, two-dimensional patterns of light at the back of

the eyes into stable perceptions of three-dimensional objects in three-dimensional

space.’ (ibid., p. 9, original emphasis). This interpretive process is, it is widely asserted,

an inferential process. Marr (1982) states that ‘the true heart of visual perception is the

inference from the structure of an image to the structure of the real world outside’ (ibid.,

p. 68). Fodor and Pylyshyn (1981) sketch the dominant position of visual perception

processing as the view ‘that perception depends, in several respects presently to be

discussed, upon inferences.’ (ibid., p. 139)

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Rule-following is frequently cited as the central mechanism by which the inference is

performed. Since the visual system cannot rely on the stimulus alone, it must be

following an internally-represented rule, which ensures the delivery of a stable percept.

Rock (1983) makes this appeal to rule-following explicit (ibid., p. 15, emphasis added):

Perception makes use of assumptions and of internalized rules and applies these to

novel situations. […] … inferences occur on the basis of such rules. ‘Rule following’

means more than ‘lawful’. An object in a free fall follows the law of gravitational

attraction, but it does not ‘know’ any law. Perception, on the other and – or any

mental or behavioral act for that matter – that is based on a rule implies that some

lawful relation is ‘known’ or internally represented.

In other words, it is not the case that the transitions merely look lawful; the perceptual

system internally represents its rules of operation, for Rock, and hence, it can be

considered as a genuinely inferential system. This idea occurs elsewhere in

inferentialist accounts as well. Palmer, quoted in Rock (1997), says that ‘the inferential

approach hypothesizes that observers make very rapid and unconscious inferences

based jointly on optical information in their retinal images and internally stored

knowledge of the likelihood of various real-world situations given particular kinds of

image structure.’ (Rock 1997, p. xiii, emphasis added) Predictive accounts of perception

also appeal to the notion that the system has access to internal representations of the

rules that govern its predictions. Maloney and Mamassian (2009) claim that a Bayesian

observer—such as the human visual system—is one that has access to ‘separate

representations of gain, likelihood, and prior.’ (ibid., , p. 150)

What about the auditory system, then? It, too, faces the problem of under-

determination of the stimulus. In particular, it must confront what Bregman (1993) calls

the ‘problem of mixtures’. The stimulus that reaches the ear is just an undifferentiated

pattern of pressure changes over time. The auditory system must somehow parse this

into individual streams, such that a perceiver may make sense of his or her

environment. ‘Auditory scene analysis’, initially described by Bregman, is the set of

processes whereby the auditory system manages to solve the problem of mixtures.

The existence of the problem of mixtures gives rise to inferentialist explanations of

the workings of the auditory system, particularly in the music cognition literature, where

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Bregman’s work in auditory scene analysis is given a decidedly inferentialist reading.

Daniel Levitin (2006) explicitly aligns himself with the inferentialists (ibid., pp. 99-100):

According to the great perception psychologists Hermann von Helmholtz, Richard

Gregory, Irvin Rock, and Roger Shepard, perception is a process of inference, and

involves an analysis of probabilities. The brain’s task is to determine what the most

likely arrangement of objects in the physical world is, given the particular pattern of

information that reaches the sensory receptors—the retina for vision, the eardrum

for hearing.

The brain, says Levitin, has to make a ‘calculated guess’ (ibid., p. 103), or unconscious

inference, as to what is really out there. It does this on the basis of principles (ibid., p.

77):

Through this process of “unconscious inference” (as von Helmholtz called it), our

brains assume that it is highly unlikely that several different sound sources are

present, each producing a single component of the harmonic series. Rather, our

brains use the “likelihood principle” that it must be a single object producing these

harmonic components.

Deutsch (2013, p. 183) also describes the principles governing auditory scene analysis

as ‘rules’: ‘When presented with a complex pattern, the auditory system groups

elements according to some rule based on frequency, amplitude, timing, spatial

location, or some multidimensional attribute such as timbre.’

However, the inferentialist reading is complicated by the frequent appeals made by

Bregman, Levitin and Deutsch to the central role played by environmental regularities. If

the rules originate in environmental patterns, it could be the case that the system’s

states have merely developed such that they respond directly to these external

regularities, without representing them. If so, the processes of auditory scene analysis

might just have the appearance of being rule-following processes. Bregman (2008)

himself suggests this interpretation when he emphasizes, as he does frequently, that

the principles upon which auditory scene analysis relies are provided by regularities in

the acoustic environment. For instance, he says (ibid., p. 864, my italics):

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ASA takes advantage of regularities that cut across environments regardless of the

provenance of meaning in the signal. For example, the various frequency

components arising from an individual acoustic source (such as a voice or an

instrument) tend to start and stop at the same time. (…) It is plausible that over

evolutionary time, the auditory system has incorporated processes into its hard

wiring that detect the general cues that signal the presence of an independent

acoustic source within a mixture. These can be called primitive, or built-in processes

of ASA.

Here, Bregman seems to be suggesting that the principles are not in the system, but in

the world; and that the system is just responding to these environmental regularities in a

causal fashion. We may well ask, when it comes to audition, should we prefer

inferentialist explanations, or direct ones?

I do not propose to settle the issue, with regard to the auditory system as a whole.

Nor do I propose to conduct an exegesis of Bregman, or Levitin or Deutsch for that

matter, in order to discover whether they are really inferentialists or not. What I propose

to do is much more modest. I will examine one phenomenon from auditory perception,

which is commonly given an inferentialist explanation, and investigate whether that

explanation is warranted. In the next section, then, I consider the ‘missing fundamental’.

4 DOES THE MISSING FUNDAMENTAL REQUIRE AN INFERENTIALIST

EXPLANATION?

Consider a sound, which consists of a wave of frequency 200 Hz, along with its

harmonics—that is, waves whose frequencies are integer multiples of 200. The 200 Hz

wave is known as the fundamental. The sound, or complex tone, thus contains waves

with frequency 200 Hz, 400 Hz, 600 Hz, 800 Hz, … and so on. It is heard as having a

low pitch, corresponding to the pitch of the wave with frequency 200 Hz, and a sharp

timbre. However, if this sound is filtered, such that the 200 Hz component is removed

entirely, the timbre alters, but the pitch does not change. In fact, it is possible to remove

all the harmonics, apart from a small group of mid-frequency ones—for example, 1800,

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2000 and 2200 Hz—and the low pitch will still be heard, even though the timbre will be

markedly different. The question is, how is the continued perception of the low pitch

possible, if there is no component in the stimulus with the right corresponding

frequency? This phenomenon is known as the ‘missing fundamental’.

Many contemporary discussions of the phenomenon deploy inferentialist

explanations. Patel and Demorest (2013), for instance, describe auditory perception as

‘constructive’, when it comes to the missing fundamental (ibid., p. 651, italics in original):

When humans process a complex periodic sound consisting of integer harmonics of

a fundamental frequency (such as a vowel or cello sound), they perceive a pitch at

the fundamental frequency, even if that frequency is physically absent (the “missing

fundamental”). Hence the nervous system constructs the percept of pitch from

analysis of a complex physical stimulus.

Levitin argues that the brain ‘fills in’ the missing component, and our pitch percept is

stable ‘because our brain “knows” that a normal, harmonic sound with a pitch of 200 Hz

would have an overtone series of 200 Hz, 400 Hz, 600 Hz, 800 Hz etc.’ (op. cit., p. 40-

41, italics in original). If this account of the missing fundamental is right, then it seems

that, when we hear a pitch in the absence of the fundamental being present in the

stimulus, our auditory system is telling us something about the world that isn’t true.

Recall that we distinguished between two sorts of cases, where an appeal to

representational content seems to be warranted, and where an inferentialist explanation

of the delivery of that content may be on point. These were the cases of perceptual

illusion and impoverishment of the stimulus, respectively. In what follows, I deny both

that the missing fundamental is a case of perceptual illusion, and also that it is a case of

a crucially impoverished stimulus. Hence, the motivation for an appeal to perceptual

inference is undermined.

4.1 The ‘missing fundamental’ is not a perceptual illusion

If the perception of pitch is solely dependent on the fundamental frequency, then the

missing fundamental seems to be a perceptual illusion. This is certainly what von

Helmholtz thought; but it is no longer thought to be the case. Modern theories do not tie

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pitch exclusively to the perception of the fundamental. If modern theories are right, then,

the missing fundamental is not an anomalous percept, where the fundamental is

erroneously perceived despite its absence; instead, it is a normal, representative case

of pitch perception. This is because pitch perception never relies on the fundamental

being present.

Ohm (1843) argued that the fundamental frequency determines the pitch, and all

other frequencies determine timbre2. Von Helmholtz agreed, suggesting that, in cases

where the fundamental frequency is absent, or lacking in power, the auditory system

‘fills in’ the missing component, on the basis of an inference as to what the most likely

cause of the stimulus is. Later, however, Schouten (1938) produced evidence

discrediting this theory. New technologies enabled him to generate complex tones,

consisting of a group of high harmonics whose fundamental had been artificially

removed (in previous experiments with sirens, the fundamental had been present, albeit

weakly). These tones were still heard as having the same pitch as the fundamental.

Schouten referred to the pitch perceived in the absence of the fundamental as the

‘residue pitch’.

In general, when a complex harmonic tone elicits the perception of a residue pitch, or

‘low pitch’ as it is sometimes called, that residue pitch will have the same pitch quality

as that which would be experienced, were one to be presented with a signal possessing

the fundamental frequency of the complex tone. As Pierce (1999) puts it, residue pitch

is ‘just a musical pitch corresponding to the frequency of the fundamental in the

absence of a component at the fundamental frequency.’ (ibid., p. 150) So, the residue

pitch corresponds to the pitch at the fundamental, but it is not produced by a detection

of this component, because it can be heard regardless of whether there is a component

present at the fundamental or not. ‘Even when the fundamental component of a

complex tone is present,’ says Moore (2003, p. 207), ‘the pitch of the tone is usually

determined by harmonics other than the fundamental. Thus, the perception of a residue

pitch should not be regarded as unusual. Rather, residue pitches are what we normally

hear when we listen to complex tones.’ Hence, pitch perception does not depend on the

2 See Houtsma (1995) for a review of the history of this debate, conducted between Seebeck (1841) and Ohm (1843).

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fundamental, even when it is present. The ‘missing fundamental’ is therefore not a case

of the fundamental component being erroneously perceived when it is, in fact, absent.

The missing fundamental is unlike other perceptual illusions in the following respect.

In familiar visual illusions, such as the Müller Lyer illusion, the lines appear to be of

unequal length, even though they are, as a matter of objective, discoverable fact, equal.

Similarly, when we view a white wall in yellow light, the wall appears yellow. Again, the

yellow appearance of the wall is at odds with its actual whiteness (assuming that colour

is a mind-independent feature of objects). Representational theorists and relational

theorists must, respectively, account for this divergence between the appearance and

the fact of the matter. However, the case of the missing fundamental is different. When

we are presented with the stimulus, with the component at the fundamental removed,

the sound appears to have the pitch of C, say. Still more frequencies are removed; we

continue to hear the pitch of C, even though the timbre alters. But this does not mean

that the appearance of pitch diverges from the actual pitch of the sound. It is not the

case that we continue to perceive a C, even though we know that the actual pitch is

some X, say. The situation is not analogous to the Müller Lyer case, where we may

know that the lines are of equal length, or to the wall case, where we may know that the

wall is white. Even if we know that the fundamental is missing, we have no idea what

the actual pitch of the altered stimulus could be, if not a C. This is because, unlike the

wall, which is really white, and not yellow, the tone really does have the pitch of a C. It

has the same pitch as the one we hear. In other words, the ‘missing fundamental’ is not

an illusion at all. The appearance does not diverge from the fact of the matter.

Many auditory psychologists agree that the missing fundamental is not a case of

perceptual error. It does not indicate a discrepancy between experience and world.

However, they move from this observation to the implication that pitch perception is, in a

sense, never veridical. Pitch is commonly described as an imperfect tracker of

frequency; it is subjective, rather than a physical attribute of sounds. For instance,

Gelfand (1998, p. 353) describes pitch as the ‘psychological correlate of frequency’. The

standard view implies that pitch is a feature, not of the mind-independent world, but of

psychological appearances. And if pitch does not really track physical reality, then every

instance of pitch perception now seems to be a perceptual illusion.

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This does not follow, however. Even though pitch does not depend, in a simple way,

upon frequency, that does not mean that it fails entirely to depend on mind-independent,

physical properties of the stimulus. O’Callaghan (2007) points out that pitch depends,

not on frequency simpliciter, but on periodicity. Pitched tones are, in general, periodic

sounds. ‘The class of sounds with pitch,’ says O’Callaghan (ibid, p. 80), ‘is the class of

periodic sounds.’ Within the set of periodic sounds, pitch can be somewhat influenced

by intensity (see, for instance, Verschuure and van Meeteren (1975)). However, it is still

only the set of periodic sounds that have pitch in the first place. Describing pitch as a

subjective phenomenon runs the risk of characterising it as a large-scale illusion,

floating free of physical stimuli; it loses sight of the fact that pitch still relies in a lawful

way on the physical, mind-independent characteristics of sounds. Hence, the missing

fundamental is not a non-veridical anomaly, nor is it a representative example of a

perceptual capacity that never delivers veridical percepts. It is not, in other words, a

perceptual illusion.

The missing fundamental illustrates how a more detailed understanding, arrived at by

empirical research, of the precise physical parameters being detected by a perceptual

system can lead to a revision of what counts as an instance of proper functioning of that

system, and what counts as an error. Given the centrality of instances of perceptual

error, in arbitrating between representational and relational theories of perception, care

ought to be taken that the phenomena being advanced as paradigm cases of perceptual

error are, in fact, properly understood as such.

Even if it is not an illusion, however, it could still meet the other criterion for the

ascription of inferential status: that is, it could still be a case where an impoverished

stimulus is resolved by means of an inference. I counter that claim in the next section.

4.2 The stimulus is not crucially impoverished; hence, an appeal to

perceptual inference is unnecessary

We have seen that pitch perception does not require the fundamental to be present. But

how is it, then, that the system can deliver the same pitch percept upon being presented

with two complex tones, A and B, with no overlapping spectral components? It could be

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because it ‘knows’ that, despite the fact that they are different, the pattern of harmonics

in A and B respectively must have been produced by something vibrating at the same

fundamental frequency, F. That is, even if the missing fundamental is not, after all, a

case of illusion, it could be the case that every instance of pitch perception, where the

auditory system is presented with less than the complete spectral profile of a given

harmonic tone, requires perceptual inference to deliver the pitch percept.

Recall that when a process appears to be rule-following, it could be the case that it is

representing a rule and following it, or it could be merely co-varying with some

environmental regularity, without representing that regularity. A consideration of how

complex sounds tend to occur in the environment helps us to see that the auditory

system could, in this case, just be relying on an environmental regularity, in accordance

with which it developed in the first instance, rather than representing a rule. When a

given pattern of harmonics, A, consisting of components that are integer multiples of

some frequency F, is encountered in the environment, this is almost always

encountered as part of a complete complex tone with fundamental frequency F, of

which A is a subset. The same is true for B, a separate subset of the same complete

complex tone. The system does not need to ‘know’, or to represent to itself the rule that

this regularity obtains. It just needs to have evolved in a world where this regularity

exists. We ought to consider that the functioning of a given perceptual system is

determined, not just by the stimulus with which it is presented at a given time t, but with

the types of stimulus with which it evolved to cope in the first instance. When these

extra environmental considerations are taken into account, then the stimulus presented

at time t does not need to be considered crucially impoverished after all, and we do not

need to suppose that a perceptual inference is being made.

An observation from Oxenham (2013) helps to illustrate this point. Oxenham points

out that pitch perception does not rely on all of the harmonics in the stimulus equally.

The upper, unresolved harmonics seem to be less important for ensuring pitch

perception than the lower ones; the first five to eight harmonics have been found to be

the primary determinants of pitch. However, Oxenham notes that this phenomenon is

not arbitrary: it makes sound ecological sense, because complex tones have more

energy at the lower harmonics (ibid., p. 17):

16

Most sounds in our world, including those produced by musical instruments, have

more energy at low frequencies than at high; on average, spectral amplitude

decreases at a rate of about 1/f, or -6 dB/octave. It therefore makes sense that the

auditory system would rely on the lower numbered harmonics to determine pitch, as

these are the ones that are most likely to be audible. Also, resolved harmonics –

ones that produce a peak in the excitation pattern and elicit a sinusoidal temporal

response – are much less susceptible to the effects of room reverberation than are

unresolved harmonics.

The crucial point here is that the system does not have to ‘know’ about any of these

regularities, or to represent the ‘rule’ that governs the behaviour of sound-emitting

sources. It just needs to have evolved in an environment where such regularities obtain.

We would need more evidence in order to ascribe inferential capacity to the system,

given the existence of this environmental regularity. We would require some evidence,

for instance, that there exists, in the system, a representation of a rule like ‘When A or B

is encountered, output F’. In its absence, I suggest that it is prudent to assume that the

system merely looks as though it is following a rule. There is a ‘buttered slide’, to borrow

a term from Dennett (1978), from truly cognitive systems to automatic ones. Being too

liberal in our ascriptions of inferential capacity robs the notion of its explanatory value. It

should only be done when there is no way to explain the workings of the system

otherwise.

Some objections to this approach suggest themselves, however, which I discuss in

the next section.

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5 OBJECTIONS AND REPLIES

5.1 Objection 1: Inference in general might not be best characterized

in terms of rule-following; hence, subpersonal inference may not be,

either.

As Boghossian notes, the nature of inference has not been given much philosophical

attention. His account also yields something of a vicious circle: inference is needed to

follow rules, and rule-following is needed to perform inferences. He is willing, however,

to bite the bullet and to accept that rule-following may be an ‘unanalyzable primitive’.

(op. cit., p. 17) Wright (2014) suggests that, even though the notion of rule-following is

tempting, what he terms ‘basic inference’ (which is roughly what happens when we act

for a reason, as opposed to performing a random movement, say) does not require rule-

following in Boghossian’s sense. He suggests that we need to abandon the ‘taking

condition’ for these cases. Rules are still involved, but they play a different role, in cases

of basic inference: their role is not of principles that are followed, but rather of

constitutive norms. That is, conformity to these rules—understood to play the role of

norms in this case—‘sets a defeasible but prevailing standard for the very possession of

the acceptances among which they mandate movement. It is accordingly unhappy (…)

to view them as rules which the rational (sub-personally) follow or tacitly know.’ (ibid., p.

37) Wright notes, however, that there is much more to be said on the issue than has

attempted to say: ‘I am under no illusion that these remarks do anything more than point

in a direction,’ he says. (ibid., p. 36)

There might be room, then, to characterize inference at the personal level in a way

that still incorporates rules, but not in Boghossian’s sense. If such an account can be

made clear, then perhaps there is room for an alternative job description for

subpersonal inference—one that does not involve the representation of a rule. As

Wright’s remarks only gesture toward such an account, however, there is much more to

be done in order to (1) develop a satisfactory theory of person-level inference; and then

(2) extend that theory to the subpersonal level. The onus is on the inferentialists to

18

provide this kind of account. In the meantime, I think that it is plausible to assume that

something like Boghossian’s notion of rule-following, with associated rule-

representation, is what underlies the remarks of theorists like Rock, who explicitly

appeals to the internal representation of lawful relations.

5.2 Objection 2: Person-level inference has nothing to do with how

we should think of subpersonal inference.

That there will have to be alterations, when we move from person-level inference to

subpersonal inference, is without question. It could even be that subpersonal inference

need not have any features in common with person-level inference. That being said, it

seems to me that if we abandon all of the salient features of person-level inference in

the move to the subpersonal level, then we no longer have any reason to think that

anything like inference is happening at the subpersonal level. It is unclear why we would

want to call it subpersonal inference at all, in that case. I submit that the job description

for subpersonal inference will have to maintain some features of person-level inference.

What those features are will depend on a combination of analysis of person-level

inference, and empirical evidence as to the functioning of the subpersonal states in

question.

5.3 Objection 3: ‘Inferentialists’ don’t really think that perceptual

systems are rule-following. Talk of ‘inference’ is merely an exercise

of poetic license.

It might be objected that I am misrepresenting the inferentialist camp. I may be doing

this in one of two ways. Firstly, inferentialists might hold that subpersonal systems do

really perform inferences, but that those inferences do not involve rule-following. Or,

secondly, they might deny that perceptual systems are performing inferences at all, but

maintain that, despite this fact, inferentialist language is useful and ought to be retained.

In relation to the first possible misrepresentation, there is reason to think that many

inferentialists do think that perceptual systems are rule-following. After all, many

19

inferentialists often describe systems explicitly in those terms. Recall, from section 3,

Rock’s statement that perception ‘makes use of assumptions and of internalized rules

and applies these to novel situations’. This is a robust statement to the effect that

perceptual systems follow internally-represented rules. It seems to me that we should

take exponents of rule-following, such as Rock and Levitin, at their word. At the very

least, these descriptions give us prima facie reason to think that some inferentialists do

hold that the systems in question are following rules and, in virtue of this, possess

inferential capacity.

However, perhaps Rock is an outlier: maybe other inferentialists are not so

committed. Perhaps moderate inferentialists don’t really think that perceptual systems

follow rules, or even perform inferences in any robust sense. The first thing to say here

is that, if inferentialists don’t really think that perceptual systems are rule-following, and

they don’t actually think that they perform inferences either, then the inferentialist view

just collapses into the direct view. That is, inferentialism fails to amount to a substantive

position, if it doesn’t take its central premise seriously. The remaining way in which it

could be distinct, however, might consist in the view that it is still useful to describe

perceptual systems in inferentialist terms. It might be a convenient way of generating

hypotheses to test, analysing complex phenomena, and so on. In fact, Bregman (1994)

suggests that he holds such a view, when he says (ibid., p. 225):

… the process being carried out by the auditory system differs from ordinary thought

in that we are not conscious of any inference process going on. Helmholtz called it

unconscious inference and we can do so too, as long as we realize that the

metaphor that we are using is based only on the results of the process and not on

its detailed inner nature, which may or may not work like a thinking process. It is

best, perhaps, to think of it as a process that has evolved in our nervous systems to

deal with problems in the accurate use of sensory information.

Bregman obviously thinks that perceptual systems may well not be inferential, but it

might be useful to talk of them as though they are; and he is explicit about this, at least

in this passage. It is, in fact, common for authors to appeal to ‘poetic license’ in their use

of intentional concepts to describe brain processes. For example, Colin Blakemore

(1990) argues that scientists know full well that brains do not really use maps, but the

20

term is useful nonetheless: it is just that scientists lack the words to describe the

topographic organisation in the brain in any other way. They are using the terms

metaphorically, as an exercise of poetic license. Dennett (2007) thinks that using terms metaphorically in this way is useful. Firstly, it

enables us to generate predictions and hypotheses about how the system will behave;

and secondly, it constitutes a crucial step toward seeing how we could begin to get

something like ‘real’ belief and desire out of subpersonal processes (ibid., p. 89):

Far from it being a mistake to attribute hemi-semi-demi-proto-quasi-pseudo

intentionality to the mereological parts of persons, it is precisely the enabling move

that lets us see how on earth to get whole wonderful persons out of brute

mechanical parts. That is a devilishly hard thing to imagine, and the poetic license

granted by the intentional stance eases the task substantially.

However, Dennett emphasizes that a commitment to ‘hemi-semi-demi-proto-quasi-

pseudo intentionality’, rather than fully-fledged intentionality, must be made explicit; and

he also stresses that it must be clear just how this attenuated version is different to its

‘real’ counterpart. He gives the example of belief (ibid., p. 87): we do not attribute full-

blown belief to a brain process, but rather a version of belief that is stripped of many of

its everyday connotations, such as, he suggests, responsibility and comprehension.

This is what legitimises the adoption of the intentional stance. However, the problem

with inference is that we have very little idea what even the everyday connotations of

regular, person-level inference are. If we don’t know what those everyday connotations

are, then we cannot, with confidence, identify the ones that we should abandon when

we move to the subpersonal level.

Perhaps the adoption of the intentional stance facilitates the generation of testable

hypotheses about the perceptual system. This leads me to my final, and most important,

point: using inferentialist language is not the only way to generate testable hypotheses.

Explorations of the regularities that exist in auditory environments, and the realization

that perceptual systems have evolved to mirror those regularities, can lead to the

formulation of hypotheses, too. For instance, take again the example from Oxenham

(2013). We know, from studying environmental sounds, that the lower harmonics

possess more power, are less susceptible to effects from room reflection, and so on.

21

We can thus hypothesise that a system that evolved in an environment such as this

might deliver a percept of pitch on the basis of co-variation with this regularity. Tests

have indicated that, indeed, this is the case. We do not need to talk in terms of the

system ‘knowing’ about these regularities to generate testable hypotheses. In other

words, we can analyse complex perceptual systems, and make useful predictions,

without deploying inferentialist language, and without opening the door to the

confusions I have indicated above.

6 CONCLUSION

Let us sum up. I began by observing that disagreement between representational and

relational views of perception often centres on cases where perceptual content appears

to outstrip the information present in the stimulus. The ‘missing fundamental’ is

commonly thought to involve this kind of discrepancy between mind and world, and the

content of the pitch percept is often described by inferentialists as having been

generated by an act of perceptual inference. However, just what these subpersonal acts

of inference are supposed to consist in is unclear. I suggested that we should start by

getting a feel for person-level inference, in order to formulate a ‘job description’ for

subpersonal inference. I argued that rule-following, which involves the internal

representation of the rule, should feature in this job description.

I then identified two criteria that generally motivate an appeal to perceptual inference:

the existence of a perceptual illusion, on one hand, and the existence of a crucially

impoverished stimulus, on the other. I argued that, contrary to the way it is commonly

discussed in the literature, the missing fundamental satisfies neither criterion. I argued

that, in the absence of clear evidence that the ‘rule’ governing the system is being

represented, and in the presence of a simpler explanation in terms of environmental

regularities, we should avoid ascribing inferential capacity. I closed by considering

objections, and offering replies.

To emphasise, I am not attempting to claim that the auditory system in general fully

submits to a relational explanation—that the auditory system is best thought of as an

‘embedded’ system, to use Orlandi’s term (Orlandi 2013, 2014). That would be an

22

ambitious project. I have confined myself to arguing that explaining the missing

fundamental, at least, does not require an appeal to inferential capacity on the part of

the auditory system. I also acknowledge that much more could be—and, indeed, ought

to be—said on the subject of subpersonal inference. For now, I hope to have made

some contribution to the extension, toward other sense modalities, of a debate that has

heretofore been overwhelmingly dominated by discussions of vision.

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