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\title {Systems, Models and Signature Limits \\ Systems, Models and Signature Limits}

Systems, Models and Signature Limits

\def \ititle {Systems, Models and Signature Limits}
\def \isubtitle {Systems, Models and Signature Limits}
\textbf{\ititle}: \isubtitle
\iemail %
Readings refer to sections of the course textbook, \emph{Language, Proof and Logic}.
I want to start with two questions about the development of mindreading in humans.

first question

The first question starts from an observation. There are different ways to measure performance in tracking false beliefs. For instance, we can measure verbal predictions or we can measure anticipatory looking. For some pairs of measures, there is an interaction between age and measure on performance which looks like this ...
Whereas age appears to have no affect on performance measured one way---by anticipatory looking, say--- when tracking false beliefs is measured another way there is gradual improvement in peformance with age. Why does this apparent discrepancy between the measures exist? Is it because some of the measures used are defective? Or might the interaction reflect something about the processes underlying belief tracking---and if so, what exactly does it reflect?
Relatedly, improvements in performance as measured by the second type of measure---the one on which performance improves gradually with age---are correlated with developments in working memory and inhibitory control (*refs). Why is this?

second question

The discrepancy just mentioned appears to be specific to tracking beliefs as opposed to tracking mental states generally ...
If you look at three-year-olds, you see that they can track anothers' incompatible desires or perceptions on just about any reasonable measure, including explicit verbal predictions (*ref: Rakoczy et al, Gopnik et al). So the discrepancy between measures of belief tracking ability does not extend to other mental states. Why not?
So these are two questions about the development of mindreading. I'm not going to fully answer these questions in my 15 minute talk. Instead I want to introduce some conjectures which, if true, would provide a partial answer to these questions and explain the predictions that follow from them. In this way I hope to set the scene for the science you'll be enjoying in about 12 minutes from now.


  1. two questions [done]
  2. a conjecture about systems
  3. predictions
  4. a conjecture about models
  5. predictions
My first conjecture is about the systems involved in mindreading ...
Consider the conjecture that

first conjecture:

There are two (or more) systems for tracking others’ beliefs.

What is a system? This, I think, is problematic ...
Fortunately, we don't need a detailed theoretical account of systems. It should be almost uncontroversial that humans have multiple systems for tracking numbers (counting vs subetizing), causes, colours and actions. I use ‘system’ as a placeholder for whatever turns out to be the right theoretical notion for understanding these cases. The conjecture that there are multiple systems is not tied to a particular notion of system. Rather, the conjecture is that belief is like number, colour, causation and actions in that there are multiple systems for tracking all of these things.
(Caffeine free diet coke is coke and alcohol free, gluten free beer is beer.)
What predictions follow from the conjecture that there are multiple systems?
The first prediction is that different belief-tracking processes have different processing characteristics. For example, some belief-tracking processes should be automatic in the sense that whether they occur is to a significant extent independent of the subjects' task and motvation. Whereas other belief-tracking processes should not be automatic. Different belief-tracking processes should also vary with respect to the demands they place on cognitive resources like working memory and inhibitory control.


  1. different processing characteristics (e.g. automaticity).
  2. doubly inconsistent responses.
  3. The second prediction is that, where different measures reflect different belief-tracking processes, it should be possible for these measures to yield evidence of inconsistent responses to the same stimuli. This is best illustrated with a diagram ...
Here the idea is that, on the first task, one measure indicates that the subjects predict an action that would be rational given a false belief whereas the other measure indicates that the same subejcts predict a different action, one that would not be rational given the false belief.
So my first conjecture was that

first conjecture:

There are two (or more) systems for tracking others’ beliefs.

(And some system for tracking beliefs in infants exists unchanged in adults.)

I also want to sneak in an extra conjecture about continuity across development. The conjecture is that some system for tracking beliefs in infants exists unchanged in adults. One prediction of this additional conjecture is that, on some measures, adults should perform like infants on belief tracking tasks; in particular, adults’ failures should match infants’. This will be clearer when we add the next conjecture.
So far I have only been talking about tracking beliefs. By saying that a system tracks beliefs I mean that, within limits, it enables a subject to respond differently depending on what others believe.
Now tracking beliefs does not necessarily involve representing beliefs, nor representing any mental states at all. Next I want to move towards a bolder conjecture: I want to say not just that there are multiple belief-tracking systems but, further, that there are multiple mindreading systesm, that is multiple systems that track beliefs by means of representing mental states. To explain this idea, I need the notion of a model.

How do mindreaders’ belief-tracking systems model minds?

The notion of model complements that of system. The idea is going to be that different belief-tracking systems rely on different models of the mental. But I'm getting ahead. Let me start with a simple question.
How do mindreaders model minds?
Or, if you accept the conjecture, we should really be asking about how different belief-tracking systems model minds.
This question needs explanation. Let me explain the question by analogy with the physical.

- compare -

How do physical thinkers model the physical?

To say that a certain group of subjects can represent physical properties like weight and momentum leaves open the question of how they represent those things.
In asking how the subjects, infants say, weight or momentum, we are aiming to understand these things as infants understand them; we are aiming to see them as infants see them. (NB: I'm going to focus on human adults!)
How can we do this? We need a couple of things [STEPS: (1) theories; (2) models; (3) signature limits; (4) trade-offs]


theories of the physical/mental

The first thing we need is theories of the physical or mental.
It is a familiar idea, from the history of science, that there are multiple coherent \textbf{theories} of the physical: impetus and Newtonian mechancs, for example. The impetus theory says that moving objects have something, impetus, that they gradually loose. When they loose their impetus they stop moving. If you push them you impart impetus to them, and that is why they move. With Newtonian mechanics this is not the case; there is no impetus.
Theories specify models. A theory isn't a model, and to say that someone relies on a model is not to say that they necessarily know any theory. But the theory describes a way the universe could be, and so specifies one way that a process could model it as being.
The impetus theory isn't right but it is (broadly) coherent; it describes a way the world could be. This is why it specifies a model.


efficiency-flexibility trade offs

The second thing we need (in order to answer the question about how humans model the physical) is to understand why different humans use different models of the physical.
What makes it useful to have different models of the physical for, say putting up a garden fence or landing a robot on a comet, is that some models allow you to get answers quickly without too much effort whereas other models, although harder to use, will provide accurate answers even in situations far from the mundane.
So because we're interested in actual thinkers, we are looking for pairs (or sets) of models that allow different trade-offs between efficiency and flexibiliy. It's for just this reason that the contrast between impetus and Newtonian models is interesting ...
Why do human cognitive and perceptual processes use different models of the phyiscal? One reason may be that different models permit differet trade offs between efficiency and flexibility.
Different models of the physical permit different trade offs between cognitive efficiency (necessary for achieving the speeds needed to stay ahead of one or more moving objects) with flexibility (necessary for accuracy across the widest range of situations), (\citealp[p.\ 640]{hubbard:2013_launching}; \citealp[p.\ 450]{kozhevnikov:2001_impetus}).

‘an impetus heuristic could yield an approximately correct (and adequate) solution ... but would require less effort or fewer resources than would prediction based on a correct understanding of physical principles.’

Hubbard (2014, p. 640)

\citet[p.\ 450]{kozhevnikov:2001_impetus}: ‘To extrapolate objects’ motion on the basis of physical principles, one should have assessed and evaluated the presence and magnitude of such imperceptible forces as friction and air resistance operating in the real world. This would require a time-consuming analysis that is not always possible. In order to have a survival advantage, the process of extrapolation should be fast and effortless, without much conscious deliberation. Impetus theory allows us to extrapolate objects’ motion quickly and without large demands on attentional resources.’
\citep[p.\ 640]{hubbard:2013_launching}: ‘prediction based on an impetus heuristic could yield an approximately correct (and adequate) solution [...] but would require less effort or fewer resources than would prediction based on a correct understanding of physical principles.’
I want to say that there's an analogy with the physical: there are multiple models of the mental.


signature limits

The fourth and last thing we need is to identify signature limits if the models.
In general, a signature limit of a model is a set of predictions derivable from the model which are incorrect, and which are not predictions of other models under consideration.

Kozhevnikov & Hegarty (2001, figure 1)

In limited but common range of cases, impetus and Newtonian mechanics coincide. However, the two theories make different predictions about the acceleration of falling objects, and of ascending objects (those launched vertically, in the manner of a rocket).
Consider ascending objects. We're fixing density and shape and considering how the size of objects changes things.
According to Newtonian mechanics, if we ignore air resistance, then size makes no difference to accelleration. If we include air resistance, larger objects accellerate faster (because of the difference in ratio of mass to surface).
By contrast, according to an impetus principle: ‘More massive objects accelerate at a slower rate. An object’s initial impetus continually dissipates because it is overcome by the effect of gravity. The more massive the ascending object, the more gravity counteracts its impetus.’ \citep[p.\ 445]{kozhevnikov:2001_impetus}

How do mindreaders’ belief-tracking systems model minds?

- compare -

How do physical thinkers model the physical?

I've just been looking at a question about phyiscal cognition with the aim of trying to illustrate a question about mindreaders.
The idea is this. Where someone is a mindreader, that is, is capable of identifying mental states, we need to understand what model of the mental underpins her abilities.
In the case of the mental we need the same three ingredients: theories which specify competing models, efficiency-flexibility trade offs which justify interest in particular types of model, and the signature limits that allow conjectures about models to generate predictions.
Start with the theories first. Where do we get theories that might allow us to identify how mindreaders model the mental? As in the case of the physical, we're interested in simple theories that only need to be approximately correct.
Theories are the things that philosophers create when they do things like trying to explain what an intention is, (Bratman says he is giving a theory of intention, for example). Or when they try to explain how belief differs from supposing, guessing and the rest. Most of these theories are highly sophisticated and concern propositional attitudes only. But what about the bad theories, the mental analogues of impetus theories?
Instead of going to the history of science for our bad theories, we turn to early philosophical attempts to characterise mental states. My favourite is Jonathan Bennett's. These theories are hopeless considered as accounts of adult's explicit thinking about mental states. But, like impetus theories of the physcial, they provide inspiration for very simple theories about the mental which make correct predictions of action in a limited but important range of circumstances.
One attempt to codify a the core part of a theory of the mental analogous to impetus mechanics is provided in Butterfill and Apperly's paper about how to construct a minimal theory of mind.

theories of the mental:

full-blown vs minimal theory of mind

I can't explain it in detail here, but minimal theory of mind is like impetus mechanics. It's obviously flawed and gets things quite wildly wrong but still useful in a limited range of circumstances.
Butterfill and Apperly's minimal theory of mind identifies a model of the mental.
I'm not going to describe the construction of minimal theory of mind, but I've written about it with Ian Apperly and outlined the idea on your handout.
The construction of minimal theory of mind is an attempt to describe how mindreading processes could be cognitively efficient enough to be automatic. It is a demonstration that automatic belief-tracking processes could be mindreading processes.
For this talk, the details don't matter. What matters is just that it's possible to construct minimal models of the mental which are powerful enough that using them would enable you to solve some false belief tasks.
\section{Minimal theory of mind\citep{butterfill_minimal}} An agent’s \emph{field} is a set of objects related to the agent by proximity, orientation and other factors. First approximation: an agent \emph{encounters} an object just if it is in her field. A \emph{goal} is an outcome to which one or more actions are, or might be, directed. %(Not to be confused with a \emph{goal-state}, which is an intention or other state of an agent linking an action to a particular goal to which it is directed.) \textbf{Principle 1}: one can’t goal-directedly act on an object unless one has encountered it. Applications: subordinate chimps retrieve food when a dominant is not informed of its location;\citep{Hare:2001ph} when observed scrub-jays prefer to cache in shady, distant and occluded locations.\citep{Dally:2004xf,Clayton:2007fh} First approximation: an agent \emph{registers} an object at a location just if she most recently encountered the object at that location. A registration is \emph{correct} just if the object is at the location it is registered at. \textbf{Principle 2}: correct registration is a condition of successful action. Applications: 12-month-olds point to inform depending on their informants’ goals and ignorance;\citep{Liszkowski:2008al} chimps retrieve food when a dominant is misinformed about its location;\citep{Hare:2001ph} scrub-jays observed caching food by a competitor later re-cache in private.\citep{Clayton:2007fh} %,Emery:2007ze \textbf{Principle 3}: when an agent performs a goal-directed action and the goal specifies an object, the agent will act as if the object were actually in the location she registers it at. Applications: some false belief tasks \citep{Onishi:2005hm,Southgate:2007js,Buttelmann:2009gy}

efficiency / flexibility?

Unlike the full-blown model, a minimal model distinguishes attitudes by relatively simple functional roles, and instead of using propositions or other complex abstract objects for distinguishing among the contents of mental states, it uses things like locations, shapes and colours which can be held in mind using some kind of quality space or feature map.
What about signature limits?

signature limits

(This is for illustrating mistakes about identity.) You might not realise that your bearded drinking pal ‘Ian’ and the author ‘Apperly’ are one and the same person.

signature limits

full-blown theory of mindminimal theory of mind
false beliefs about location, colour, ...YY
false beliefs about identity, appearance ...YN
level-1 perspective takingYY
level-2 perspective takingYN

second conjecture:

So my second conjecture is that

Some relatively efficient belief-tracking systems rely on minimal models of the mental.

This conjecture generates two predictsion.


  1. In adults, some relatively efficient belief-tracking processes are subject to the signature limits.
  2. And, second,
  3. 0- and 1-year-olds’ belief-tracking is subject to the signature limits.
  4. (The signature limits are on tracking false beliefs which essentially involve mistakes about identity, incompatible appearances or quantification.)
So the first prediction says this is what we should observe in adults.
And this is what we should observe in 0/1-year-old infants.
In conclusion, I've offered two conjectures and outlined the predictions they generate. Perhaps there is an interaction of age and measure on performance in belief-tracking because humans have two or more belief-tracking systems which rely on different models of the mental.

1. Systems

Conjecture: There are ≥ two belief-tracking systems

Predictions: different processing characteristics; doubly inconsistent responses.

2. Models

Conjecture: Some efficient* belief-tracking systems rely on minimal models.

I use ‘efficient*’ because this should say relatively efficient.

Predictions: Infants’ & adults’ efficient belief-tracking are subject to signature limits.


(Background on models)

There are ≥ two models of the mental, full-blown and minimal.

There are full-blown models, in which mental states are propositional attitudes. And there are minimal models, in which mental states have contents individuated by actual physical objects, locations, colours, shapes and others that can be identified by points in a quality space.

Mindreading is the use of any model of the mental.

To be a mindreader is to track beliefs or other mental states using a model of the mental.

Different models allow different flexibility-efficiency trade offs.

We can distinguish models by signature limits.