Microsoft Word - levellism and the method of abstraction3.doc

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

Abstract

The use of “levels of abstraction” in philosophical analysis (

levellism

) has recently come

under attack. In this paper, we argue that a refined version of

epistemological levellism

should be retained as a fundamental method, which we call

the method of abstraction

After a brief introduction, in section two we make clear the nature and applicability of the

(epistemological) method of levels of abstraction. In section three, we show the

fruitfulness of the new method by applying it to five case studies: the concept of

agenthood, the Turing test, the definition of emergence, quantum observation and

decidable observation. In section four, we further characterise and support the method by

distinguishing it from three other forms of “levellism”: (i) levels of organisation; (ii)

levels of explanation and (iii) conceptual schemes. In this context, we also briefly address

the problems of relativism and antirealism. In the conclusion, we indicate some of the

work that lies ahead, two potential limitations of the method and some results that have

already been obtained by applying the method to some long-standing philosophical

problems.

Keywords

Abstraction, Level of Abstraction, Gradient of Abstraction, Levellism, Observable.

LORIDI

ANDERS

Levellism and the Method of Abstraction

, IEG Research Report

22.11.04, digital editing by G. M. Greco, Information Ethics Group, Oxford University – University of
Bari,

http://web.comlab.ox.ac.uk/oucl/research/areas/ieg

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

“If we are asking about wine, and looking for the kind of

knowledge which is superior to common knowledge, it

will hardly be enough for you to say “wine is a liquid

thing, which is compressed from grapes, white or red,

sweet, intoxicating” and so on. You will have to attempt

to investigate and somehow explain its internal substance,

showing how it can be seen to be manufactured from

spirits, tartar, the distillate, and other ingredients mixed

together in such and such quantities and proportions.”

Gassendi,

Fifth Set of Objections to Descartes’ Meditations

1. Introduction

Reality can be studied at different levels, so forms of “levellism” have often been

advocated in the past.

In the seventies, levellism nicely dovetailed with the

computational turn and became a standard approach both in science and in philosophy

(Dennett [1971], Mesarovic et al. [1970], Simon [1969], see now Simon [1996], and

Wimsatt [1976]). The trend reached its acme at the beginning of the eighties (Marr

[1982], Newell [1982]) and since then levellism has enjoyed great popularity

and

textbook status (Foster [1992]). However, after decades of useful service, levellism seems

to have come under increasing criticism.

Consider the following varieties of levellism available in the literature:

epistemological, e.g., levels of observation or interpretation of a system (see

section 4);

ontological, e.g., levels (or rather layers) of organization, complexity, or causal

interaction etc. of a system;

methodological, e.g., levels of interdependence or reducibility among theories

about a system; and

an amalgamation of (1)-(3), e.g., as in Oppenheim and Putnam [1958].

See for example Brown [1916]. Of course the theory of ontological levels and the “chain of being” goes

as far back as Plotin and it is the basis of at least one version of the ontological argument.

The list includes Arbib [1989], Bechtel and Richardson [1993], Egyed and Medvidovic [2000], Gell-

Mann [1994], Kelso [1995], Pylyshyn [1984], Salthe [1985].

Poli [2001] provides a reconstruction of ontological levellism; more recently, Craver [2004] has analysed

ontological levellism, especially in biology and cognitive science, see also Craver [forthcoming].

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

The current debate on multirealizability in the philosophy of AI and cognitive science has

made (3) controversial (Block [1997]). And two recent articles by Heil [2003] and

Schaffer [2003] have seriously and convincingly questioned the plausibility of (2). Since

criticisms of (2) and (3) end up undermining (4), rumours are that levellism should

probably be decommissioned.

In general, we agree with Heil and Schaffer that

ontological levellism

may be

untenable, but we also contend that

epistemological levellism

should be retained as a

proper method of conceptual analysis, if in a suitably refined version. Fleshing out and

defending epistemological levellism is the main task of this paper. We shall proceed in

two stages. First, we shall clarify the nature and applicability of what we shall call

the

method of

(

levels of

)

abstraction

. We shall then distinguish it from other level-based

approaches, which may not, and indeed need not, be rescued. Here is a more detailed

outline of the paper.

In section two, we provide a definition of the basic concepts fundamental to the

method. Although the definitions require some rigour, only the rudiments of

mathematical notation are presupposed and all the main concepts are introduced without

assuming any previous knowledge. The definitions are illustrated by several intuitive

examples, which are designed to familiarise the reader with the method.

In section three, we show how the method of abstraction may be fruitfully applied

to several philosophical topics.

In section four, we further characterise and support the method of abstraction by

distinguishing it from three forms of “levellism”: (i) ontological levels of organisation;

(ii) methodological levels of explanation and (iii) conceptual schemes. In this context we

also briefly address the problems of relativism and antirealism.

In the conclusion, we indicate some of the work that lies ahead, two potential

limitations of the method and some results that have already been obtained by applying

the method to some long-standing philosophical problems in different areas.

Before starting, an acknowledgement of our intellectual debts is in order.

Levellism has been of essential importance in science since antiquity. Only more recently

has the concept of simulation been used in computer science to relate levels of abstraction

(for example de Roever and Engelhardt [1998], Hoare and He [1998]), to satisfy the

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

requirement that systems constructed in levels (in order to tame their complexity)

function correctly. Our definition of

Gradient of Abstraction

(GoA, see section 2.6) has

been inspired by this approach. Indeed, we take as a definition the property established by

simulations, namely the conformity of behaviour between levels of abstraction (more on

this later). Necessarily, our definition of GoA remains mathematical and for this reason

we do not follow through all details in the examples. However, we hope that the

discussion of Turing's imitation game in section 3.2 will make clear not just the use of

levels of abstraction but also their conformance in a GoA.

2. Definitions and preliminary examples

In this section, we define six concepts (“typed variable”, “observable”, “level of

abstraction”, “behaviour”, “moderated level of abstraction” and “gradient of abstraction”)

and then the “method of abstraction” based on them.

2.1. Typed variable

A variable is a symbol that acts as a place-holder for an unknown or changeable referent.

A “typed variable” is a variable qualified to hold only a declared kind of data.

Definition. A

typed variable

is a uniquely-named conceptual entity (the

variable

)

and a set, called its

type

, consisting of all the values that the entity may take. Two

typed variables are regarded as

equal

if and only if their variables have the same

name and their types are equal as sets. A variable that cannot be assigned well-

defined values is said to constitute an

ill-typed variable

(see the example in

section 2.3).

When required, we shall write

to mean that

is a variable of type

. Positing a typed

variable means taking an important decision about how its component variable is to be

conceived. We shall be in a position to appreciate this point better after the next

definition.

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

2.2. Observable

The notion of an “observable” is common in science, occurring whenever a (theoretical)

model is constructed. Although the way in which the features of the model correspond to

the system being modelled is usually left implicit in the process of modelling, it is

important here to make that correspondence explicit. We shall use the word “system” to

stand for the object of study. This may indeed be what would normally be described as a

system in science or engineering, but it may also be a domain of discourse, of analysis or

of conceptual speculation: a purely semantic system, as it were.

Definition. An

observable

is an interpreted typed variable, that is, a typed variable

together with a statement of what feature of the system under consideration it

represents. Two observables are regarded as

equal

if and only if their typed variables

are equal, they model the same feature and, in that context, one takes a given value if

and only if the other does.

Being an abstraction, an observable is not necessarily meant to result from quantitative

measurement or even empirical perception. The “feature of the system under

consideration” might be a physical magnitude – we shall return to this point in section

3.4, when talking about quantum observation – but it might also be an artefact of a

conceptual model, constructed entirely for the purpose of analysis.

An observable, being a typed variable, has specifically determined possible values. In

particular:

Definition. An observable is called

discrete

if and only if its type has only finitely

many possible values; otherwise it is called

analogue

In this paper, we are interested in observables as a means of describing behaviour at a

precisely qualified (though seldom numerical) level of abstraction; in general, several

observables will be employed.

The distinction is really a matter of topology rather than cardinality. However, this definition serves our

present purposes.

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

2.3. Five examples

Let us now consider some simple examples.

1) Suppose we wish to study some physical human attributes. To do so we, in Oxford,

introduce a variable,

, whose type consists of rational numbers. The typed variable

becomes an (analogue) observable once we decide that the variable

represents the

height of a person, using the Imperial system (feet and parts thereof). To explain the

definition of equality of observables, suppose that our colleague, in Rome, also interested

in observing human physical attributes, defines the same typed variable but declares that

it represents height in metres and parts thereof. Our typed variables are the same, but they

differ as observables: for a given person, the two variables take different representing

values. This example shows the importance of making clear the interpretation by which a

typed variable becomes an observable.

2) Consider next an example of an ill-typed variable. Suppose we are interested in the

rôles played by people in some community; we could not introduce an observable

standing for those beauticians who depilate just those people who do not depilate

themselves, for it is well-known that such a variable would not be well typed (Russell

[1902]). Similarly, each of the standard antinomies (Hughes and Brecht [1976]) reflects

an ill-typed variable. Of course, the modeller is at liberty to choose whatever type befits

the application, and if that involves a potential antinomy then the appropriate type might

turn out to be a non-well-founded set (Barwise and Etchemendy [1987]). However, in this

paper we shall operate entirely within the boundaries of standard naive set theory.

3) Suppose we follow Gassendi and wish to analyse wine. Observables relating to tasting

wine include the attributes that commonly appear on “tasting sheets”:

nose

(representing

bouquet),

legs

tears

(viscosity),

robe

(peripheral colour),

colour

clarity

sweetness

acidity

fruit

tannicity

length

and so on, each with a determined type.

If two wine

tasters choose different types for, say,

colour

(as is usually the case) then the observables

Despite a recent trend towards numeric values, these have not been standardised and so we leave to the

reader the pleasant task of contemplating appropriate types; for a secondary source of inspiration we refer
to tasting-related entries in Robinson [1994].

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

are different, despite the fact that their variables have the same name and represent the

same feature in reality. Indeed, as they have different types they are not even equal as

typed variables.

Information about how wine quality is perceived to vary with time – how the wine

“ages” (Robinson [1989]) – is important for the running of a cellar. An appropriate

observable is the typed variable

, which is a function associating to each year

Years

perceived quality

(

Quality

, where the types

Years

and

Quality

may be assumed to

have been previously defined. Thus,

is a function from

Years

Quality

, written

Time

→

Quality

. This example shows that, in general, types are constructed from more

basic types, and that observables may correspond to operations, taking input and yielding

output. Indeed, an observable may be of arbitrarily complex type.

4) The definition of an observable reflects a particular view or attitude towards the entity

being studied. Most commonly, it corresponds to a simplification, in which case

nondeterminism, not exhibited by the entity itself, may arise. The method is successful

when the entity can be understood by combining the simplifications. Let us consider

another example.

In observing a game of chess, we would expect to record the moves of the game.

Other observables might include the time taken per move; the body language of the

players; and so on. Suppose we are able to view the chessboard by looking just along

files

(the columns stretching from player to player). When we play “files-chess”, we are

unable to see the ranks (the parallel rows between the players) or the individual squares.

Files cannot sensibly be attributed a colour black or white, but each may be observed to

be occupied by a set of pieces (namely those that appear along that file), identified in the

usual way (king, queen and so forth). In “files-chess”, a move may be observed by the

effect it has on the file of the piece being moved. For example, a knight moves one or two

files either left or right from its starting file; a bishop is indistinguishable from a rook,

which moves along a rank; and a rook that moves along a file appears to remain

This is done by recording the history of the game: move by move the state of each piece on the board is

recorded – in English algebraic notation – by rank and file, as are recorded the piece being moved and the
consequences of the move.

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

stationary. Whether or not a move results in a piece being captured, appears to be

nondeterministic. “Files-chess” seems to be an almost random game.

Whilst the “underlying” game is virtually impossible to reconstruct, each state of

the game and each move (i.e., each operation on the state of the game) can be “tracked”

with this dimensionally-impoverished family of observables. If one then takes a second

view, corresponding instead to rank, we obtain “ranks-chess”. Once the two views are

combined, the original game of chess can be recovered, since each state is determined by

its rank and file projections, and similarly for each move. The two disjoint observations

together (“files-chess” + “ranks-chess”) reveal the underlying game.

5) The degree to which a type is appropriate depends on its context and use. For example,

to describe the state of a traffic light in Rome we might decide to consider an observable

colour

of type {

red

amber

green

} that corresponds to the colour indicated by the light.

This option abstracts the length of time for which the particular colour has been

displayed, the brightness of the light, the height of the traffic light, and so on. This is why

the choice of type corresponds to a decision about how the phenomenon is to be regarded.

To specify such a traffic light for the purpose of construction, a more appropriate type

would comprise a numerical measure of wavelength (see section 2.6). Furthermore, if we

are in Oxford, the type of colour would be a little more complex, since – in addition to

red, amber and green – red and amber are displayed simultaneously for part of the cycle.

So, an appropriate type would be {

red

amber

green

red

amber

2.4. Level of abstraction

We are now ready to introduce the basic concept of

level of abstraction

(LoA).

Any collection of typed variables can, in principle, be combined into a single “vector”

observable, whose type is the Cartesian product of the types of the constituent variables.

In the wine example, the type

Quality

might be chosen to consist of the Cartesian product

of the types

Nose

Robe

Colour

Acidity

Fruit

and

Length

. The result would be a single,

more complex, observable. In practice, however, such vectorisation is unwieldy, since the

expression of a constraint on just some of the observables would require projection

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

notation to single out those observables from the vector. Instead, we shall base our

approach on a

collection

of observables, that is, a level of abstraction:

Definition. A

level of abstraction

(

LoA

) is a finite but non-empty set of observables.

No order is assigned to the observables, which are expected to be the building blocks

in a theory characterised by their very definition. A LoA is called

discrete

(respectively

analogue

) if and only if all its observables are discrete (respectively

analogue); otherwise it is called

hybrid

Consider the wine example. Different LoAs may be appropriate for different purposes.

To evaluate a wine, the “tasting LoA”, consisting of observables like those mentioned in

the previous section, would be relevant. For the purpose of ordering wine, a “purchasing

LoA” – containing observables like

maker

region

vintage

supplier

quantity

price

, and

so on – would be appropriate; but here the “tasting LoA” would be irrelevant. For the

purpose of storing and serving wine – the “cellaring LoA” - containing observables for

maker

type of wine

drinking window

serving temperature

decanting time

alcohol level

food matchings

quantity remaining in the cellar

, and so on – would be relevant.

The traditional sciences tend to be dominated by analogue LoAs, the humanities

and information science by discrete LoAs and mathematics by hybrid LoAs. We are

about to see why the resulting theories are fundamentally different.

2.5. Behaviour

The definition of observables is only the first step in studying a system at a given LoA.

The second step consists in deciding what relationships hold between the observables.

This, in turn, requires the introduction of the concept of system “behaviour”. We shall see

that it is the fundamentally different ways of describing behaviour in analogue and

discrete systems that account for the differences in the resulting theories.

Not all values exhibited by combinations of observables in a LoA may be realised

by the system being modelled. For example, if the four traffic lights at an intersection are

modelled by four observables, each representing the colour of a light, the lights cannot in

fact all be green together (assuming they work properly). In other words, the combination

in which each observable is green cannot be realised in the system being modelled,

although the types chosen allow it. Similarly, the choice of types corresponding to a rank-

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

and-file description of a game of chess allows any piece to be placed on any square, but

in the actual game two pieces cannot occupy the same square simultaneously.

Some technique is therefore required to describe those combinations of observable

values that are actually acceptable. The most general method is simply to describe all the

allowed combinations of values. Such a description is determined by a predicate, whose

allowed combinations of values we call the “system behaviours”.

Definition. A

behaviour

of a system, at a given LoA, is defined to consist of a

predicate whose free variables are observables at that LoA. The substitutions of

values for observables that make the predicate true are called the

system behaviours

moderated LoA

is defined to consist of a LoA together with a behaviour at that

LoA.

Consider two previous examples. In reality, human height does not take arbitrary rational

values, for it is always positive and bounded above by (say) nine feet. The variable

representing height, is therefore constrained to reflect reality by defining its behaviour to

consist of the predicate 0 <

< 9, in which case any value of

in that interval is a

“system” behaviour. Likewise, wine too is not realistically described by arbitrary

combinations of the aforementioned observables. For instance, it cannot be both white

and highly tannic.

Since Newton and Leibniz, the behaviours of the analogue observables, studied in

science, have typically been described by differential equations. A small change in one

observable results in a small, quantified change in the overall system behaviour.

Accordingly, it is the rates at which those smooth observables vary which is most

conveniently described.

The desired behaviour of the system then consists of the solution

of the differential equations. However, this is a special case of a predicate: the predicate

holds at just those values satisfying the differential equation. If a complex system is

approximated by simpler systems, then the differential calculus provides a supporting

method for quantifying the approximation.

It is interesting to note that the catastrophes of

chaos theory

are not smooth; although they do appear so

when extra observables are added, taking the behaviour into a smooth curve on a higher-dimensional
manifold. Typically, chaotic models are weaker than traditional models, their observables merely reflecting

average

long-term

behaviour. The nature of the models is clarified by making explicit the LoA.

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

The use of predicates to demarcate system behaviour is essential in any

(nontrivial) analysis of discrete systems because in the latter no such continuity holds: the

change of an observable by a single value may result in a radical and arbitrary change in

system behaviour. Yet, complexity demands some kind of comprehension of the system

in terms of simple approximations. When this is possible, the approximating behaviours

are described exactly, by a predicate, at a given LoA, and it is the LoAs that vary,

becoming more comprehensive and embracing more detailed behaviours, until the final

LoA accounts for the desired behaviours. Thus, the formalism provided by the method of

abstraction can be seen as doing for discrete systems what differential calculus has

traditionally done for analogue systems.

Likewise, the use of predicates is essential in subjects like information and

computer science, where discrete observables are paramount and hence predicates are

required to describe a system behaviour. In particular, state-based methods like

(Hayes

and Flinn [1993], Spivey [1992]) provide notation for structuring complex observables

and behaviours in terms of simpler ones. Their primary concern is with the syntax for

expressing those predicates, an issue we shall try to avoid in this paper by stating

predicates informally.

The time has come now to combine approximating, moderated LoAs to form the

primary concept of the method of abstraction.

2.6. Gradient of abstraction

For a given (empirical or conceptual) system or feature, different LoAs correspond to

different representations or views. A

Gradient of Abstractions

(GoA) is a formalism

defined to facilitate discussion of discrete systems over a range of LoAs. Whilst a LoA

formalises the scope or granularity of a single model, a GoA provides a way of varying

the LoA in order to make observations at differing levels of abstraction.

For example, in evaluating wine we might be interested in the GoA consisting of the

“tasting” and “purchasing” LoAs, whilst in managing a cellar we might be interested in

the GoA consisting of the “cellaring” LoA together with a sequence of annual results of

observation using the “tasting” LoA.

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

In general, the observations at each LoA must be explicitly related to those at the

others; to do so, we use a family of relations between the LoAs. For this, we need to

recall some (standard) preliminary notation.

Notation. A

relation

from a set

to a set

is a subset of the Cartesian product

is thought of as relating just those pairs (

) that belong to the relation. The

reverse

is its mirror image: {(

) | (

)

∈



}. A relation

from

translates any predicate

to the predicate

(

) on

that holds at just those

which are the image through

of some

satisfying

(

)(

) =

∃

(

)

∧

(

We have finally come to the main definition of the paper.

Definition. A

gradient of abstractions

GoA

, is defined to consist of a finite set

{

| 0

≤

} of moderated LoAs

and a family of relations

i,j

⊆

, for 0

≤

≠

relating the observables of each pair

and

of distinct LoAs in such a way that:

1. the relationships are inverse: for

≠

i,j

is the reverse of

j,i

2. the behaviour

is at least as strong as the translated behaviour

(

)

⇒

(

(1)

Two GoAs are regarded as

equal

if and only if they have the same moderated LoAs (i.e.,

the same LoAs and moderating behaviours) and their families of relations are equal. A

GoA is called

discrete

if and only if all its constituent LoAs are discrete.

Condition (1) means that the behaviour moderating each lower LoA is

consistent

with that specified by a higher LoA. Without it, the behaviours of the various LoAs

constituting a GoA would have no connection to each other. A special case, to be

elaborated below in the definition of “nestedness”, helps to clarify the point.

If one LoA

extends another

by adding new observables, then the relation

i,j

is the inclusion of the observables of

in those of

and (1) reduces to this: the

The case of infinite sets has application to analogue systems but is not considered here.

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

constraints imposed on the observables at LoA

remain true at LoA

, where “new”

observables lie outside the range of

i,j

A GoA whose sequence contains just one element evidently reduces to a single

LoA. So our definition of “LoA” is subsumed by that of “GoA”.

The consistency conditions imposed by the relations

i,j

are in general quite weak. It

is possible, though of little help in practice, to define GoAs in which the relations connect

the LoAs cyclically. Of much more use are the following two important kinds of GoA:

“disjoint” GoAs (whose views are complementary) and “nested” GoAs (whose views

provide successively more information). Before defining them we need a little further

notation.

Notation. We recall that a

function

from a set

to a set

is a relation, i.e., a subset

of the Cartesian product

, which is single-valued

∀

((

)

∈

∧

(

∈

)

⇒

(this means that the notation

(

) =

is a well-defined alternative to (

)

∈

), and total

∀

∃

A f

(

) =

(this means that

(

) is defined for each

). A function is called

surjective

if and only if

every element in the target set lies in the range of the function:

∀

∃

(

) =

Definition. A GoA is called

disjoint

if and only if the

are pairwise disjoint (i.e.,

taken two at a time, they have no observable in common) and the relations are all

empty. It is called

nested

if and only if the only nonempty relations are those between

and

, for each 0

≤

−

1, and moreover the reverse of each

i, i

is a surjective

function from the observables of

to those of

A disjoint GoA is chosen to describe a system as the combination of several non-

overlapping components. This is useful when different aspects of the system behaviour

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

are better modelled as being determined by the values of distinct observables. This case is

rather simplistic, since the LoAs are more typically tied together by common

observations. For example, the services in a domestic dwelling may be represented by

LoAs for electricity, plumbing, telephone, security and gas. Without going into detail

about the constituent observables, it is easy to see that, in an accurate representation, the

electrical and plumbing LoAs would overlap whilst the telephone and plumbing would

not.

A nested GoA (see Fig.1) is chosen to describe a complex system exactly at each

level of abstraction and incrementally more accurately. The condition that the functions

be surjective means that any abstract observation has at least one concrete counterpart. As

a result, the translation functions cannot overlook any behaviour at an abstract LoA:

behaviours lying outside the range of a function translate to the predicate

false

. The

condition that the reversed relations be functions means that each observation at a

concrete LoA comes from at most one observation at a more abstract LoA (although the

converse fails in general, allowing one abstract observable to be refined by many

concrete observables). As a result the translation functions become simpler.

Fig. 1 Nested GoA with four Levels of Abstraction

For example, the case of a traffic light which is observed to have colour

colour

of type

{

red

amber

green

} is captured by a LoA,

, having that single observable. If we wish

to be more precise about colour, perhaps for the purpose of constructing a new traffic

light, we might consider a second LoA,

, having the variable

whose type is a

positive real number corresponding to the wavelength of the colour. To determine the

behaviour of

, Suppose that constants

red

red'

delimit the wavelength of red, and

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

similarly for amber and green. Then the behaviour of

is simply this predicate with free

variable

(

red



≤

≤ λ

red'

)

∨

(

amber

≤

≤ λ

amber'

)

∨

(

green

≤

≤ λ

green'

The sequence consisting of the LoA

and the moderated LoA

forms a nested GoA.

Informally, the smaller, abstract, type {

red

amber

green

} is a projection of the larger.

The relevant relation associates to each value

red

amber

green

} a band of

wavelengths perceived as that colour. Formally,

(

colour

) is defined to hold if and

only if, for each

red

amber

green

colour

↔

≤

≤ λ

In the wine example, a first LoA might be defined to consist of the variable “kind” having

type consisting of

red

white

rose

under the obvious representation. A second LoA might

be defined to consist of the observable “kind” having type

{

stillred

sparklingred

stillwhite

sparklingwhite

stillrose

sparklingrose

Although the second type does not contain the first, it produces greater resolution under

the obvious projection relation. Thus, the GoA consisting of those two LoAs is nested.

Those two important forms of GoA – disjoint and nested – are in fact theoretically

interchangeable. For if

and

are disjoint sets then

and their union

∪

are

increasing sets and the former is embedded in the latter. Thus, a disjoint GoA can be

converted to a nested one. Conversely, if

and

are increasing sets with the former

embedded in the latter, then

and the set difference

A \ B

are disjoint sets. Thus, a nested

GoA can be converted to a disjoint one.

Following the technique used to define a nested GoA, it is possible to define less

restricted but still hierarchical GoAs. Important examples include tree-like structures, of

which our nested GoAs are a special, linear case.

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

For theoretical purposes, the information captured in a GoA can be expressed

equivalently as a single LoA of more complicated type, namely one whose single LoA

has type equal to the sequence of the LoAs of the complex interface. However, the

current definition is better suited to application.

2.7. The method of abstraction

Models are the outcome of the analysis of a system, developed at some LoA(s) for some

purpose. An important contribution of these ideas is to make precise the commitment to a

LoA/GoA before further elaborating a theory. We call this the

method of abstraction

Three main advantages of the method can be highlighted here.

First, specifying the LoA means clarifying, from the outset, the range of questions

that (a) can be meaningfully asked and (b) are answerable in principle. One might think

of the input of a LoA as consisting of the system under analysis, comprising a set of

data

;

its output is a

model

of the system, comprising

information

. The quantity of information

in a model varies with the LoA: a lower LoA, of greater resolution or finer granularity,

produces a model that contains more information than a model produced at a higher, or

more abstract, LoA. Thus, a given LoA provides a quantified commitment to the kind and

amount of information that can be “extracted” from a system. The choice of a LoA pre-

determines the type and quantity of data that can be considered and hence the information

that can be contained in the model. So, knowing at which LoA the system is being

analysed is indispensable, for it means knowing the scope and limits of the model being

developed.

Second, being explicit about the LoA adopted provides a healthy antidote to

ambiguities, equivocations and other fallacies or errors due to level-shifting, such as

Aristotle’s “metabasis eis allo genos” (shifting from one genus to another) and Ryle’s

“category-mistakes”.

Third, by stating its LoA, a theory is forced to make explicit and clarify its

ontological commitment. The ontological commitment of a theory is best understood by

distinguishing between a

committing

and a

committed

component. A theory commits

itself ontologically by opting for a specific LoA. Compare this to the case in which one

has chosen a specific kind (better not speak of “model” here, to avoid confusion) of car

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

but has not bought one yet. A theory becomes ontologically committed in full through its

model, which is therefore the bearer of the specific commitment. The analogy here is

with the specific car one has actually bought. So LoAs commit a theory to types, while

their ensuing models commit it to the corresponding tokens.

3. Case studies

The advantages of the method of abstraction can be shown by applying it to some more

substantial examples. In this section, we shall consider five of them: agenthood, the

Turing test, emergence, quantum observation and decidable observation.

3.1. Agents

agent

can be thought of (see Floridi and Sanders [2004]) as a

transition system

(i.e.,

a system of states and transitions between them) that is

interactive

(i.e.,

can respond to

a stimulus by a change of state),

autonomous

(i.e.,

is able to change state without any

stimulus) and

adaptable

(i.e.,

is able to change the transition rules by which it changes

state). However, each of these properties, and hence the definition of agenthood, makes

sense only at a prescribed LoA. Consider the following examples.

1) Whether or not a rock is deemed to be interactive depends on the length of time and

level of detail of observation. Over a long period it erodes and hence changes state. By

day, it absorbs solar radiation, which it emits at night. However, if one relies on

observables resulting from scrutiny over a period of ten seconds by the naked eye from

ten metres, a stone can be deemed not to be interactive.

2) If the LoA adopted abstracts gravity and resistance, a swinging pendulum appears to

be autonomous but neither interactive nor adaptive. By extending the LoA to incorporate

air resistance, it becomes adaptive. By observing also the whistling sound it makes with

the air, it becomes interactive.

3) If a piece of software that exhibits machine learning (Mitchell [1997]) is studied at a

LoA that registers its interactions with its environment, then the software will appear

interactive, autonomous and adaptive, i.e., to be an agent. However, if the program code

is revealed, then the software is shown to be simply following rules and hence not to be

adaptive. Those two LoAs are at variance. One reflects the “open source” view of

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

software: the user has access to the code. The other reflects the commercial view:

although the user has bought the software and can use it at will, she has no access to the

code. At stake is whether or not the software forms an (artificial) agent. For further

examples we refer to Floridi and Sanders [2004].

3.2. The Turing test

Turing [1950] took the incisive step of arguing that the ability to think (“intelligence”)

can be satisfactorily characterised by means of a test, rather than by explicit definition. In

retrospect, that step may seem a small one. After all, we are quite familiar with contexts

in which no explicit definition is possible or sensible. Society makes no attempt to

characterise what it means to be an acceptable driver in terms of vision, response times,

coordination, experience and other physical attributes. Instead, it relies on a driving test.

Likewise, society does not attempt to define what it means for a school student to have

reached an acceptable academic standard by the end of school; it relies on final school

examinations. Nevertheless, incisive that step certainly must have been in view of the

vast number of attempts (even to this day) to characterise intelligence explicitly.

Opponents of Turing’s approach usually object that his test functions at the wrong

level of abstraction: perhaps it ought to include a component of creativity, of spontaneity,

of embodiment, of emotional involvement and so forth. However, without concepts like

those introduced above, it is hard to make one’s objections precise or defend Turing’s

approach. So let us see, first, how the Turing test can be expressed using

phenomenological LoAs and, second, how it can be analysed using conceptual LoAs.

3.2.1. Turing’s imitation game

Let us start, as did Turing, by considering an imitation game, in which a man

and a

woman

are placed in a room separate from an interrogator

, who communicates with

each by teleprinter (or these days by computer).

poses questions to

and

, who are

known only as

and

’s task is to identify

and

or, conversely,

and

, by considering their responses.

On the history of the Turing Test, see Shieber [2004].

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

We might describe this scenario by taking a first, extremely abstract, LoA to

reflect just the correctness of

’s identification. The LoA

consists of a single variable

ans

of type {

right

wrong

}, which becomes an observable under the correspondence:

ans

takes the value

right

is correct and the value

wrong

is incorrect. By choosing this

LoA, we intentionally abstract the actual answer (whether

was

), the questions

and answers, response times, and so on, in order to capture just the outcome of the

imitation game.

We might reveal

’s actual identification by defining a second, disjoint, LoA

whose single variable,

, is of type {(

), (

)}, which is made into an observable

under the correspondence that the first component of

is the putative identity of

and

the second component that of

. Combining the two LoAs

and

gives a disjoint GoA.

Of course, there are alternative approaches, which is why it is important to be

precise about the one adopted. One might define a GoA by replacing the LoA

with a

LoA containing two observables, the first corresponding to the identity of

and the

second corresponding to the identity of

. This would be more involved, since each

would have type {

} and one would have to moderate it with the behaviour that the

values of

and

differ. Our choice of

avoids this complication by building that

behaviour into the type of

but, with several observables, in general such moderating

behaviours cannot be circumvented.

In order to model

’s questions, the addressees and their responses, we define a

third LoA,

. Let

and

denote the sets of possible (well-posed) questions and

responses respectively (an example where the type of text strings may be considered

appropriate). Then each “question, addressee and response” triplet is a variable whose

type is the Cartesian product of

, {

} and

. It becomes an observable under the

correspondence just established. The observable we seek now consists of a sequence (of

arbitrary but finite length) of such triplets, corresponding to the sequence of interactions

in temporal order; and

contains that single observable (an alternative would be to have

an observable for the number of questions, an observable for each question and an

observable for each response).

can be added to either GoA

' to obtain a GoA

which is still disjoint but has higher resolution.

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

More detailed LoAs are possible and easy to define but, following Turing, we stop

here. It is clear that any discussion of the imitation game can be accurately “calibrated”,

according to its level of abstraction, with a GoA.

3.2.2. Turing’s test analysed

In the Turing test,

is replaced by a “machine” (nowadays a “computer”). Turing

proposed that the question “can machines think?” be replaced by the question: “will the

interrogator decide wrongly as often when the game is played like this as he does when

the game is played between a man and a woman?”.

These days the test is normally stripped of its sex-specific nature and the

interrogator is simply asked to determine the human from the machine.

Appropriate

GoAs are defined as above, but with

representing a computer and

a human.

Although Turing did not make it explicit, the phrase “as often” in his description

implies repetition of the test, and hence a conclusion reached by statistical analysis.

Suppose that

initiates a pair of question/answer sessions of the type used in the

imitation game. A list of questions is put in two situations, one containing a man and a

woman, the other containing a machine and a woman. We suppose that the answers in the

first situation are of type

and those in the second of type

, thus avoiding here the

question of whether or not

. As before,

makes an identification. The appropriate

LoA has type, call it

, equal to the Cartesian product of the type {(

), (

)} and the

type of all sequences of elements of the type

{

}

The observable corresponding to repetition of that situation

times, though not

necessarily with the same questions, has type consisting of the Cartesian product of

many copies of type

, namely

. The LoA incorporating that observation plus the

answer to the ultimate question is then the Cartesian product of the type

and the type

{

right

wrong

}. Likewise, a more complex type can be constructed to reveal the nature of

the statistical test; in this case too, we follow Turing and omit the details.

For a summary of the Turing test today, and its incarnation in competitive form (the Loebner prize), see

Moor [2001].

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

3.2.3. Turing’s test discussed

The previous two sections have shown how to formalise the Turing test using

phenomenologically motivated GoAs, but the method of abstraction can be used to

discuss and compare variations on the test. Indeed, it is difficult to imagine how such an

analysis could be formulated accurately without a concept equivalent to LoA. Of course,

details of the LoAs need not be given as long as it is clear that they could be.

Turing couched his test in terms of a single human interrogator. In the Loebner

test (Moor [2001]), the interrogation is carried out by a panel of humans interacting via

computer interface in real time. Alternatively, the interrogator could be a machine; or

instead of real-time interaction, a list of pre-arranged questions might be left, and the list

of answers returned to the interrogator; or instead of serial questions and answers the

interrogator might hold a “general conversation” (Moor [2001]). Each alternative

modifies the power of the test. All can be formalised by defining different GoAs.

The Turing test might be adapted to target abilities other than “thinking”, like:

interpretation of text, game playing, puzzle solving, spatial reasoning, creativity, and so

on. Contemplating the possible GoAs provides a way to formalise the variant test clearly

and elegantly, and promotes simple comparison with contending treatments.

3.3. Emergence

The method of abstraction is ideally suited to the study of systems so complex that they

are best understood stepwise, that is, by their gradual disclosure at increasingly fine levels

of abstraction.

A key concept in such an approach to complex systems is that of “emergent

behaviour”, that is, behaviour that arises in the move from one LoA to a finer level.

Gell-Mann [1994] has suggested that the study of such phenomena be called

plectics

. He introduces it

using an idea he calls

granularity

, which is conveniently formalised by LoA. The concept of emergence is

coupled not only to that of complexity but also to that of reduction. However, for reasons of space, we shall
limit ourselves here only to the first, leaving the analysis of the other two to the reader. Damper [2000]
discusses reductionism and emergent properties from a LoA perspective.

See Hendriks-Jansen [1989], section 2 for a discussion.

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

3.3.1. Tossing a coin

Emergence, not surprisingly for such an important concept, can take various forms

(Cariani [1991], Nagel [1974]). Epistemologically, the concept refers to the fact that

“properties at higher levels are not necessarily predictable from properties at lower

levels”.

Here, we shall be concerned only with the concept of emergence, not with its

empirical counterparts. We shall see that it is neatly captured using a GoA containing two

nested LoAs.

Let us begin with an example.

The process of tossing a coin may be modelled abstractly, with an observable

outcome

of type {

head

tail

} corresponding to the side of the coin that faces upwards

after the toss. This LoA abstracts any other result, like the coin’s landing on its edge or

becoming lost, and all other features, like type of agent tossing the coin, manner of

tossing, number of spins, time taken and so on. In particular, it models just one toss of the

coin and so it cannot account for the “fairness” of the coin, which can be revealed

statistically only after a large number of tosses.

Now, suppose we wish to model the repetition of that process with the explicit

aim of discussing the fairness of a coin. We introduce a more concrete LoA, whose

observables are: a natural number

, corresponding to the number of tosses of the coin;

and a list of

values from {

head

tail

}, corresponding to successive tosses as modelled

above. At this second LoA, it is possible to assess the fairness of the coin – using

standard statistics, for example – on the basis of the frequency of the outcomes. So this is

an example of an emergence property in the following sense: in many repeated tosses of

the coin, the more abstract model applies toss by toss, but does not allow frequency of

outcome to be observed, as it is in the finer model. We say that the notion of the coin’s

fairness is emergent at the finer LoA. We are now ready for a definition.

3.3.2. Emergence defined

Suppose that a system

is under consideration using a nested GoA consisting of two

LoAs. Let us also assume that the more abstract LoA,

, is moderated by behaviour

which describes the abstract view of

, and that the more concrete LoA,

, is moderated

Hendriks-Jansen [1989], p. 283; note that, in the quotation, “lower levels” are more abstract and “higher

levels” more detailed or concrete.

For a similar line of reasoning see Damper [2000].

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

by behaviour

, which describes the concrete view of

. The abstract and concrete

observables are related by the total and one-to-many relation

. Recall that a behaviour

of the system

at LoA

is a tuple of values for the observables of

that satisfies

Definition. A behaviour of a system

at LoA

is said to be

emergent

(with respect to

that nested GoA) if and only if its translation under the relation

fails to satisfy

Emergence

is said to hold in the GoA if and only if there is some emergent behaviour.

There is frequently confusion about emergence. This is scarcely surprising since, without

the notion of LoA, the various levels at which a system is discussed cannot be formalised.

Emergence arises typically because the concrete LoA embodies a “mechanism”, or rule

for determining an observable, which has been overlooked at the abstract LoA, usually

quite deliberately, in order to gain simplicity at the cost of detail. Frequently, the

breakthrough in understanding some complex phenomenon comes by accounting for

emergent behaviour, and this results from considering the process by which it occurs,

rather than by taking a more static view of the ingredients or components involved.

the coin example, we have avoided incorporating any mechanism, but any of the

multitude of pseudo-random number generators could be used to generate lists of

head

and

tail

and hence to account for the emergent phenomenon (Knuth [1997]).

3.3.3. Process observed

The majority of observables considered so far have been “static”, but operations too

constitute vital observables. Indeed, the importance of “process” may be indicated by the

example of a sponge cake. With only the ingredients as observables (i.e., the amount of

each ingredient) the sponge-like nature of the cake is, as many a novice cook has found,

emergent. However, if the manner of aeration (a variable indicating the aeration effect of

bicarbonate of soda under the right conditions) is also an observable, then sponginess is

explicable. In other words, the behaviour of sponginess emerges at the finer level of

abstraction only.

This is well documented by Emmeche et al. [1997], who, however, go on to analyse emergence in terms

of ontological or

de re

levels.

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

3.4. Quantum observation

It is remarkable that the disparate disciplines of quantum mechanics and social

anthropology share a fundamental feature: in each, observation inevitably involves

interference. Observing a quantum or anthropological system is possible only at the

expense of a change to the system. By contrast, our definition of observable makes no

assumptions about how the system is (capable of being) measured (effectively) in

practice. In this section we address this issue.

In quantum mechanics, “observable” has a much more restricted meaning than in

this paper. There it is to be distinguished from the state that is posited to exist in order to

explain the frequency with which observables take their values. Such “beables” (Bell

[1987]) are for us also observables as is, for that matter, the frequency with which an

observable takes on its values. The latter might be regarded as unachievable in practice,

since any finite number of readings can achieve only an approximation to it, but that need

be of no concern to us. Our only requirement is that an observable be well-typed. When

desired, the stricter “observation as measurement” from quantum mechanics can be

modelled as a certain kind of observation in our sense: the change in behaviour associated

with an “observation as measurement” event is simply specified to conform to the

uncertainty principle. The same holds for the constraint of quantum mechanics that only

certain (i.e., commuting) observables may be measured simultaneously: whilst two

events, say A and B, may be observed independently, their simultaneous observation

constitutes a third event, AB say, with the different behavioural consequences dictated by

quantum mechanics.

3.5. Decidable observation

In the theory of computation, an observable is called decidable, or effective, if and only if

its behaviour is given by a computable function. For example, it is known to be

undecidable whether or not a program terminates i.e., there is no algorithm for its

determination (Boolos et al. [2002]). We make no assumption about the decidability of an

observable. The reason is simple. The field of

Formal Methods

within computer science

concerns the mathematical specification and development of information systems.

Typically, a specification embodies a twofold constraint: the required program must

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

conform to such-and-such a functional specification and terminate. Without the last

conjunct, undesirable programs which execute forever, never yielding a result, might be

allowed (in some models of computation). However, such a specification is no more than

a behaviour phrased in terms of observables for input, output (appearing in the functional

specification) and termination (appearing in the second conjunct). And we have just seen

that termination cannot be assumed to be decidable.

The consequence of allowing an observable to be undecidable is that some

ingenuity is required to prove that an implementation meets a specification phrased in

terms of its observables: no program can possibly achieve that task in general.

4. The philosophy of the method of abstraction

The time has come to provide further conceptual clarification concerning the nature and

consequences of the method of abstraction. In this section we relate the relevant work of

Marr, Pylyshyn, Dennett and Davidson to the method of abstraction, and discuss the

thorny issues of relativism and antirealism. A word of warning may be in order. When

confronted with a new theory or method, it is natural to compare it and perhaps

(mistakenly) identify it with something old and well-established. In particular, previous

theories or methods can work as powerful magnets that end by attracting anything that

comes close to their space of influence, blurring differences. So this section aims at

putting some distance between some old acquaintances and our new proposal.

4.1. Levels of organization and of explanation

Several important ways have been proposed for speaking of levels of analysis of a

system. The following two families can be singled out as most representative:

1) Levels of organization (LoOs) support an ontological approach, according to which the

system under analysis is supposed to have a (usually hierarchical) structure in itself, or

, which is allegedly uncovered by its description and objectively formulated in some

neutral observation language (Newell [1990], Simon [1996]). For example, levels of

communication, of decision processing (Mesarovic et al. [1970]) and of information flow

can all be presented as specific instances of analysis in terms of LoOs.

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

There is a twofold connection between LoOs and LoAs. If the hierarchical

structure of the system itself is thought of as a GoA, then for each constituent LoA there

is a corresponding LoO. Alternatively, one can conceive the analysis of the system, not

the system itself, as being the object of study. Then the method of abstraction leads one to

consider a GoA whose constituent LoAs are the LoOs. Note that, since the system under

analysis may be an artefact, knowledge of its LoO may be available in terms of

knowledge of its specifications.

2) Levels of explanation (LoEs) support an epistemological approach, quite common in

cognitive and computer science (Benjamin et al. [1998]). Strictly speaking, the LoEs do

not really pertain to the system or its model. They provide a way to distinguish between

different epistemic approaches and goals, such as when we analyse an exam question

from the students’ or the teacher’s perspectives, or the description of the functions of a

technological artefact from the expert’s or the layperson’s point of view.

A LoE is an important kind of LoA. It is pragmatic and makes no pretence of

reflecting an ultimate description of the system. It has been defined with a specific

practical view or use in mind. Manuals, pitched at the inexpert user, indicating “how to”

with no idea of “why”, provide a good example.

The two kinds of “structured analysis” just introduced are of course interrelated.

Different LoEs – e.g., the end-user’s LoE of how an applications package is to be used

versus the programmer’s LoE of how it is executed by machine – are connected with

different LoAs – e.g., the end-user’s LoA represented by a specific graphic interface

versus the programmer’s code – which in turn are connected with different LoO – e.g.,

the commonsensical WYSIWYG versus the software architecture. However, LoAs

provide a foundation for both, and LoOs, LoEs and LoAs should not be confused. Let us

consider some clarifying examples.

One of the most interesting and influential cases of multi-layered analysis is

provided by Marr’s three-levels hypothesis. After Marr [1982], it has become common in

cognitive and philosophical studies (McClamrock [1991]) to assume that a reasonably

complex system can be understood only by distinguishing between levels of analysis.

Here is how Marr himself put it: “Almost never can a complex system of any kind

be understood as a simple extrapolation from the properties of its elementary

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

components. Consider for example, some gas in a bottle. A description of thermodynamic

effects – temperature, pressure, density, and the relationships among these factors – is not

formulated by using a large set of equations, one for each of the particles involved. Such

effects are described at their own level, that of an enormous collection of particles; the

effort is to show that in principle the microscopic and the macroscopic descriptions are

consistent with one another. If one hopes to achieve a full understanding of a system as

complicated as a nervous system, a developing embryo, a set of metabolic pathways, a

bottle of gas, or even a large computer program, then one must be prepared to

contemplate different kinds of explanation at different levels of description that are

linked, at least in principle, into a cohesive whole, even if linking the levels in complete

detail is impractical. For the specific case of a system that solves an information-

processing problem, there are in addition the twin strands of process and representation,

and both these ideas need some discussion.” (Marr [1982], pp. 19–20).

In particular, in the case of an information-processing system, Marr and his

followers suggest the adoption of three levels of analysis (all the following quotations are

from Marr [1982]):

the computational level

. This is a description of “the abstract computational theory of

the device, in which the performance of the device is characterised as a mapping from

one kind of information structures, the abstract properties of this mapping are defined

precisely, and its appropriateness and adequacy for the task at hand are demonstrated” (p.

24);

the algorithmic level

. This is a description of “the choice of representation for the input

and output and the algorithm to be used to transform one into the other” (p. 24-25);

the implementational level

. This is a description of “the details of how the algorithm

and representation are realized physically – the detailed computer architecture, so to

speak.” (p. 25).

The three levels are supposed to be loosely connected and in a one-to-many mapping

relation: for any computational description of a particular information-processing

problem there may be several algorithms for solving that problem, and any algorithm

may be implemented in several ways.

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

Along similar lines, Pylyshyn [1984] has spoken of the semantic, the syntactic,

and the physical levels of description of an information-processing system, with the (level

of) functional architecture of the system playing the role of a bridge between Marr’s

algorithmic and implementational levels. And Dennett [1987] has proposed a hierarchical

model of explanation based on three different “stances”: the intentional stance, according

to which the system is treated as if it were a rational, thinking agent attempting to carry

out a particular task successfully; the design stance, which concerns the general principles

governing the design of any system that might carry out those tasks successfully; and the

physical stance, which considers how a system implementing the appropriate design-level

principles might be physically constructed.

The tripartite approaches of Marr, Pylyshyn and Dennett share three important

features. First, they are each readily formalised in terms of GoAs with three LoAs.

Second, they do not distinguish between LoO, LoE and LoA; and this because (third

feature) they assign a privileged role to explanations. As a result, their ontological

commitment is embedded and hence concealed. The common reasoning seems to be the

following: “this is the right level of analysis because that is the right LoO”, where no

justification is offered for why that LoO is chosen as the right one. Nor is the

epistemological commitment made explicit or defended; it is merely presupposed. This is

where the method of abstraction provides a significant advantage. By starting from a

clear endorsement of each specific LoA, a strong and conscious effort can be made to

uncover the ontological commitment of a theory (and hence of a set of explanations),

which now needs an explicit acceptance on the part of the user, and requires no hidden

epistemological commitment, which now can vary depending on goals and requirements.

4.2. Conceptual schemes

The resemblance between LoAs and conceptual schemes (CSs) is close enough to require

clarification. In this section, we shall briefly compare the two. The aim is not to provide

an exegetical interpretation or a philosophical analysis of Davidson’s famous criticism of

the possibility of irreducible CSs, but rather to clarify further the nature of LoAs and

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

explain why LoAs can be irreducible, although in a sense different from that liked by

supporters of the irreducibility of CSs.

According to Davidson, all CSs share four features (the following quotations are

from Davidson [1974]):

1) CSs are clusters or networks of (possibly acquired) categories. “Conceptual schemes,

we are told, are ways of organizing experience; they are systems of categories that give

form to the data of sensation; they are points of view from which individuals, cultures, or

periods survey the passing scene” (p. 183).

2) CSs describe or organise the world or its experience for communities of speakers.

“Conceptual schemes (languages) either organize something, or they fit it”, and as “for

the entities that get organized, or which the scheme must fit, I think again we may detect

two main ideas: either it is reality (the universe, the world, nature), or it is experience (the

passing show, surface irritations, sensory promptings, sense-data, the given)” (p. 192).

3) CSs are inescapable, in the sense that communities of speakers are entrapped within

their CSs.

4) CSs are not intertranslatable.

Davidson argues against the existence of CSs as inescapable (from within) and

impenetrable (from without) ways of looking at the world by interpreting CSs

linguistically and then by trying to show that feature (4) is untenable. Could the strategy

be exported to contrast the existence of equally inescapable and impenetrable LoAs? Not

quite.

Let us examine what happens to the four features above when LoAs are in

question:

a) LoAs are clusters or networks of observables. Since they deal with observables, LoAs

are not an anthropocentric prerogative but allow a more general (or indeed less biased)

approach. We do not have to limit ourselves to human beings or to communities of

speakers. Different sorts of empirical or abstract agents – not only human beings but also

computers, animals, plants, scientific theories, measurement instruments etc. – operate

and deal with the world (or, better, with the data they glean from it) at some LoAs. By

Newell reached similar conclusions, despite the fact that he treated LoA as LoO, an ontological form of

levellism that allowed him to escape relativism and antirealism more easily, see Newell [1982] and Newell
[1993].

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

neatly decoupling LoAs from the agents that implement or use them, we avoid the

confusion between CSs, the languages in which they are formulated or embodied, and the

agents that use them. We shall return to this point presently.

b) LoAs model the world or its experience. LoAs are anchored to their data, in the sense

that they are constrained by them; they do not describe or organise them, they actually

build models out of them. So the relation between models and their references (the

analysed systems) is neither one of discovery, as in Davidson’s CSs, nor one of invention,

but one of design, to use an equally general category. It follows that, contrary to

Davidson’s CSs, it makes no sense to speak of LoAs as Xerox machines or personal

organisers of some commonly shared ontology (the world or its experience). Ontological

commitments are initially negotiated through the choice and shaping of LoAs, which

therefore cannot presuppose a metaphysical omniscience.

Because of the differences between (1)–(2) and (a)–(b), the remaining two

features acquire a significantly different meaning, when speaking of LoAs. Here is how

the problem is reformulated. LoAs generate, and commit the agent to, information spaces.

In holding that some LoAs can be irreducible and hence untranslatable we are not arguing

that:

i) agents using LoAs can never move seamlessly from one information space to another.

This is false. They obviously can, at least in some cases: just imagine gradually replacing

some observables in the LoAs of an agent. This is equivalent to arguing that human

beings cannot learn different languages. Note, however, that some agents may have their

LoAs hardwired: imagine, for example, a thermometer;

ii) agents using LoAs can never expand their information spaces. This is also false. Given

the nested nature of some LoAs and the possibility of constructing supersets of sets of

observables, agents can aggregate increasingly large information spaces. This is

equivalent to arguing that human speakers cannot expand their languages semantically,

another obvious nonsense.

So, if we are talking about the agents using or implementing the LoAs, we know

that agents can sometimes modify, expand or replace their LoAs, and hence some degree

of intertranslatability, understood as the acquisition or evolution of new LoAs, is

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

guaranteed. The point in question is another one, however, and concerns the relation

between the LoAs themselves.

LoAs are the place at which (diverse) independent systems meet and act on or

communicate with each other. If one reads carefully, one will notice that this is the

definition of an interface. The systems interfaced may adapt or evolve their interfaces or

adopt other interfaces, as in (i) and (ii), yet different interfaces may still remain mutually

untranslatable. Consider, for example, the “tasting LoA” and the “purchasing LoA” in our

wine example. But if two LoAs are untranslatable, it becomes perfectly reasonable to

assume that:

iii) agents may inhabit only some types of information spaces in principle.

Some information spaces may remain inaccessible not just in practice but also in

principle, or they may be accessible only asymmetrically, to some agents. Not only that,

but given the variety of agents, what is accessible to one or some may not be accessible to

all. This is easily explained in terms of modal logic and possible worlds understood as

information spaces. The information space of a child is asymmetrically accessible from

the information space of an adult, and the information space of a bat overlaps

insufficiently with the information space of any human agent to guarantee a decent

degree of translatability (Nagel [1974]).

In principle, some information spaces may remain forever disjoint from any other

information spaces that some agents may be able to inhabit. When universalised, this is

Kant’s view of the noumenal world, which is accessible only to its creator. Does this

imply that, after all, we are able to say what a radically inaccessible information space

would be like, thus contradicting ourselves? Of course not. We are only pointing in the

direction of the ineffable, without grasping it. It is a bit like drawing a figure without ever

being able to paint it.

To return to Davidson, even conceding that he may be successful in criticising the

concept of CSs, his arguments do not affect LoAs. The problem is that Davidson limits

his considerations to information spaces that he assumes, without much reason, to be

already linguistically and ontologically delimited. When this is the case, one may

concede his point. However, LoAs do not vouch for the kind of epistemic realism,

verificationism, panlinguism and representationist view of knowledge that Davidson

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

implicitly assumes in analysing CSs. And once these fundamental assumptions are

eliminated, Davidson’s argument loses most of its strength. Incommensurable and

untranslatable LoAs are perfectly possible. We shall see that this provides no good

ground for a defence of some form of radical conceptual relativism (section 4.3) or anti-

realism (section 4.4).

Davidson’s criticism ends by shaping an optimistic approach to the problem of the

incommensurability of scientific theories that we cannot share, but then, what conclusions

can be drawn, from our analysis of LoAs, about the anti-realist reading of the history of

science? An unqualified answer would fall victim to the same fallacy of un-layered

abstraction we have been denouncing in the previous pages. The unexciting truth is that

different episodes in the history of science are more or less comparable depending on the

LoA adopted. Consider the great variety of building materials, requirements, conditions,

needs and so on, which determine the actual features of a building. Does it make sense to

compare a ranch house, a colonial home, a town house, a detached house, a semidetached

house, a terraced house, a cottage, a thatched cottage, a country cottage, a flat in a single-

storey building, and a Tuscan villa? The question cannot be sensibly answered unless one

specifies the LoA at which the comparison is to be conducted. Likewise, our answer

concerning the reading of the history of science is: given the nature of LoAs, it is always

possible to formulate a LoA at which comparing different episodes in the history of

science makes perfect sense. But do not ask absolute questions, for they just create an

absolute mess.

4.3. Pluralism without relativism

A LoA qualifies the level at which a system is considered. In this paper, we have argued

that it must be made clear before the properties of the system can sensibly be discussed.

In general, it seems that many disagreements might be clarified and resolved if the

various “parties” make explicit their LoA. By structuring the explanandum, LoAs can

reconcile the explanans. Yet, another crucial clarification is now in order. It must be

stressed that a clear indication of the LoA at which a system is being analysed allows

pluralism without falling into relativism or “perspectivism”, a term coined by Hales and

Welshon [2000] in connection with Nietzsche’s philosophy. As remarked above, it would

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

be a mistake to think that “anything goes” as long as one makes the LoA explicit, because

LoAs can be mutually comparable and assessable, in terms of inter-LoA coherence, of

their capacity to take full advantage of the same data and of their degree of fulfilment of

the explanatory and predictive requirements laid down by the level of explanation. Thus,

introducing an explicit reference to the LoA makes it clear that the model of a system is a

function of the available observables, and that it is reasonable to rank different LoAs and

to compare and assess the corresponding models.

4.4. Realism without descriptivism

For a typed variable to be an observable it must be interpreted, a correspondence that has

inevitably been left informal. This interpretation cannot be omitted: a LoA composed of

typed variables called simply x, y, z and so on and treated rather formally, would leave

the reader (or the writer some time later) with no hint of its domain of application. Whilst

that is the benefit of mathematics, enabling its results to be applied whenever its axioms

hold, in the method of abstraction it confers only obscurity. Does the informality of such

interpretation hint at some hidden circularity or infinite regress? Given the distinction

between LoO and LoA, and the fact that there is no immediate access to any LoO that is

LoA-free, how can an observable be defined as “realistic”? That is, must the system

under consideration already be observed before a “realistic” observation can be defined?

The mathematics underlying our definitions of typed variable and behaviour has been

indicated (even if it is not always fully used in practice) to make the point that, in

principle, the ingredients in a LoA can be formalised. There is no circularity: the

heuristically appreciated system being modelled never exists on the same plane as that

being studied methodically.

The point might be clarified by considering Tarski’s well-known model-theoretic

definition of truth (Tarski [1944]). Is there circularity or regress there? Might it be argued

that one needs to know truth before defining it, as Meno would have put it? Of course

not, and the same resolution is offered here. Tarski’s recursive definition of truth over

syntactic construction is based on an appreciation of the properties truth is deemed to

have, but that appreciation and the rigorous definition exist on “different planes”. So

circularity is avoided.

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

More interesting is the question of infinite regress. Tarski’s definition formalises

certain specific properties of truth; a regress would obtain only were a complete

characterisation sought. So it is with the interpretation required to define an observable.

Some property of an undisclosed system is being posited at a certain level of abstraction.

An unending sequence of LoAs could possibly obtain were a complete characterisation of

a system sought.

It is implicit in the method of abstraction that a GoA is to be chosen that is

accurate or “realistic”. How, then, is that to be determined without circularity? We give

the answer traditionally offered in mathematics and in science: it is determined by

external adequacy and internal coherence or, in computer jargon, validation (the GoA

satisfies its operational goals) and verification (each step in the development of the GoA

satisfies the requirements imposed by previous steps). First, the behaviours at a

moderated LoA must adequately reflect the phenomena sought by complying with their

constraints; if not, then either the definition of the behaviour is wrong or the choice of

observables is inappropriate. When the definition of observables must incorporate some

“data”, the latter behave like constraining affordances and so limit the possible models.

We refer to Floridi [2004] for further details and examples. Second, the condition

embodied in the definition of GoA is a remarkably strong one, and ensures a robust

degree of internal coherence between the constituent LoAs. The multiple LoAs of a GoA

can be thought of as interlocking like the answers to a multidimensional cross-word

puzzle. Though such consistency does not guarantee that one’s answer to the cross-word

is the same as the originator’s, it drastically limits the number of solutions, making each

more likely.

Adequacy/validation and coherence/verification neither entail nor support naive

realism. GoAs ultimately construct models of systems. They do not describe or portray or

uncover the intrinsic nature of the systems they analyse. We understand systems

derivatively, only insofar as we understand their models. Adequacy and coherence are the

most we can hope for.

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

5. Conclusion

Feynman once remarked that “if we look at a glass of wine closely enough we see the

entire universe. […] If our small minds, for some convenience, divide this glass of wine,

this universe, into parts – physics, biology, geology, astronomy, psychology, and so on –

remember that nature does not know it!”.

In this paper, we have shown how the analysis

of the glass of wine may be conducted at different levels of epistemological abstraction

without assuming any corresponding ontological levellism. Nature does not know about

LoAs either.

In the course of the paper we have introduced the epistemological method of

abstraction and applied it to the study, modelling and analysis of phenomenological and

conceptual systems. We have demonstrated its principal features and main advantages.

Yet one may object that, by providing a few simple examples and some tailored case-

based analyses, the method really predates its applications, which were merely chosen

and shaped for their suitability. In fact, it is exactly the opposite: we were forced to

develop the method of abstraction when we encountered the problem of defining the

nature of agents (natural, human and artificial) in Floridi and Sanders [2004]. Since then,

we have been applying it to some long-standing philosophical problems in different areas.

We have used it in computer ethics, to argue in favour of the minimal intrinsic value of

informational objects (Floridi [2003]); in epistemology, to prove that the Gettier problem

is not solvable (Floridi [forthcoming]); in the philosophy of mind, to show how an agent

provided with a mind may know that she has one and hence answer Dretske’s question

“how do you know you are not a zombie?” (Floridi [forthcoming]); in the philosophy of

science, to propose and defend an informational approach to structural realism that

reconciles forms of ontological and epistemological structural realism (Floridi

[forthcoming]); and in the philosophy of AI, to provide a new model of telepresence

(Floridi [forthcoming]). In each case, the method of abstraction has been shown to

provide a flexible and fruitful approach. Clearly, the adoption of the method of

abstraction raises interesting questions, such as why certain LoAs, e.g. the so-called

“naive physics” view of the world and the “folk psychology” approach to the mind,

Feynman [1995], the citation is from the Penguin edition, p. 66.

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

appear to be “privileged”, or whether artificial life (ALife) can be defined in terms of

GoA. So much work lies ahead.

The method clarifies implicit assumptions, facilitates comparisons, enhances

rigour and hence promotes the resolution of possible conceptual confusions. It also

provides a detailed and controlled way of comparing analyses and models. Yet, all this

should not be confused with some neo-Leibnizian dream of a “calculemus” approach to

philosophical problems. Elsewhere (Floridi [2004]), we have argued that genuine

philosophical problems are intrinsically

open

, that is, they are problems capable of

different and possibly irreconcilable solutions, which allow honest, informed and

reasonable differences of opinion. The method we have outlined seeks to promote explicit

solutions, which facilitate a critical approach and hence empower the interlocutor. It does

not herald any sort of conceptual “mechanics”.

The method is not a panacea either. We have argued that, for discrete systems,

whose observables take on only finitely-many values, the method is indispensable.

Nevertheless, its limitations are those of any typed theory. Use of LoAs is effective in

precisely those situations where a typed theory would be effective, at least informally.

Can a complex system always be approximated more accurately at finer and finer levels

of abstraction, or are there systems which can simply not be studied in this way? We do

not know. Perhaps one may argue that the mind or society – to name only two typical

examples – are not susceptible to such an approach. In this paper we have made no

attempt to resolve this issue.

We have also avoided committing ourselves to determining whether the method

of abstraction may be exported to ontological or methodological contexts. Rather, we

have defended a version of epistemological levellism that is perfectly compatible with the

criticisms directed at other forms of levellism.

Introduction of LoAs is often an important step prior to mathematical modelling

of the phenomenon under consideration. However, even when that further step is not

taken, introduction of LoAs remains a crucial tool in conceptual analysis. Of course, care

must be exercised in type-free systems, where the use of the method may be problematic.

Such systems are susceptible to the usual paradoxes and hence to inconsistencies, not

only when formalised mathematically but also when considered informally. Examples of

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

References

Arbib, M. A. 1989,

The Metaphorical Brain Ii: Neural Networks and Beyond

(New

York: John Wiley & Sons).

Barwise, J., and Etchemendy, J. 1987,

The Liar: An Essay on Truth and Circularity

(New

York; Oxford: Oxford University Press).

Bechtel, W., and Richardson, R. C. 1993,

Discovering Complexity: Decomposition and

Localization as Strategies in Scientific Research

(Princeton: Princeton University

Press).

Bell, J. S. 1987, "The Theory of Local Beables" in

Speakable and Unspeakable in

Quantum Mechanics - Collected Papers on Quantum Mechanics

(Cambridge:

Cambridge University Press), 52-62.

Benjamin, P., Erraguntla, M., Delen, D., and Mayer, R. 1998, "Simulation Modeling and

Multiple Levels of Abstraction" in

Proceedings of the 1998 Winter Simulation

Conference

, edited by D. J. Medeiros, E. F. Watson, J. S. Carson, and M. S.

Manivannan (Pistacaway, New Jersey: IEEEPress), 391-398.

Block, N. 1997, "Anti-Reductionism Slaps Back" in

Philosophical Perspectives 11:

Mind, Causation, and World

, edited by J. E. Tomberlin (Oxford - New York:

Blackwell), 107-133.

Boolos, G., Burgess, J. P., and Jeffrey, R. C. 2002,

Computability and Logic

4th ed.

(Cambridge: Cambridge University Press).

Brown, H. C. 1916, "Structural Levels in the Scientist's World",

The Journal of

Philosohy, Psychology and Scientific Methods

, 13(13), 337-345.

Cariani, P. 1991, "Emergence and Artificial Life" in Langton et al. [1992], 775-797.

Craver, C. F. 2004, "A Field Guide to Levels",

Proceedings and Addresses of the

American Philosophical Association

, 77(3).

Craver, C. F. forthcoming,

Explaining the Brain: A Mechanist's Approach

Damper, R. I. 2000, "Emergence and Levels of Abstraction",

International Journal of

Systems Science

, 31(7), 811-818.

Davidson, D. 1974, "On the Very Idea of a Conceptual Scheme",

Proceedings and

Addresses of the American Philosophical Association

, 47. Reprinted in

Inquiries

into Truth and Representation

(Oxford: Clarendon Press, 1984): 183-98. All page

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

numbers to the quotations in the text refer to the reprinted version.

de Roever, W.-P., and Engelhardt, K. 1998,

Data Refinement: Model-Oriented Proof

Methods and Their Comparison

(Cambridge: Cambridge University Press).

Dennett, D. C. 1971, "Intentional Systems",

The Journal of Philosophy

, (68), 87-106.

Dennett, D. C. 1987,

The Intentional Stance

(Cambridge, Mass; London: MIT Press).

Egyed, A., and Medvidovic, N. 2000, "A Formal Approach to Heterogeneous Software

Modeling" in

Proceedings of the Third International Conference on the

Fundamental Approaches to Software Engineering (Fase 2000, Berlin, Germany,

March-April) - Lecture Notes in Computer Science, No. 1783

, edited by Tom

Mailbaum (Berlin/Heidelberg: Springer-Verlag),

Emmeche, C., Køppe, S., and Stjernfelt, F. 1997, "Explaining Emergence: Towards an

Ontology of Levels",

Journal for General Philosophy of Science

, 28, 83-119.

Feynman, R. P. 1995,

Six Easy Pieces

(Boston, MA.: Addison-Wesley).

Floridi, L. 2003, "On the Intrinsic Value of Information Objects and the Infosphere",

Ethics and Information Technology

, 4(4), 287-304.

Floridi, L. 2004, "Information" in

The Blackwell Guide to the Philosophy of Computing

and Information

, edited by L. Floridi (Oxford - New York: Blackwell), 40-61.

Floridi, L. 2004, "Open Problems in the Philosophy of Information",

Metaphilosophy

35(4), 554-582.

Floridi, L. forthcoming, "Consciousness, Agents and the Knowledge Game".

Floridi, L. forthcoming, "The Informational Approach to Structural Realism".

Floridi, L. forthcoming, "On the Logical Unsolvability of the Gettier Problem",

Synthese

Floridi, L. forthcoming, "Presence: From Epistemic Failure to Successful

Observability"",

Presence: Teleoperators and Virtual Environments

Floridi, L., and Sanders, J. W. 2004, "On the Morality of Artificial Agents",

Minds and

Machines

, 14(3), 349-379.

Foster, C. L. 1992,

Algorithms, Abstraction and Implementation: Levels of Detail in

Cognitive Science

(London: Academic Press).

Gell-Mann, M. 1994,

The Quark and the Jaguar: Adventures in the Simple and the

Complex

(London: Little Brown).

Hales, S. D., and Welshon, R. 2000,

Nietzsche's Perspectivism

(Urbana: University of

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

Illinois Press).

Hayes, I., and Flinn, B. 1993,

Specification Case Studies

2nd ed (New York; London:

Prentice Hall).

Heil, J. 2003, "Levels of Reality",

Ratio

, 16(3), 205-221.

Hendriks-Jansen, H. 1989, "In Praise of Interactive Emergence: Or Why Explanations

Don't Have to Wait for Implementations" in Langton

et al.

[1989], 282-299.

Hoare, C. A. R., and He, J. 1998,

Unifying Theories of Programming

(London: Prentice

Hall).

Hughes, P., and Brecht, G. 1976,

Vicious Circles and Infinity: A Panoply of Paradoxes

(London: Cape). Originally published: Garden City, N.Y.: Doubleday, 1975.

Kelso, J. A. S. 1995,

Dynamic Patterns: The Self-Organization of Brain and Behavior

(Cambridge, Mass; London: MIT Press).

Knuth, D. E. 1997,

The Art of Computer Programming

3rd ed. (Reading, Mass.; Harlow:

Addison-Wesley). 3 vols.

Marr, D. 1982,

Vision: A Computational Investigation into the Human Representation

and Processing of Visual Information

(San Francisco: W.H. Freeman).

McClamrock, R. 1991, "Marr's Three Levels: A Re-Evaluation",

Minds and Machines

, 1,

185-196.

Mesarovic, M. D., Macko, D., and Takahara, Y. 1970,

Theory of Hierarchical,

Multilevel, Systems

(New York: Academic Press).

Mitchell, T. M. 1997,

Machine Learning

International edition (New York; London:

McGraw-Hill).

Moor, J. H. (ed.) 2001,

The Turing Test: Past, Present and Future

(Special issue of

Minds and Machines

, 11(1)).

Nagel, T. 1974, "What Is It Like to Be a Bat?"

The Philosophical Review

, 83(4), 435-450.

Newell, A. 1982, "The Knowledge Level",

Artificial Intelligence

, 18, 87-127.

Newell, A. 1990,

Unified Theories of Cognition

(Cambridge, Mass; London: Harvard

University Press).

Newell, A. 1993, "Reflections on the Knowledge Level",

Artificial Intelligence

, 59, 31-

38.

Oppenheim, P., and Putnam, H. 1958, "The Unity of Science as a Working Hypothesis"

L. Floridi – J. W. Sanders,

Levellism and the Method of Abstraction

——————————————————————————————————————————————-——————

——————— Information Ethics Group – Research Report 22.11.04 ————

Minnesota Studies in the Philosophy of Science. Concepts, Theories, and the

Mind-Body Problem.

, edited by H. Feigl, Michael Scriven, and Grover Maxwell

(Minneapolis: University of Minnesota Press), vol. 2, 3-36.

Poli, R. 2001, "The Basic Problem of the Theory of Levels of Reality",

Axiomathes

, 12,

261–283.

Pylyshyn, Z. W. 1984,

Computation and Cognition: Toward a Foundation for Cognitive

Science

(Cambridge, Mass: MIT Press).

Robinson, J. 1989,

Vintage Timecharts: The Pedigree and Performance of Fine Wines to

the Year 2000

(London: Mitchell Beazley).

Robinson, J. (ed.) 1994,

The Oxford Companion to Wine

(Oxford: Oxford University

Press).

Russell, B. 1902, "Letter to Frege" In

From Frege to Gödel: A Source Book in

Mathematical Logic, 1879-1931

, ed. by J. van Heijenoort (Harvard University

Press: Cambridge, MA, 1967), 124-125.

Salthe, S. N. 1985,

Evolving Hierarchical Systems: Their Structure and Representation

(New York: Columbia University Press).

Schaffer, J. 2003, "Is There a Fundamental Level?"

Nous

, 37(3), 498-517.

Shieber, S. 2004,

The Turing Test - Verbal Behavior as the Hallmark of Intelligence

(Cambridge, Mass.: MIT).

Simon, H. A. 1969,

The Sciences of the Artificial

1st ed. (Cambridge, Mass. - London:

MIT Press). The text was based on the Karl Taylor Compton lectures, 1968.

Simon, H. A. 1996,

The Sciences of the Artificial

3rd ed. (Cambridge, Mass.; London:

MIT Press).

Spivey, J. M. 1992,

The Z Notation: A Reference Manual

2nd ed (New York; London:

Prentice-Hall).

Tarski, A. 1944, "The Semantic Conception of Truth and the Foundations of Semantics",

Philosophy and Phenomenological Research

, 4, 341-376. Reprinted in L. Linsky

(ed.)

Semantics and the Philosophy of Language

(Urbana: University of Illinois

Press, 1952).

Turing, A. M. 1950, "Computing Machinery and Intelligence",

Minds and Machines

, 59,

433-460.