the system of carnap's 1aufbau - osnabrueck/germanygraduate/aufbau/systemprint.pdf · goal and...

Concepts from Similarity. - The Second Day: TheAufbau System

Klaus Robering

IFKI, Syddansk Universitet

December 21th, 2003

Klaus Robering Concepts from Similarity - 2nd Day

Goal and Structure of the SystemBuilding up the System

Part I

Presenting the System of the Aufbau



1 Goal and Structure of the SystemGoalsA Gross Overview of the SystemMethods of Constitution

2 Building up the SystemSetting up the BasisThe Lower Levels of ConstitutionThe Next Steps



GoalsA Gross Overview of the SystemMethods of Constitution

Goal and Structure of the System




Goals




The Goal of the Project

Carnap’s goal in the Aufbau is to explain all “objects” - both thatof everydaylife an that of science (der “vorwissenschaftlichen undder wissenschaftlichen Erkenntnis”, §68) - in terms of certain basicconcepts.

The complete system presents a “family tree of all concepts” (“einStammbaumm der Begriffe”, §1).

The system is eventually used for the solution - or ratherdissolution - of certain philosophical problems.




Explanations

The term object is used in a very general sense. Everythingabout which one can talk is an object. Thus objects includethings, properties, relations, states, processes. There are bothreal and unreal objects.

The term explain is meant in a special technical sense whichincludes both standard explicit definition and more liberalforms of definitions (called definition in use orcontext-definitions).

The family tree of concepts mirrors simultaneously logical andepistemological relationships.




Two Types of Problems

According to the double aim of the system - namely, to representlogical as well as epistemological relationships - there are two kindsof problems concerning the whole project:

formal problems and

material problems




Carnap’s “Four Main Problems”

1 The problem of the basis: What are the basic individuals andrelationships? �Explanation

2 The problem of the ascension form [Germ. Problem derStufenformen]: Which formal operations effect the transitionfrom one level to the next? �Explanation

3 The problem of the forms of the objects: The objects have tobe actually defined on their respective levels. �Explanation

4 The problem of the form of the system: What overallarchitecture fits best the complete stystem of levels.�Explanation

�Continue




The Problem of the Basis

“Erstens muß eine Ausgangsbasis gewahlt werden, eine erste Stufe,auf die sich alle weiteren grunden”; §26.

“To begin with, a basis must be chosen, a lowest level upon whichall others are founded”.

�Go back




The Problems of the Forms of the Levels

“Zweites sind die immer wiederkehrenden Formen zu bestimmen,in denen sich der Ubergang von einer Stufe zur nachsten vollziehensoll”; §26.

“Secondly, we must determine the recurrent forms through whichwe ascend from one level to the next.”

�Go back




The Problem of the Forms of the Objects

“Drittens ist fur die Gegenstande der verschiedenen Arten zuuntersuchen, wie sie durch schrittweise Anwendung derStufenformen konstituiert werden konnen”; §26.

Thirdly, we must investigate how the objects of various types canbe constructed through repeated applications of the ascensionforms”.

�Go back




The Problem of the Form of the System

“Die vierte Frage ist die nach der Gesamtform des Systems, wie siesich aus der Ubereinanderschichtung der verschiedenenGegenstandsarten ergibt”; §26.

“The fourth question concerns the over-all form of the system asits resutls form the stratified arrangement of the object types”.

�Go back




Classifying the Problems

Whereas the first, third and fourth problem involve materialquestions, the second one is a “formal-logical problem”; Aufbau§26.




Type Theory and Constitution Theory

Actually, Carnap treats the 2nd problem as a question how asuitable type-theoretic language can be set up for theconstitutional system.

More precisely (and expressed in modern terminology): Whattype-constructors should be assumed for such a language?

Carnap adopts two “ascension forms”: classes and relations.




Carnapian Type Theory

Carnap is not very explicit about the grammar of his formallanguage. But in modern terms, we could say that this is atype-theoretic language which admits of two type constructor:

1 the class constructor: ( ) : τ 7→ (τ).

2 the pair constructor: 〈〉 : τ1τ2 〈7→ τ1, τ2〉.

By means of the pair constructor, relations (of any arity) can behandled (as usual) as classes of tuples.




A Gross Overview of the System




The Main Kinds of Objects

objects

mental physical cultural

private of others




Relations of “Constitutionability”

Cultural

Of Others

Private

Physical

Documentation

Manifestation

Psycho-physical Parallelism

phen. Red.

Exp.




System Architectures

This leaves several possibilites for a system. One may base it on:

either the “autopsychological”, construct the physical byphenomenological reduction to the autopsychological, and usethe expression relation to construct the “heteropsychological”;finally, the cultural objects are constructed by means of therelationships of documentation and manifestation;

or the physical, reduce the psychological objects to physicalones via the expression relation; then again approaching thecultural objects using documentation and manifestation.




Methods of Constitution




Overview

There are three basic methods of constitution:

1 quasi-analysis of the first kind (i. e., in terms of a relation ofpartial identity),

2 quasi-analysis of the second kind (i. e., in terms of a relationof partial similarity),

3 and definition by abstraction (i. e., setting up equivalenceclasses modulo an equivalence relation).




Methods of Type Theoretic Modelling

In order to construct new objects from the given basic ones andthe basic relationships between them, one has the wholetype-theoretic apparatus at one’s disposal.

Though this apparatus is very powerful, as was surely clear toCarnap from his study of the Principia Mathematica, there remainsnevertheless a special problem relating to the basic elements.




A Problem Arising from Type Theory

No matter how one choses the basic object of one’s system, thetype theoretic approach - at least that which is confined to the twochosen type-constructors - makes it unpossible to conceive of themas entities with an internal structure.

“Even if we were to suppose that the basic elements arethemselves again classes of other elements, classes of ’fundamentalelements’, we could not construct these fundamental elements withthe aid of the given ascension forms. The basic elements of theconstructional system cannot be analyzed through construction”;Aufbau, §68.




The Reason for the Problem

Ultimately, the reason for this problem is just that the chosentype-constructors are both synthetic:

They build up more complex entities (= entities on higher level ofthe type hierarchy) from more simple one.

What is thus lacking is an analytic procedure!




A Problem Arising from Ontology

As described so far, our problem is purely formal - arising from theuse of type theory.

This, however, is not the complete story - at least not according toCarnap.

With reference to

Gestalt psychology and

holistic philosophy (Driesch)

he points out that there are, for instance, “many psychological,escpecially phenomenal [sinnesphanomenale], objects, whichtraditional psychology thought of as being compounds” but whichreally are “unanalyzable units”; Aufbau, §71.




A Special Method: Quasi-Analysis

In order to solve the problem of the lacking analytic procedure,Carnap develops the method of Quasi-Analysis, which is the keyprocedure of his whole project.

It consist in setting up by means of the purely synthetic means ofthe system formal substituents for the unreachable components ofunanalysable objects.

These formal substituents bear relationships to their “parentobject” which are formally analogous to those which itscomponents - recognized in a proper analysis - bear to it.




Proper and Quasi-Analysis

Since a quasi-analysis is supposed to lead to results analogous tothose of a proper analysis, it suggests itself to look first at themethod of analysis ...

... and then to generalize from its formal properties in order toreach at a description of quasi-analysis.




Analysis

Explanation

According to Carnap, analysis is a procedure to find the unknowncomponents of some objects by investigating into their mutual rela-tionships.




Analysis: An Example

Consider the following example (cf. Aufbau, §70):

Assume that there is a number of things, each of which hasone or more of five colours.

It is, however, unknown which things have which colours.

But we are informed (just by means of a pair list) about therelation of “color kinship” between these things.

Two things bear this relation to each other if they share acommon colour.

Here the task of analysis is to recognize from our knowledgeof colour kinship which thing has which colours.




Quasi-Analysis of the First Kind

Note that the relation in question here - namely, color kinship - is akind of partial identity.

Two objects bear this relation to one another iff they agree, iffthey are identical with respect to one component (namely, in acolor quality).

This reliance on a relationship of partial identity is characteristicfor a quasi-analysis of the first kind.




Attaining the Goal of Analysis: Extensions are Enough

To each of the five colour qualities corresponds a colour class as itsextension: the class of all objects having the colour in question.

Carnap explains that “[t]he task of analysis is attained once wesucceed in determining the ’color classes”’; Aufbau, §70.




The Two Essential Properties of Colour Classes

Each colour class is distinguished by two properties:

1 “... any two elements of a color class stand in the relation ofcolor kinship to one another (because the members of the pairboth have the color which determines the color class)”; ibid.

2 “the color classes are the largest possible classes all of whosemembers are color-akin (i. e., there is no thing outside of acolor class which stands in the relation of color kinship to allthe things in the class)”; ibid.




Similarity Circles

Formally, the two mentioned characteristics determine what Carnapcalls the similarity circles of the relationship of color kinship.

Explanation

Given a similarity relation R (i. e., a reflexive and symmetric rela-tion). A subclass N of the field of R is called a similarity circle (withrespect to R) iff N is

1 consistent in the sense that R holds between all members ofN and

2 maximal in the sense that there is no object in the field of Rbut outside of N which bears this relation to a member of thisclass.




Unfavourable Circumstances

Carnap has to concede, however, that the analytical methods ofsetting up similarity circles only works under “favourablecircumstances”.

In the colour example, for instance, it does not work under thoseunfavourable circumstances when one colour is a companion ofanother one.

Maximality, then, will not hold for the companion class: “Forexample, if blue is a companion of red, then the blue color classdoes not have this second property, for a thing which is red but notblue does not belong to this color class and is nevertheless colorakin to all things in this class, since all of them are also red;”Aufbau, §70.




Favourable Circumstances

Carnap explains: “If there are no systematic connections betweenthe distributions of the different colors, then this unfavorable case,namely, that the second property is missing in a color class,becomes the less likely the smaller the average number of colors ofthe thing and the larger the total numer of things is;” Aufbau, §70.




Generalizing

Generally speaking then, we should make sure in applying thisanalytical method

that we deal with many different objects,

that the number of qualities is even higher, and

that the qualities are independent of each other, and

that each object has only a very limited number of qualities(as compared with the huge number of all qualities).




From Proper Analysis to Quasi-Analysis

Quasi-analysis shares with proper analysis the formal core, namely:the setting up of similarity circles modulo a reflexive andsymmetric relationship.

We have to keep in mind, however, that - at least in the case ofontologically unanalyzable basic units - quasi-analysis does not leadup to the rediscovery of qualitative constituent features.

Rather “... quasi-analysis of an essentially unanalyzable entity intoseveral quasi-constituents means placing the entity in severalkinship contexts on the basis of a kinship relation, where the unitremains undivided”; Aufbau §71.




Quasi-Analysis of the Second Type

Up to now, we have only dealt with quasi-analyses of the first kind,which are based on “partial identities”.

But there is a second type of quasi-analysis, “which does not havethe same general importance as the first one, but which must beexplained because it is later applied in the constructional system”;Aufbau, §72.

Quasi-analyses of this second type are based on relationships ofpartial resemblance (rather than partial identity).




An Example Again

In order to explain quasi-analysis of the second type, Carnapmodifies the colour example.

Now we assume much more objects and much more colourswhich might come from all parts of the colour solid.

“We call two things colour similar if, among other colors, theyeach have a color whicdh is similar to that of the other (i. e.,which, on the color solid, has a distance from the other whichis smaller than a certain arbitraritly chosen magnitude)”;Aufbau, §72.




Colour Spheres and Colour Similarity Circles

a single colour

xy z

colour solida colour sphere

a colour similarity circle

�Back




Type-2-Quasi-analysis: Summing up

The colour similarity circles still have the properties of consistencyand maximality with respect to the base relation, which now, ofcourse, is colour similarity.

Technically: The colour similarity circles are the similarity circles(in the logical sense) modulo colour similarity.

But, of course, these do no more correspond to the individualcolours (= points of the colour solid) but rather to balls includedin this solid.

The diameter of such a globe corresponds to the maximal distanceyet allowing for similarity.




Generalizing from the Example

Let us in the following generally speak of qualities instead of themore special colours.

We shall assume that the qualities build up a metrical space, aquality solid (remember Fechner) with dimension n.

We call the solid’s points quality points.

Let d be the maximal distance within the solid still admittingfor similarity.

The n-dimensional balls with radius d are our quality spheres.

Maximal sets of objects sharing some one quality within agiven quality sphere make up a similarity circle.




The Basic Idea of Type-2-Quasi-Analysis

Similarity circles corresponding tointersecting quality spheres overlap.

Corresponding to single qualitypoints we have the sets of all objectspossessing this quality.

These sets are subsets of similaritycircles; ...

they will, however, never (?) bedissected into two areas by theboundary of a similarity circlecrossing throug them.




Defining Quality Classes: The First Attempt

Of course, each subset of a set of objects corresponding to a singlequality point, will also never be crossed by the boundary of asimilarity circle.

Thus we shall, in a first attempt, explain:

Explanation

A quality class is a set of objects

1 which is included in every similarity circle which is not disjointfrom it

2 and which is maximal beneath all the sets fulfilling the firstcondition.




The Meaning of Maximality: Separation

a quality point x 6∈ Q

quality class Q

a similarity circle M

for each x 6∈ Q there has to be a similarity circleM such that x 6∈ M but Q ⊆ M



Setting up the BasisThe Lower Levels of Constitution

Building up the System




Setting up the Basis




The Problem of Elementism

“The obvious objection to psychological elementism is the factthat phenomenal experience is a constant flux.

It is not even a kaleidoscopic change of parts, for there are noseperate parts. It is, as James made clear, like the flow of a streamthat can not properly be thought of as grouping of elements”,

Boring, History of Experimental Psychology, p. 344.

“Wundt sought to emphasize this fact by naming the element a’mental process’. The force of this term is that it persistentlyasserts that experience is active in the sense of changing process,although not in the sense of an activity that requires an agent”;ibd.




Chosing Basic Relations

Carnap (Aufbau §75) lists some requirements which the basicrelations of his constitutional system should fulfil:

1 they should be of the same logical type,

2 the members of their domains should be exclusivelyelementary experiences,

3 all recognizable state of affairs should be describable in termsof them.

As regards to item no. 3, the possibility of constituting the physicalthings is of special importance.




Motivating Partial Identity

“In order to be able to construct the physical world, we needcertain pconstitutents of elementary experiences, especiallysensations with their determinations of quality and intensity, lateron also spatial and temporal order ...p”; Aufbau, §76.

Note that this is completely in line with the Mach-concept (things= complexes of elements/sensations).

Since elementary experiences are “indivisibel units”, we thus haveto chose our basic relationships in such a way that the requiredcomponents are definable by means of quasi-analysis.




Motivating Partial Identity

This leads up to consider a relation which may be described, fromoutside the system, as “sharing a common component”.

“The pconstitutents of elementary experiencesp will have to bequasi constituents, since in our system the celementaryexperiencesc are indivisible units”; Aufbau §76.

“pEvery sensation quality, whether it is a color, a tone, a fragrance,etc.p, will have to be a pcommon property of those elementaryexperiences p in which it occurs as a pconstituentp (i. e., as a quasiconstituent);” ibid.




Using Part Identity for the Constructions Required

“This pcommon propertyp is constructionally represented as theclass of the appropriate celementary experiencesc (’cqualityclassc’);” Aufbau, §76.

“This class could be constructed, for example, for every psensationqualityp through the procedure of quasi analysis on the basis of therelation pagreement of two elementary experiences in such aqualityp”; ibid.




Part identity Explained

Explanation

Thus we consider that relation which pholds between two exlemen-tary experiences, x and y , if and only if in x there occurs an experi-ence constituent a and in y an experience constituent b such that aand b agree in all characteristics, namely, in quality in the narrowersense, in intensity, and in the location sign[...]p”; ibid.




Quasi Analysis in Action

...

elex x

a

elex y

b

elex z

c

Pi

Pi

Pi

{x , y , z , ...}




From Identity to Similarity

Though the relation Pi of partial identity suffices to isolate theneeded parts of elementary experiences, it is not possible to definethe different ordering relations between sensory qualities in termsof this notion.

“The dimensions pof the sensation qualities of a sense modality,namely quality solid, [German: “Die Ordnungen PderEmpfindungsqualitaten eines Sinnesgebietes als Qualitatskorper ...](e. g., color solid, tone scale), intensity scale, and sensory field (e.g., visual field, tactile field) are not recognizable on the basis of therelation of part identityp (i. e., they are not constructable from thecpart identityc)”; Aufbau, §77.

This motivates the consideration of the relation Ps of partsimilarity.




Part Similarity Explained

Explanation

“pTwo elementary experiences x and y are called ’part similar’ ifand only if an experience constituent (e. g., a sensation) a of xand an experience constituent b of y agree, either aproximately orcompletely, in their characteristics (quality in the narrower sense,inensity, local sign)p”; Aufbau, §77.




Similarity

The ordering relation between sensory qualities is called similaritysimpliciter - as opposed to partial similarity, which is a relationbetween elementary experiences.

We use the abbreviation “Sim” for similarity (simpliciter).

Example: “We say, for example, that ptwo color sensations a and bare similarp (a Sim b), pif they argee approxiamtely or completelyin hue, saturation, brightness, (or hue, whit content, blackcontent) [remember Ostwald’s colour system, K. R.]; and locationsign (i. e., place in the visual field),...”; Aufbau, §77.




Partial Similarity and Similarity

elexes type: ι

...

PsPs

Ps

sensory qualities type: (ι)

...

Sim

Sim

Sim

constitution




Accounting for Temporal Order

In order to catch the temporal series, which is covered neither byPi nor by Ps, Carnap makes use of recollection ...

... apparently assuming that the only way to recognize theresemblance between two elementary experiences is to compare amemory image of the one to the other, which (seemingly) must bethe present experience.

“pIf it is recognized that two elementary experiences x and y arepart similar, then a memory image of the earlier of the two, of say,x must have been compared with yp;” Aufbau, §78.




Recollection of Similarity

“Rs” is used as an abbreviation for the relationship of recollectionof similarity.

Obviously, Rs is an asymmetric relation - as is to be expected froma temporal order.

Rs is explained as follows

Explanation

“’x Rs y ’ [...] means ’px and y are elementary eperiences whichare recognized as part similar through the comparison of a memoryimage of x with yp”’




Problems

1 What’s about elementary experiences which are neverrembered?

2 If x is remembered to be similar to y and the relation ofrecollection of similarity also holds true for y and z , may wesuppose that x bears this very relation to z , too?

3 As the transitivity of temporal order seems to require?

Actually, Carnap uses “x Rs y” simply as: “x is earlier than yand both experiences are partly similar to each other”.




Categories

Obviously, there is a close connection between Carnap’srelationship Rs of recollection of similatrity and traditional lists ofcategories.

Remember Hume’s prinicples of assosiation: similarity, contiguity(in space and time), causality.




The Lower Levels of Constitution




Getting Back Partial Similarity

The first step in the system is the redefinition of partial similarityin terms of recollection of similarity.

Explanation

x is partially similar to y iff either x is the same as y or x is remem-bered to be similar to y or, finally, y is remembered to be similar toy .




The Similarity Circles

Now we start the process of quasi-analysis with the relationship ofpartial similarity. With a slight abuse of terminology, we call thesimilarity circles of this relationships just the similarity circles (thussuppressing their dependence on the relation).

Explanation

A class α of objects is a similarity circle iff

1 every two items from α are partially similar to each other,

2 there is no object outside of α which is paritally similar toeach element of this class.




Quality Classes

Our next aim is to construct those classes of elementaryexperiences which correspond to single sensory qualities - say apatch of color of a certain color at a specific place of the visualfield or a certain fragrance or taste or whatsoever.

Since we are working now with a relation of partial similarity(rather than with a relation of partial identity), we should applynow a quasi-analysis of the second type.

There is, however, still an obstacle to such a direct approach.




Companionship Again

Again, we have to recognize the possibility of accidentialcompanionships.

When we discussed the quasi-analysis of the second type, weconceived of objects as assigned to those quality points whichrepresent qualities which the assigned objects share. �Example

Since our objects are “extended” now and may thus correspond bydifferent parts to different quality points, the same object may beassigned to different quality points.

As a consequence, similarity circles may intersect by “pureaccidence” which may well lead up to a dissection of a quality class.




The Carnap-Example

The Visual Field

S1

S2

SC: “bluish at S1

SC: “reddish at S2

QS: “royal blue at S1




Quality Classes Explained

Thus we arrive at the following explanation.

Explanation

A class α of objects is a quality class iff

1 α is totally contained in any similarity circle which containsalready a substantial part of α;

2 and for every elementary experience x outside of α there is asimilarity circle β which seperates α from x in that β includesall of α but excludes x .




Order

The quality classes correspond to quality points in quality spaces(quality solids): “We have previously seen that the cquality classesc

are constructional representations of the psensation qualitiesp (inthe widest sense, including the emotion qualities, etc.;” Aufbau,§81.

They are orderable according to the following explanation.

Explanation

Two quality classes are similar to each other iff every elementaryexperience of the first resembles every such experience of the second.




Sense Classes

“... we can now proceed with a division into sense modalities;”Aufbau, §85.

The procedure for this follows Helmholtz’s explanation (which inturn was inspired by Fichte).

Explanation

The quality classes α and β belong to the same sense modalityiff there is a chain α = γ0, γ1, ..., γn = β (n ≥ 0) such thatconsecutive members of this chain are similar to each other.

A sense modality is a maximal class of quality classesbelonging to the same sens modality.




Matters of Dimensionality

The next step makes essential use of a new theoretical tool,namely: topological dimension theory.

One of the “fathers” of this discipline, the mathematician KarlMenger, was a member of the Vienna Circle.

In psychology, it has been quite common to speak of the dimensionof a sense modality.

Carnap follows this tradition; cf. the next slide.




Carnap on the Dimensions of the Sense Modalities

“We have mentioned above that the sense class of tone sensationshas Dn [dimension number] 2, that of the visual sense, of colorsensations, Dn 5 (§80). For the senses of the skin, the locationsigns are orderable in two dimensions. Since their qualities arefurthermore differentiable through intensity and perhaps alsothrough a quality series, the Dn of each of them (tactile sense,sense of warmth, sense of cold, sense of pain) is 3 or 4. The Dn ofthe other sensess, including the domain of the emotions, is forsome of them 2, for others 3”; Aufbau, §86.




Psychology and Topology

Carnap re-interprets the claims of psychology in terms oftopological dimension theory.

Of course, the concept of dimension builds up of that of atopological space which in turn makes use of the notion of aneighbourhood of a point.

Carnap takes the neighbourhood of quality classes as determinedby the similarity relation between them.

The formal details do not need us to concern here (they are notexplained in the Aufbau).




The Visual Sense

As we have seen, Carnap considers the visual modality asdistinguished by having the maximal dimension number 5 amongall sense modalities. Thus we may explain:

Explanation

The visual sense is that class of qualities which has the dimension 5if similarity is taken as determining the neighbourhood relationships.




Analyzing Visual Qualities

The next step is to seperate the “chromatic” aspects of a visualquality - vulgo: its colour - from its geometrical: its locational sign.Visual qualities are called colour identical iff they agree in colour,place identical iff they agree in their locational sign.

Note that place identical qualities cannot simultaneously occurwithin the same elementary experience.

I. e., the same “spot” of the visual field cannot simultaneously beoccupied by two different colours, say, e. g., blue and red.




Blue and Red at the Same Place

Thus take α := blue at S1 and β := red at S2. If then S1 = S2, αand β cannot share even a single elementary experience.

Note that being disjunct (sharing no elementary experience) is asymmetric relation. If we join it with identity (between qualityclasses), we get a similarity relation! We shall call it therelationship of exclusion.




A Difficulty Again

Unfortunately, being disjunct is only a necessary condition for placeidentity, not a sufficient one.

Suppose that for some, perhaps physiological reasons, a spot ofcolour C1 at place S1 never co-occurs with a spot of colour C2 atplace S2 (where, of course, S1 6= S2).

Then C1 at S1 would not share any experience with C2 at S2. Butnevertheless, the two qualities would not be place identical.




Circumventing the Problem by Quasi-Analysis

If we already had constituted the relation of place identity, it wouldbe rather simple indeed to define the places of the visual field (thelocational signs).

A place could be taken as a maximal class of mutually placeidentical visual qualities.

Since place identity is not yet available to us, we use the abovementioned similarity relation of exclusion as a surrogate for it ...

and base a quasi-analysis of the first kind on it.




Gauging the Adequateness of the Solution

... ... ... ... ...

S0 S0 S0 S0

...

S1

“... there are two necessary conditions for an erroneous assignmentof an elment to a given place class, namely,

first, that the visual field place in question should beunoccupied in at least one elementary experience andsecondly, that the element to be assigned, which actuallybelongs to a different place, should occur only in suchexperiences as leave that other place unoccupied;”

Aufbau, §88.According to Carnap:

If the number of “unoccupied places” is fairly low, the numberof similarity circles modulo exclusion may still outrun thenumber of visual field places.Nevertheless, the probability, then, that the number ofsimilarity circles surpasses that of the places, is relativelysmall.Furthermore, one may exclude as “suspicious” those similaritycircles which contain elements also belonging to other circles.




Visual Field Places Explained

Thus we arrive at:

Explanation

A visual field place is a non-empty class of visual qualities whichonly contains those elements of a similarity circle modulo exclusionwhich do not belong to other similarity circles of this relation.




The Geometrical Order of the Visual Field

Now, the topological order of places redurces just to similaritybetween their members.

Explanation

Two places of the visual field are proximate iff the one contains aquality similar to a quality of the other.

Remark: This reduces the Humean category of local contiguity tosimilarity (leaving just similarity and temporal contiguity behind).




Colours: The Basic Idea

Turning to colours now, we first note that proximate visual fieldplaces ( = qualities belonging to proximate places) may, of course,still differ with respect to colour.

They are, however, colour identical if they bear the relation ofsimilarity to exactly the same qualities at some definite place intheir vicinity.

Since these qualities are in the local vicinity of both of the twoqualities to be compared, differences, if there were any, could notbe due to spatial circumstances at all but, rather, would bedifferences in colour.




Colours at Proximate Places

Thus we arrive at the following explanation:

Explanation

1 Two visual qualities are colour identical at proximate places iffthey belong to proximate places and if there is a third placeproximate to the both the places of the two qualities suchthat these two qualities are similar to exactly the samequalities within this third place.

2 Two visual qualities are colour identical (simpliciter) iff thereis a chain of such qualities connecting them such that twoconsecutive members of this chain are colour identical atproximate places.




The Colour Solid

We may now constitute the colours and their topological order inthe colour solid.

Explanation

1 A colour is a maximal class of mutually colour identical visualqualities.

2 Two colours are proximate iff one of them contains a qualitysimilar to a quality contained by the other.




Sensations

Elementary experiences belonging to a common quality class, agreein one of their respective components: e. g., they are equal insharing some particular fragrance or taste or feeling or they havesome particular colour at a certain place of the visual field incommon.

In order to identify a specific experience, it does not suffice to referto its quality (such and such a fragrance, feeling, taste, colourspot), but we have to fix its position within the “stream ofconsciousness”.




Sensations Explained

Explanation

1 A sensation is an ordered pair consisting of an elementaryexperience and a quality class containing it.

2 Sensations are simultaneous iff they agree in their firstcomponent (i. e., belong to the same elementary experience).




Analyzing Elementary Experiences

Elementary experiences may now be analyzed into their (quasi-)”components”. Such analyses are called division classes by Carnap.

They differ according to the status of the components belonging tothem.

A component may be

either “general”; then it re-occurs in many differentexperiences (e. g., “blue at S1”;

or “individual”; then it is bound to a specific occurence withina certain elementary experience (e. g., blue at S1 as it appearsin the elementary experience x .




Division Classes

Explanation

1 A division class of the first type is a maximal set ofsimultaneous sensations.

2 A division class of the second type of an elementary experienceis the set of all qualities to which this experience belongs.




Temporal Order

Explanation

The elementary experiences x is earlier in time than the elementaryexperience y iff there is a chain x = z0, z1, ..., zn = y (n > 0)between them such that the relation of recollection of similarityobtains between consecutive members of this chain.

This is only a preliminary time order since there will be “gaps”(dreamless sleep, lost of consciousness) in it. “A completetemporal sequence can be constructed only later with the aid of theregularities of the processes of the outside world;” Aufbau §120.


The Chain of Definitions

Part II

The Formal System on Its Lower Levels


The Chain of DefinitionsThe Type SystemDefinitions in the System’s Lower PartsThe Next Step

The Chain of Definitions



The Type System



Types and their Constructors

Basic types: the type ι of individuals; these include especiallythe elementary experiences

Constructors:

the set constructor: ( ) : τ 7→ (τ),the pair constructor: 〈〉 : τ1τ2 7→ 〈τ1, τ2〉

Special Logical Constant: Found (Foundation) of type (〈ιι〉).



Foundation

The constant Found denotes the set of “real” binary relations.These are relations which are not just arbitrary sets of orderedpairs but correspond to actual relationships between individuals.

Found serves to define (and thus to eliminate) the relationalconstant Rs for recollection of similarity.



Defining Recollection of Similarity

Using the constant Rs one can formulate a sentence A[Rs]containing just Rs as the only constant and expressing the(empirical) fact that the colour solid is of dimensionality 3.

We replace in A[Rs] the constant Rs by a variable X of the sametype (〈ιι〉) which yields the constant-free sentence A[X ].

Now we may define:

Rs =def

ιX .[X ∈ Found ∧ A[X ]

]



Definitions in the System’s Lower Parts



The Basic Elements

Constitution

elex =def

F(Rs)

Translation: The class elex is the field of the relation Rs.

Fictitious Operation: Add to the description of each elementaryexperience that it belongs to the class elex.



Partial Similarity

Constitution

Ps =def

Rs ∪ Rs ∪ Rs0

Translation: The relation of partial similarity is the union of therelations of (a) recollection of similarity, (b) its converse, and (c)the identity relation restricted to the field of these relations.



The Similarity Circles

Constitution

similcirc =def

elex/Ps



The Quality classes

Constitution

qual =def

{α|

(1) ∀γ.[γ ∈ similcirc ∧ Nc′(α∩γ)Nc′α

> 12 → α ⊆ γ] ∧

(2) ∀x .[x 6∈ α → ∃δ.[δ ∈ similcirc ∧ α ⊆ δ ∧ x 6∈ δ]

]}



Partial Identity

Constitution

Pi =def

∈ � qual | ∈



Similarity Between Qualities

Constitution

Sim =def

{(α, β)|α, β ∈ qual ∧ α× β ⊆ Ps}



Sense Classes

Constitution

sense =def

qual/Sim∗



Visual Sense

Constitution

sight =def

{α|∃λ.[λ ∈ sense ∧Dnp(5, λ, α,Vcin(Sim)]}



Sensations

Constitution

sen =def

{(x , α)|x ∈ α ∧ α ∈ qual}



Simultaneity of Sensations

Constitution

Simul = {(p, q) ∈ sen2|π1(p) = π1(q)}



Divisions of the First Kind

Constitution

Div1 =def

sen/Simul



Divisions of the Second Kind

Constitution

1 Div2 =def

{(λ, x)|x ∈ elex ∧ λ = {α ∈ qual|x ∈ α}}

2 div2 =def

DI(Div2)



The Visual Field Places

First we need an auxiliary definition:

Excl =def

{(α, β) ∈ sight2|α ∩ β = ∅ ∨ α = β}

Now we define:

Constitution

place =def

{κ|κ 6= ∅ ∧ ∃λ.[λ ∈ sight/Excl ∧ κ = λ \⋃

((sight/Excl) \{λ})]}



Geometry of the Visual Field

Constitution

1 Plid =def

∈ � place | ∈

2 Proxpl =def

(∈ | Sim | ∈) ∩ place2



Colours at Proximate Places

Constitution

1

Colidprox =def

{(α, β)|∃χλ µ.[

(1) χ Proxpl λ ∧ χ Proxpl µ ∧ λ Proxpl µ ∧(2) α ∈ χ ∧ β ∈ λ ∧(3) µ ∩ Sim(α) = µ ∩ Sim(β)

2 Colid =def

Colidprox∗



Colours in General

Constitution

1 color =def

sight/Colid

2 Proxcol =def

(∈ | Sim | ∈) ∩ color2



The Next Step



Constituting the Physical World

The next important step in building up the constitutional systemconsists in the definition of a “thing”, a physical object.

This makes use of the assignment of colour spots to geometricalpoints in the four-dimensional space-time-continuum.



Colours to World Points

Constitution Explanation

“1. There is a series of promi-nent world points which we callthe points of view. They form acontinuous cruve in such a waythat each of the n − 1 spacecoordinates is a single-valued,continues function of the timecoordinate”.

“1. The particuar point in theinterior of the head from whichthe world seems to be seen hasas its world line a continuouscurve in the space-time world”.


“2. The straight lines whichproceed from a given point ofview and which form with thenegative direction of time, theangle γ, we call the lines ofview”.

“2. The optical medium be-tween the eye and the seenthings can generally be con-sidered homogeneous. Underthis assumption, the light rayswhich impinge upon the eyeform straight lines which en-close the angle arc tg c withthe negative direction of time (cdesignates the speed of light)”.


“4. A one-to-one correspon-dence is established between el-ementary experiences and someof the points of view in such away that an experience whichis later in time [...] correspondsto a point of view with a largertime coordinate”.

“4. Each visual perception isbased upon an act of seeingfrom one of the points of view”.


“5. [...] (a) to sensations withproximate visual field places [...]we assign lines of view whichform only a small angle withone another, and vice versa; [...]

“5.a. Visual field places that lienext to one another alaways de-pict only points of the outsideworld whose lines of view forma small angle at the eye.

(b) the pairs of lines of viewwhich are assigned to the visualsensations of two definite placesin different elementary experi-ences all form the same angle,and conversely”.

5.b. A given pair of visual fieldplaces always has the same vi-sual angle.


“6. The color of a visual sen-sation is assigned to a worldpoint of the corresponding lineof view. Points which are oc-cupied in this way are called’world points seen from thegiven point of view’ or, in short,seen color spots.”

“6. We conclude, from a visualsensation, that a point of theoutside world which lies on thecorresponding line of view hasthe colour of the visual sensa-tion”.


PreliminariesConsistency and Maximality

Quasi-Analysis

Part III

The Formal Theory of Quasi-Analysis



Quasi-Analysis

4 PreliminariesNotation and Basic DefinitionsQuasi-Analysis

5 Consistency and MaximalityGenerating SystemsConsistent SetsMaximal SetsSimilarity-Circles

6 Quasi-AnalysisThe Representation TheoremConstructing the Similarity CirclesQuestions of Cardinality



Quasi-Analysis

Notation and Basic DefinitionsQuasi-Analysis

Preliminaries



Quasi-Analysis


Notation and Basic Definitions



Quasi-Analysis


Basic Notation

In the following we shall use the variables

B = {x , y , ....} for the set of our basic individuals,

B × B = {(x , y), ...} for its Cartesian product with itself,

PB = {N,P,Q, ...} for its power set,

RB = {R,S ,T , ...} for the power set of B × B,

SB = {Π,Σ, ...} for the power set of PB .

Q = {a, b, c , ...} for set of qualities of elements



Quasi-Analysis


Individuals and Their Qualities

Let us write “x : a” in order to express that the individual x hasthe quality a.

Thus, “x : a, y : a” indicates that x and y share the property a.

We define f : B → ℘(Q) by f (x) =def

{a ∈ Q|x : a}.

Thus for x ∈ B: x : a ↔ a ∈ f (x).



Quasi-Analysis


Qualities and Their Extensions

The set of all individuals possessing a given quality is called thequality’s extension.

We use “a∗” to denote the extension of the quality a. For a ∈ Qwe have: a∗ ∈ PB .

And define a∗ =def

{x ∈ B|x : a}.

Thus ∗ : Q → PB .



Quasi-Analysis


Domain, Range and Field

Definition

Let R ∈ RB .

The domain of R is the set of all items which bear R tosomething: DI(R) =

def{x ∈ B|∃y ∈ B.xRy}.

The range of R is the set of all items to which somethingbears the relation R: DII(R) =

def{y ∈ B|∃x ∈ B.xRy}.

The field of R ist the union of its domain and field:F(R) =

defDI(R) ∪ DII(R).



Quasi-Analysis


Similarity Relations

Definition

We call a relation R ∈ RB a similarity relation (over B) iff it isboth reflexive (in its field) and symmetric. We use “R+

B” to denotethe set of similarity relations over B.

R ∈ R+B ⇔

def(1.) ∀x ∈ F(R).xRx ∧(2.) ∀x , y ∈ B.[xRy → yRx ].

The structure (B,R) is also called a similarity structure or atolerance space.



Quasi-Analysis


Graphs

We may depict a tolerance T = (B,R) space by a graph :

The nodes of the graph represent the individuals from B.An edge indicates that the individuals represented by its endpoints are similar to each other, i. e., related to each other bythe relation R.

A Simple Example

1 2 3

4 5



Quasi-Analysis


(Half-) Matrices

We can represent the same information also by a halfmatrix:

1 2 3 4 5 R

+ - + - 1+ + + 2

- - 3+ 4

5

We may omit the diagonal because of the similarity relation’sreflexivity and the lower half because of its symmetry.



Quasi-Analysis


Adding Qualities - Colouring Graphs

If the similarity relations is explained in terms of qualities, namelyby

x and y are similar to each other if they share aquality (formally, xRy ⇔ f (x) ∩ f (y) 6= ∅)

we may add the qualities to the graph by “colouring” its edges.Our Example Continued

1 2 3

4 5

�Neighbourhoods �Quasi-Analysis �Algorithm



Quasi-Analysis


Compatability Sets

Definition

Let x be a member of the tolerance space T = (B,R). Thecompatibily set of x in T is the set of all y ∈ B which bear therelation R to x (which are similar to x).

More formally, we set for x ∈ B and N ∈ PB :

CR(x) =def

⋃(x ,y)∈R{y}

CR(A) =def

⋂x∈N CR(x)

Compatability sets are also called similarity neighbourhoods.



Quasi-Analysis


Our Example - Continued

�Graph

CR(1) = {1, 2, 4}CR(2) = {1, 2, 3, 4, 5}CR(3) = {2, 3}CR(4) = {1, 2, 4, 5}CR(5) = {2, 4, 5}



Quasi-Analysis


Quasi-Analysis



Quasi-Analysis


Quasi-Analysis

Definition

A weak quasi-analysis of the tolerance space T = (B,R) is afunction f : B → ℘(Q) which fulfils the following two conditions:

(i) If two elements are similar to each other, then they share aquality: xRy ⇒ f (x) ∩ f (y) 6= ∅.

(ii) If two elements share a common property, then they aresimilar to each other: f (x) ∩ f (y) 6= ∅ ⇒ xRy .



Quasi-Analysis


Our Example - Continued

For the tolerance space of our example, we set up the following setof qualities: �Graph

Q =def

{Blue,Red,Green}

Furthermore, we set:

Object : o Quality Set : f (o)

1 {Blue}2 {Blue,Red,Green}3 {Green}4 {Blue,Red}5 {Red}

�Strong Quasi-Analysis

This, obviously, is a weak quasi-analysis.For instance, we have

(1, 2) ∈ R ⇔ f (1) ∩ f (2) 6= ∅⇔ {Blue} ∩ {Blue,Red,Green} 6= ∅⇔ {Blue} 6= ∅⇔ >

Thus, object 1 resembles object 2 since they share the qualityBlue.



Quasi-Analysis


Strong Quasi-Analysis

Definition

If, furthermore, the following conditions are also satisfied:

(iii) Being similar to exactly the same individuals implies havingthe same qualities: CR(x) = CR(y) ⇒ f (x) = f (y), and

(iv) Q is minimal in that sense, that no quality can be removedfrom this set, without falsifying at least one of the conditions(i) to (iii),

then f is called a strong quasi-analysis.



Quasi-Analysis


The Example - Continued

Obviously, our example is also a strong quasi-analysis.�Quasi-Analysis

1 Trivially, there are no objects having the same similarityneighbourhood but differing in their qualities - since there arejust no objects having the same similarity neightbourhood.

2 Furthermore,

if one removed the quality Blue, objects 1 and 2 would besimilar to each other without sharing a common quality.They same would hold true with respect to 4 and 5 if Red wereremoved.Finally, this would happen to 2 and 3 if green were removed.



Quasi-Analysis


Extensional Quasi-Analysis

Taking in our example the extensions of the qualities instead of thequalities themselves does not seem to make any difference.This motivates the following:

Definition

Let f : B → ℘(Q) be a quasi-analysis (weak or strong) of thetolerance space T = (B,R). Then define the functionf ∗ : B → SB by f ∗(x) =

def{a∗|x : a, a ∈ Q}

Then we have the following theorem

Theorem

If f : B → ℘(Q) is a quasi-analysis (weak or strong) of thetolerance space T = (B,R), then so is f ∗.

Definition

We shall call a quasi-analysis f : B → ℘(Q) of T = (B,R) anextensional quasi-analysis iff

1 Q ⊆ ℘(B), thus if the qualities are just sets of individuals;

2 x ∈⋂

f (x) for each x ∈ B, i. e., x really possesses all thequalities assigned to it;

3 if x ∈ P ∈ Q, so P ∈ f (x), i. e., each quality which xpossesses is actually assigned to it.



Quasi-Analysis


Introducing Generating Systems

The qualities used by an extensional quasi-analysis (of (B,R))make up an element of the collection SB . Thus, following ournotational conventions, we shall denote such a system of qualitiesby a capital Greek letter: Π :=

⋃x∈B f (x).

We may then reconstruct the relation R by collecting all the pairsfrom the products N × N where N ∈ Π.

That is: R =⋃

N∈Π N × N.



Quasi-Analysis


Weak Quasi-Analysis Reconsidered: The 1st Condition

The first condition in the definition of a weak quasi-analysisguarantees that R ⊆

⋃N∈Π N × N.

For let xRy ,

then - according to this condition - there is a P ∈ Π such thatP ∈ f (x) ∪ f (y).

But, since the quasi-analysis f is supposed to be extensional, wehave x , y ∈ P

and thus (x , y) ∈ P × P.



Quasi-Analysis


Weak Quasi-Analysis Reconsidered: The 2nd Condition

The second condition in the definition of a weak quasi-analysisguarantees that, conversely,

⋃N∈Π N × N ⊆ R.

For let (x , y) ∈⋃

N∈Π N × N,

then there is a P ∈ Π such that x , y ∈ P.

Since the quasi-analysis f is supposed to be extensional, we haveboth P ∈ f (x) and P ∈ f (y)

and thus xRy according to the second clause in the definition of aquasi-analysis.



Quasi-Analysis


Generating Systems Explained

Definition

We shal call a system Π ∈ SB a generating system for R ∈ RB (oran R-generating system) iff R may be reconstrcuted from Π in themanner explained - thus iff

R =⋃

N∈Π

N × N.



Quasi-Analysis


From Generating Systems to Quasi-Analyses

Let now Π be a generating system for the relation R ∈ RB anddefine fΠ : B → SB by

fΠ(x) =def

{N ∈ PB |x ∈ N}.

Then fΠ is a weak extensional quasi-analysis for (B,R).By its very definition fΠ fulfils the condition Q ∈ fΠ(x) ⇔ x ∈ Q(for all Q ∈ Π).

Assume, furthermore, that xRy . Since Π is R-generating, theremust be an N ∈ Π such that (x , y) ∈ N × N. Hence x , y ∈ N andthus N ∈ f (x) ∩ f (y) 6= ∅.

If, converseley, we have f (x) ∩ f (y) 6= ∅, then there must be anN ∈ Π such that both x and y belong to N. Hence (x , y) ∈ N ×Nand, since Π generates R, thus xRy .



Quasi-Analysis


Generating Systems and Extensional Weak Quasi-Analyses

We have just seen:

Theorem

1 If f : B → ℘(Π) is an extensional weak quasi-analysis of thetolerance space T = (B,T ), then Π is an R-generatingsystem.

2 If Π generates the relation R of the tolerance spaceT = (B,R), then fΠ is an extensional weak quasi-analyses.

Essentially, then, generating systems and extensional weakquasi-analyses are the same. We are therefore going on to studygenerating systems.



Quasi-Analysis

Generating SystemsConsistent SetsMaximal SetsSimilarity-Circles

Consistency and Maximality



Quasi-Analysis


Generating Systems



Quasi-Analysis


Systems Generating Relations

Here is again the definition:

Definition

If for R ∈ RB and Π ∈ SB it holds true that

R =⋃

N∈Π

N × N,

then we call R a relation generated by Π and, conversely, Π anR-generating system.

Π ∈ GR ⇔def

R =⋃

N∈Π N × N



Quasi-Analysis


Some Simple Facts

It is not hard to see that:

each system Π ∈ SB generates at least one relation R ∈ RB ;

no system Π ∈ SB generates more than one relation fromR ∈ RB ;

each relation generated by a system is a similarity relation;

each R ∈ R+B is generated by a Π ∈ SB ; for instance by

{{x , y}|(x , y) ∈ R}.



Quasi-Analysis


The π-Function

Thus we have the following result:

Theorem

The function π defined by

π(Π) =def

⋃N∈Π

N × N

is a mapping of SB onto the set R+B .



Quasi-Analysis


Further Facts

Furthermore:

Σ,Π ∈ GR → Σ ∪ Γ ∈ GR : The class of R-generating systemsis closed under unions

and thus it is a (join-) semilattice (with lattice-ordering ⊆).



Quasi-Analysis


Equivalent Systems

Definition

Let’s call two systems Π,Σ ∈ SB equivalent iff they generate thesame relation, i. e., iff π(Π) = π(Σ).

Π ≡ Σ ⇔def

π(Π) = π(Σ)

∗ ∗ ∗

Obviously, ≡ is an equivalence relation dissecting the set SB intomutually disjunctive classes of systems GR , GS , ... generating thesame relation R, resp. S , resp. ....



Quasi-Analysis


Similarity Relations and Their Generating Systems

ΠΣ

R

π π

Generating Systems

Symmetric Relations: R+B

Σ ≡ Π

GR



Quasi-Analysis


Consistent Sets



Quasi-Analysis


Convention

Convention:

In the following we shall always assume that

all relations, if not explicitly stated otherwise, are from R+B

and have field B.



Quasi-Analysis


R-consistent Sets

Definition

We say that N ∈ PB is R-consistent iff N × N ⊆ R.

We introduce the following abbreviation for the totality of allR-consistent sets:

ΓR =def

{N ∈ PB |B × B ⊆ R}.



Quasi-Analysis


Properties of ΓR

The set ΓR has the following properties:

∅ ∈ ΓR ;

{x} ∈ ΓR for each x ∈ B;

for (x , y) ∈ R we have {x , y} ∈ ΓR ;

if N ⊆ P and P ∈ ΓR , so also N ∈ ΓR ;

if R ⊆ S , so ΓR ⊆ ΓS ;

if for Π ⊆ ΓR and for N,P ∈ Π either N ⊆ P or P ⊆ N, so⋃Π ∈ ΓR .



Quasi-Analysis


Further Facts

It holds true that

ΓR ∈ GR : The conistent sets build up a generating system.

Π ∈ GR → Π ⊆ ΓR : Generating systems exclusively containconsistent sets.

Thus ΓR is the top-element of the semilattice GR .



Quasi-Analysis


Maximal Sets



Quasi-Analysis


R-maximal Sets

Definition

Let R ∈ RB . Then we say that N ∈ PB is R-maximal iff for everyQ ∈ PB Q × N ⊆ R implies Q ⊆ R.

We introduce the following abbreviation for the totality of allR-maximal sets:

∆R =def

{N ∈ PB |∀Q ∈ PB .[Q × N ⊆ R → Q ⊆ N]}.



Quasi-Analysis


m-Generating Systems

Definition

We say that Γ ∈ GR m-generates R if it exclusively containsmaximal sets.

Again we introduce an abbreviation for the class of systemsm-generating a relation.

G∆R =

def{Π ∈ GR |Π ⊆ ∆R}.



Quasi-Analysis


Similarity-Circles



Quasi-Analysis


Similarity-Circles

Definition

1 By a similarity circle (with respect to an R ∈ R+B ) we

understand a set which is both consistent and maximal (withrespect to R). Again we introduce an abbreviation for this:ΘR =

defΓR ∩∆R .

2 A system Γ ∈ GR sc-generates R iff it exclusively containssimilarity circles. GΘ

R =def

{Π ∈ GR |Π ⊆ ΘR} is the class of

systems which sc-generate R.

3 For an x ∈ B, we define [x ]R =def

{P ∈ ΘB |x ∈ P}. (Thus

[ ]R : B → SB .



Quasi-Analysis


Two Theorems

Theorem

There is no difference between the two modes of generation:G∆

R = GΘR

Theorem

N ∈ ΘR iff for all Q ∈ PB : Q × N ⊆ R iff Q ⊆ N.



Quasi-Analysis

The Representation TheoremConstructing the Similarity CirclesQuestions of Cardinality

Quasi-Analysis



Quasi-Analysis


The Representation Theorem



Quasi-Analysis


Carnapian Quasi-Analysis

Theorem

The similarity circles build up a generating system: ΘR ∈ GR .

We shall prove (essentially) this under the label of the“Representation Theorem”.

Note that GΘR - like its superset GR - is a (join) semilattice. ΘR is

the top element of this semilattice.



Quasi-Analysis


Aim and Strategy

We want to prove now the representation theorem according towhich each similarity relation may be represented as generated by asystem of properties.

Stated more formally, we want to prove that for each R ∈ R+B :

GR 6= ∅.

Since we know that the similarity circles build up a generatingsystem, it suffices to prove that each pair of similar elements isincluded in a similarity circle in order to prove the representationtheorem.



Quasi-Analysis


The Maximalization Lemma

Lemma

For each N ∈ ΓR there is a P ∈ ΘR such that N ⊆ P.



Quasi-Analysis


Proving the Lemma

1 We look at Π = {Q ∈ ΓB |N ⊆ Q}. This class is non-emptysince it contains at least N itself.

2 We observe that Π is partially ordered by set-inclusion (⊆).

3 We know already that every chain of the system Π has amaximal element in Π.

4 Thus we conclude by Zorn’s Lemma that the entire Π has amaximal element, too. This is the wanted P.

5 By its construction P is consistent and includes N as assubset.

6 If, furthermore, P were not maximal, there would be a Q withan x ∈ Q such that both Q × P ⊆ R but x ∈ Q \ P.

7 But then P ∪ {x} would be a proper consistent superset of Palso including N as a subset - which is impossible.



Quasi-Analysis


The Representation Theorem

Representation Theorem

For each R ∈ R+R : GR 6= ∅. Specifically ΘR ∈ GR and thus, a

fortiori, also GΘR 6= ∅.



Quasi-Analysis


Proving the Representation Theorem

1 Since each N ∈ ΘR is consistent, we clearly have⋃N∈ΘR

N × N ⊆ R. It thus only remains to prove the inverseR ⊆

⋃N∈ΘR

N × N.

2 So let (x , y) ∈ R. We shall prove that also(x , y) ∈

⋃N∈ΘR

N × N.

3 We know that {x , y} ∈ ΓR . Thus, according to theMaximalization Lemma, there is a P ∈ ΘR with {x , y} ⊆ P.

4 But then we have (x , y) ∈ P × P ⊆⋃

N∈ΘRN × N.



Quasi-Analysis


Circles and Neighbourhoods

We have the following simple

Theorem

CR(x) = CR(y) ⇔ [x ]R = [y ]R .

Proof (⇒):

Assume CR(x) = CR(y). We have to show then that [x ]R = [y ]R .

Now let P ∈ [x ]R . Since x ∈ P and P × P ⊆ R (since P isR-consistent), we have P × {x} ⊆ R and thus P ⊆ CR(x). Hence,since x and y have the same similarity neighbourhood, P ⊆ CR(y).So {y} × P ⊆ R, which implies - by the R-maximality of P -y ∈ P and thus P ∈ [y ]R . This shows [x ]R ⊆ [y ]R . The converseof this is proved analogously.Proof (⇐):

Now suppose [x ]R = [y ]R . We have to show that CR(x) = CR(y).

So assume z ∈ CR(x), which means just zRx . By therepresentation theorem there is a P ∈ ΘR such that {x , z} ⊆ P.Clearly, P ∈ [x ]R and thus by the assumption P ∈ [y ]R , too. Butthen {y , z} ∈ P and thus by the consistency of P yRz and soz ∈ CR(y). This proofs CR(x) ⊆ CR(y); the converse is againproofed analogously.



Quasi-Analysis


Similarity Circles and Extensional Strong Quasi-Analysis

If the qualities of a strong quasi-analysis f : B → ℘(Π) of (B,R)are similarity circles - and thus Π ∈ GΘ

R -, then the third clause inthe definition of a strong quasi-analysis is automatically fulfilled(and thus redundand).



Quasi-Analysis


Constructing the Similarity Cricles



Quasi-Analysis


Restrictions of Relations

Definition

The restriction of a relation R to a subdomain B ′ ⊆ B is definedby R � B ′ =

defR ∩ (B ′ × B ′).



Quasi-Analysis


Some Auxiliary Lemmas

Lemma

1 If S = R � B ′ and N ∈ ΓR , so N ∩ B ′ ∈ ΓS . Back

2 If S = R � B ′, N ∈ ΘR and N ⊆ B ′, so N ∈ ΘS . Back

3 If N ∈ ΘR . A ⊆ N, B ′ = CR(A) \ A and S = R � B ′, thenN \ A ∈ ΘS .



Quasi-Analysis


Compatibility Sets

Definition

CR(a) =def

⋃(x ,a)∈R{x}

CR(A) =def

⋂a∈A CR(a)



Quasi-Analysis


Preparing the Procedure

Assume that B is finite: B = {x1, x2, ..., xn}.

Set Θ1 =def

{{x1}}.

Furthermore let for i = 1, 2, ..., n:

Bi =def

{x1, ..., xi},

Ri =def

R � Bi , Li =def

CRi(xi ),

Γi =def

ΓRi, Θi =

defΘRi

.



Quasi-Analysis


... And Carrying It out

Let Θi = {P1, ....,Pk}. Then set

P ′j =

def(Pj ∩ Li+1) ∪ {xi+1} for j = 1, ...k;

Π′ = {P ′1, ...,P

′k};

Π∗ =def

Θi ∪ Π′;

Π =def

{N ∈ Π∗|¬∃Q ∈ Π∗.N & Q}

It is asserted: Π = Θi+1



Quasi-Analysis


Example

�Example �Step2 �Step3 �Step4

Start:

P1 := {1}Θ1 := {P1}L2 = {1, 2}

Step i = 1:

P ′1 = (P1 ∩ L2) ∪ {2} = {1, 2}

Π′ = {P ′1}

Π∗ = Θ1 ∪ Π′ = {{1}, {1, 2}}Θ2 = {{1, 2}}



Quasi-Analysis


Example: Step 2

�Example �Step 3

P1 = {1, 2}Θ2 = {P1}L3 = {2, 3}

Step i = 2:

P ′1 = (P1 ∩ L3) ∪ {3} = {2, 3}

Π′ = {P ′1}

Π∗ = Θ2 ∪ Π′ = {{1, 2}, {2, 3}}Θ3 = {{1, 2}, {2, 3}}



Quasi-Analysis


Example: Step 3

�Example �Step 4

P1 = {1, 2}, P2 = {2, 3}Θ3 = {P1,P2}L4 = {1, 2, 4}

Step i = 2:

P ′1 = (P1 ∩ L4) ∪ {4} = {1, 2, 4}

P ′2 = (P2 ∩ L4) ∪ {4} = {2, 4}

Π′ = {P ′1,P

′2}

Π∗ = Θ2 ∪ Π′ = {{1, 2}, {2, 3}, {2, 4}, {1, 2, 4}}Θ4 = {{1, 2, 4}, {2, 3}}



Quasi-Analysis


Example: Step 4

�Example

P1 = {1, 2, 4}, P2 = {2, 3}Θ4 = {P1,P2}L5 = {2, 4, 5}

Step i = 2:

P ′1 = (P1 ∩ L5) ∪ {5} = {2, 4, 5}

P ′2 = (P2 ∩ L5) ∪ {5} = {2, 5}

Π′ = {P ′1}

Π∗ = Θ2 ∪ Π′ = {{1, 2, 4}, {2, 3}, {2, 4, 5}, {2, 5}}Θ5 = {{1, 2, 4}, {2, 4, 5}, {2, 3}}



Quasi-Analysis


Correctness of the Algorithm

We want to show now the that the Brockhaus-algorithm is correct.This follows from two results:

B1: Π∗ ⊆ Γi+1;

B2: Θi+1 ⊆ Π∗.



Quasi-Analysis


Proof of Correctness: Basis of the Induction

The proof is by an inductive argument over the number n ofindividuals.

If there is only one individual, then the algorithm stops imediately(after 0 steps) and yields the obviously correct result:Θ1 = {{x1}}.

Now assume n > 1 and that the algorithm works correctly forn − 1 individuals.



Quasi-Analysis


Proof of Correctness I

We show Π ⊆ Θi+1:

Let N ∈ Π. So N ∈ Π∗, too. Furthermore then N ∈ Γi+1 by B1.Thus, according to the maximalization lemma there is a P ∈ Θi+1

such that N ⊆ P.

But then P ∈ Π∗ according to B2 above.

According to its construction, Π contains, however, only suchelements which have no proper extension in Π∗. Thus P = N andtherefore N ∈ Θi+1.



Quasi-Analysis


Proof of Correctness II

Now we show, conversely, that Θi+1 ⊆ Π.

Let N ∈ Θi+1. Then we have N ∈ Π∗ according to B2. Because ofthe maximality of similarity circles, there is no Q ∈ Θi+1 such thatN $ Q.

But then there is neither such a Q ∈ Π∗ since each element of thisset is according to B1 and the maximalization lemma a subset ofan element of Θi+1.

Thus N turns out to be a maximal element of Π∗ and so N ∈ Π.



Quasi-Analysis


Prooving B1

Per constructionem Π∗ = Θi ∪ {P ′1, ...,P

′m}.

Pj ∈ Γi+1 (for each Pj ∈ Θi ) and also P ′j ∈ Γi+1 for every one of

the newly calculated P ′j . Thus we have Π∗ ⊆ Γi+1.



Quasi-Analysis


Proving B2

Assume N ∈ Θi+1. The argument procedes by cases:

Case 1: xi+1 6∈ N. By our second Lemma on restricted relations wehave N ∈ Θi and thus N ∈ Π∗ by construction of this set.

Case2: xi+1 ∈ N. Then we have N \ {xi+1} ∈ Γi according to thefirst Lemma on restricted relations. According to the maximalizationlemma, then, there is a Q ∈ Θi such that N \ {xi+1} ⊆ Q.Furthermore, it must hold that (Q \ (N \ {xi+1})) ∩ Li+1 = ∅, forelse there would be a contradiction with the maximality of N(N ∈ ∆i+1). Thus N = (Q ∩ Li+1) ∪ {xi+1}. Since Q ∈ Θi thismeans that N ∈ Π′ ⊆ Π∗.



Quasi-Analysis


Θi+1 ⊆ Π

Let N ∈ Θi+1. There are two possibilities: Either xi+1 6∈ N orxi+1 ∈ N.Case xi+1 6∈ N:First we notice that, obviously,



Quasi-Analysis


Questions of Cardinality



Quasi-Analysis


Similarity Circles: How Many?

Our next aim is to determine the maximal number of similaritycircles a similarity relation might have. In order to do this, we firstintroduce the concept of a dissection of a relation.

Definition

{Ri}i∈I is a dissection of R ∈ RB iff for each index i Ri = R � Bi

and for the family {Bi}i∈I it holds true that

1 it is a dissection of the field B of R (i. e., it is a family ofmutually disjoint sets whose union is B),

2 and Bi × Bj ⊆ R for all i 6= j .



Quasi-Analysis


Dissectible Relations

Definition

1 An R ∈ RB is undissectible if there is no dissection of R withat least two elements.

2 A dissection of R ∈ RB is a finest dissection iff it consistsexclusively of undissectible subrelations of R.



Quasi-Analysis


Dissecting Relations

Theorem

1 Each R ∈ RB has at most one finest dissection.

2 Let R ∈ RB be the relation complementary to R ∈ RB andlet {Ci}i∈I be the elements of B/(R)∗ , then {R � Ci}i∈I is a(actually, in view of (1) the) finest dissection of R.

3 R ∈ RB is indissectible iff (R)∗ = B × B.



Quasi-Analysis


Dissections and Similarity Circles

Theorem

1 Let {Ri}i∈I a dissection of R ∈ R+B . Then N ∈ ΘR iff

N =⋃

i∈I Ni for a familiy {Ni}i∈I of sets such that Ni ∈ ΘRi.

2 If {Ri}i∈I a dissection of R ∈ R+B , then the cardinality of ΘR

equals the product of the cardinalities of all ΘRi.



Quasi-Analysis


Two Types of Indissectible Relations

We now distinguish two types of indissectible relations.

Definition

An indissectible relation R ∈ RB belongs to type I iff it fulfils bothof the following conditions (a) and (b). If it fulfils only the firstcondition (a) (but not (b)), it belongs to type II.

(a) For each x ∈ B there are at most two y ∈ B such that xRy .

(b) There is an x ∈ B such that xRy for at most one y ∈ B.



Quasi-Analysis


Depicting the Two Types

Type I Type II


the system of carnap's 1aufbau - osnabrueck/germanygraduate/aufbau/systemprint.pdf · goal and...

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