Reference as Social Interaction

Marc de Graauw


In 1988 I graduated cum laude on a thesis in Philosophy. The original thesis was in Dutch, and covered a historical account of semantics of nouns in Frege, Wittgenstein, Kripke and Putnam, as well as some aspects of evolution and a theory of my own. This is a translation of the original chapter 5, my own account of the semantics of nouns. It has lead me to believe computers will never ever master language in any significant way unless they learn to perceive the outer world and interact with humans. Most of it is a rather straightforward translation of the 1988 work, the Introduction and the Appendix are later additions. Most of the notes and bibliography are omitted out of laziness; they mostly referred to the historical parts of the thesis anyway.

My thesis was never published, though it is in the Nijmegen University library. Maybe it was an original work at the time, but no longer so. Since 1988 similar work has been done, for instance by Jim Hurford and Luc Steels, who has done some excellent experiments with perceiving robots and language. So this is no attempt to claim the ideas expressed here as solely mine, but more a short look into my intellectual past for those interested.


Introducing the Associative Rule
Individual and Social Extension
Properties of the Model
Appendix: Draft of a Formal Account


The relation between words and things is an ancient philosophical question, going back to (at least) Plato and Aristotle, and living on in medieval philosophy. This question received new attention through the work of Frege in the 19th century and has been one of the focal points in philosophy of language ever since. In this introduction I will sketch the background and influences of importance for the rest of this proposal. It is not a historical introduction: I will assume the reader is familiar with the theories presented.

Frege proposed a theory of reference based on the distinction between Sinn (or concept) and Bedeutung (or reference). According to Frege, each word has (or should have) a Sinn, and each Sinn should have a Bedeutung. Ideally, a Sinn should be the same for every speaker of a language. Frege realises quite well that natural language does not fulfil this condition. Besides Sinn and Bedeutung, he also recognises a Vorstellung, which is an individual 'concept':

"Die Vorstellung ist subjektiv: die Vorstellung des einen ist nicht die des anderen ... Die Vorstellung unterscheidet sich dadurch wesentlich von dem Sinne eines Zeichens, welcher gemeinsames Eigentum von vielen sein kann ... denn man wird wohl nicht leugnen können dass die Menschheit einen gemeinsamen Schatz von Gedanken hat ..." (Frege 1892, p. 44)

Frege states the problem quite clearly, but not without invoking new problems. It is not clear, for instance, how mental entities can be shared between individuals.

Wittgenstein pointed out some difficulties with conceptions of reference such as this one in his later work. Most important is his view that words do not have a fixed meaning, but that they have a 'family of characteristics'.

Later a new theory of reference was developed by Kripke, Putnam, Donnellan and others, sometimes called the 'causal theory of reference'. They held that proper names have no concept, but refer directly to an entity that one time was 'baptised' with this proper name. Speakers who were not present at the baptism intend to refer to the item that was baptised with this particular proper name by speakers who were. Speakers are thus connected to the baptism through a 'causal chain' of other speakers. Moreover, Kripke et al. held that this view of reference was not only valid for proper names but also for some other nouns, especially names of natural kinds such as 'tiger' and 'gold'.

I will in the following paragraphs give an informal sketch of a theory of reference which is influenced by all of the above, and which will try to explain how we can have a 'gemeinsamen Schatz von Gedanken' without reverting to a shared mental entity - a Sinn in the Fregean sense.

Introducing the Associative Rule

First I want to introduce a new concept, the associative rule. An associative rule is a mechanism which enables a speaker of a language to say a word that applies to an object when perceiving that object, and enables the speaker to interpret this word when spoken by another speaker. An associative rule is related to a word: for each word one understands one has a corresponding associative rule. For the time being, this associative rule will be a 'black box': is does not matter what mechanism is behind it, it only matters what the input and output are.

Two comments are in place. First, it might seem now I am trying to explain reference by postulating an enigmatic entity, the 'associative rule' which somehow performs reference. However, I will show that this is not the case. Second, one might wonder why I do not simply use 'concept' or 'intension' or a similar well-known term. Well, those words are certainly well-known in semantic circles, and they therefore carry far too many implicit associations. I would rather start with a clean slate, and so use the word 'associative rule', which hopefully will carry none of these associations and will not tempt the reader to make unintended assumptions about them. Particularly I hope to avoid the notion of concept as a bundle of properties, as in 'a tiger is a large black-and-yellow striped carnivore...'. I believe there is an obvious difference between such a concept and an associative rule. When I do not know tigers, and somebody explains a concept such as this to me, I might be able to identify tigers in a zoo, if the concept has been explained well. However, when I am only asked to draw a tiger, it is unlikely my picture will bear a great resemblance to real tigers, not even if I were an accomplished draftsman. And that is one important difference between such concepts and an associative rule. When I possess an associative rule, I will be able to draw a picture of a tiger whose resemblance is only inhibited by my drawing abilities, not my knowledge of what tigers look like.


Figure 1: The associative rule in action.

An associative rule then is what enables a speaker to use a word, both in speaking and in hearing. This associative rule will correspond with some neurological mechanism (or mechanisms, for the relationship is not necessarily one-to-one). The nature of this mechanism is not relevant to the theory to be enfolded. It is however clear that there is something which corresponds to an associative rule, otherwise language use would be impossible. I assume associative rules to be attached to words, i.e. there is one associative rule for 'tree', one for 'cat', one for 'chair' et cetera. I want to limit myself to words that denote observable classes of objects for the moment. Later I will explore other words and the potential relevance of the 'social interaction' theory of reference there. Further, I will only consider classes, not individuals, though I will come back to the issue later. In reality it is quite possible that the word-associative rule relationship is not as simple as sketched here. For instance, it is fair to assume that I have one associative rule for 'tree', 'Baum' and 'boom', and not three distinct associative rules. When learning a new language we do not have to relearn each and every concept we have. I will ignore this distinction however, since it is not relevant. For each language, we can still say that there is an associative rule attached directly or indirectly to each word in that particular language which I master.

One more important assumption I will make is that associative rules can change. This seems reasonable enough, considering the fact that we frequently change the way we use words when we learn a language as a child. It is also reasonable to suppose we do not do this as often anymore when we grow older, but even then we keep learning, and it is quite possible that we learn to denote slightly different things by words we are less familiar with. For instance, I might think that shrimps are not crustaceans. When a marine biologist assures me they are, I have effectively changed my associative rule for crustacean. So even when we grow older, associative rules can change.

To recapitulate, an associative rule belonging to a word is the mechanism which enables a speaker to use that word in relation to an object which fits it, and to understand what is denoted when another speaker uses this word. For brevity, I will often say 'an associative rule' when 'an associative rule belonging to a word' would be more correct.

Individual and Social Extension

The next issue is the denotation of words. Since an associative rules enables speakers to associate words with objects, they are a way to fix a denotation. So to an associative rule there belongs an extension, at least in potential. This extension is the group of objects that a speaker would denote with the particular word belonging to this associative rule if the speaker were presented with those objects and asked about them. The extension thus is an open-ended collection, since tigers that have not been born yet would be denoted by 'tiger' by me if in fact they are born sometime in the future. This open-endedness does not harm the functioning of the associative rule, but for simplicity we may ignore it and act as if the extension as fixed by the associative rule is a finite set of objects. There is also a contextual influence, which may change the objects I denote with 'tiger'. I might tell my son in a particular context of a plush toy tiger 'That is a tiger'. In another context I might say: 'That is not a real tiger', or even, 'That is not a tiger, that is just a toy'. Though important, I will also ignore such contextual influence for the moment and act as if words, through associative rules, do establish well-defined finite sets of objects, independent of the context.

To my associative rule belongs a well-defined finite set of objects, an extension. However, as noted before, associative rules can vary between speakers, and even vary in time for one speaker. We cannot therefore assume that the extension fixed by associative rules will be the same for every speaker. I will call the extension fixed by an associative rule for a particular speaker an 'individual extension', to stress the fact that it is something which belongs to an individual speaker. Individual extensions will differ, slightly for well-known words such as 'tiger', maybe more so for less well-known words such as 'crustacean'. In practice it would be safe to assume that the differences will be small, though not negligible, for accomplished speakers. In an interesting psycholinguistic experiment, Labov (1973) showed there is no identical extension for different speakers for the words 'cup' and 'bowl'. Different people will put the division between cups and bowls on different points of the 'object size' scale. The individual extension can vary in time: strictly speaking, an individual extension will be a finite and well-defined set of objects at a particular point in time. With the individual extension established, it is easy to postulate a social extension as the union of all individual extensions of all speakers of a (particular word of a) language. It is even more useful if the objects in this social extension also carry 'weights' which indicate how many speakers actually use this word for this object.


Figure 2: Individual extensions constitute a social extension. The plus-signs represent objects in the word. Three individual extensions (of a particular word) are projected on this 'world'. The social extension is the union of the three individual extensions with weights assigned to them. The grey areas represent confirmed objects: objects in the social extension with weight greater than zero. The darker grey the area, the stronger the objects within are confirmed in the social extension. Objects in the centre are have a weight of 1.

The social extension is more of an artificial construct than the associative rule and the individual extension, which clearly have counterparts in language use. Nevertheless, it is a useful construct. As has been noted before, associative rules can change. They do change often when children learn language, but also change later in life when we meet experts who show us that some of our assumptions on what is denoted with some term are not correct, or simply because we find out other people use a word different than we do. So, in a sense, the associative rule is changed by the social extension. What actually happens is that we change our associative rules through exposures to items in the individual extensions of other people, not the entire social extension. We do not and cannot see the entire social extension in all but the most limited and artificial contexts (only if the language community is so small and the number of objects a word applies to is also so small that we can know every speaker's complete individual extension). But even when we change our associative rules through exposure to items in other people's individual extensions, it is practical to assume we change them through exposure to the social extension. The items in the individual extensions are in fact a subset of the social extension, and when either we speak to a large number of people or, more realistically, the people we speak to are representative of the language community as a whole, we can safely assume that the subset of the social extension we encounter is very similar to the actual social extension. In this case, the difference between saying we change our associative rules through the social extension or through several individual extensions is not relevant.


Figure 3: Reference as interaction of associative rule, individual extensions and social extension.

To summarise the picture: individuals have associative rules, which determine their individual extensions. Their individual extensions determine the social extension, and in turn the social extension co-determines the associative rules. This is of course a circular account. I do not think this hurts, for it is quite clear that this account needs a time axis. So when we say that associative rules determine (through individual extensions) the social extension and reversely the associative rules are determined by the social extension, the associative rules which determine the social extension and those which are determined by the social extension are not necessarily the same associative rules. For instance, when Agamemnon and Clytemnestra have two associative rules, with two individual extensions, which determine their social extension, this social extension can determine the associative rules of Electra and Orestes, their children, in this mini-society, without any damaging circularity.

Properties of the Model

What properties would such a system of reference have? Without a formal model, a rigorous proof is not possible, but some properties are quite evident. First, such a model could easily explain the communicative role of reference. Since speakers continuously test their associative rules against the social extension and, if necessary, adapt their associative rules to reflect the common opinion, their individual extensions will closely resemble each other, and therefore they would be able to communicate using those words. There is no need to assume their associative rules are the same or even alike, as long as their individual extensions are. Of course it is reasonable to assume that in a community of humans the associative rules will be quite similar, since we share similar brain structures. This is not essential however, and maybe there will one day be a 'language community' of humans and computers who use quite different associative rules to fix their respective individual extensions.

Second, a theory of reference as social interaction could easily explain change of meaning. The change of meaning of words throughout the ages is a well-known phenomenon. Words do not stay the same, neither in pronunciation and spelling nor in meaning. The picture sketched above does imply the possibility of change of meaning, in fact this is an inherent property of the system. When the associative rules of individuals can change, they can and will differ. Since speakers do no have access to the entire social extension, the part of it they do know will differ, and therefore their associative rules can and will differ. Very minor deviations in the reference of a word might propagate throughout a language community. In genetics, a well-known phenomenon is 'genetic drift'. Genetic variations with no survival benefits may fluctuate in a population until one disappears and the other remains. Similarly, we could expect 'semantic drift' to occur, whereby words have two groups of individual extensions, which both make sense to the speakers. One of both, even when small, could through semantic drift become more numerous and take over the entire language community. Beside semantic drift one can also quite easily assume change of meaning where one variant does have communicative advantages over the other. Examples could be the adaptation of language to a changing society and technology.

Third, reference as social interaction can explain stability of meaning. Making a model that explains change of meaning is easy, but when this implies too random or rapid change, the communicative properties of language are lost. When different speakers adapt their associative rules by testing them against the social extension, this will effectively dampen any abrupt or sudden meaning change. So meanings, though subject to change over longer periods, will remain relatively stable for shorter time periods. Of course, when meaning change occurs because there are reasons for it, for example when technological changes cause meaning change, there will be another driver and the dampening might effectively be cancelled.

In this informal sketch the properties of the model can only be given intuitively. A formal model is needed to prove that such a model does indeed have those properties.


When the 'theory of reference as social interaction', is correct, what does it imply? Most important, it would challenge the established philosophical notion that concepts must be something shared, and identical in different people have if we want to explain communication. Frege clearly states that he thinks such a thing is necessary unless we want to deny that humans have a common wealth of ideas. The reasoning is easy: if we do not have concepts which are the same, then we cannot have the same thoughts. We are stuck in a world with purely individual notions, and do not share any common mental heritage. The reasoning is false: if we do not have exactly the same concepts, it is not necessary to assume that they are wildly different. And for communication there is no need for identical shared concepts, all we need to assume is that concepts of different people are sufficiently similar to be able to communicate. If this condition is fulfilled, we will normally be talking about the same things to the degree that possible differences are irrelevant to the discussion underway. Granted, in circumstances the differences may be relevant and hamper effective communication, but this is neither contrary to experience nor the rule (cynics may dispute the last point). The account of reference as social interaction explains precisely why we would expect meanings to be sufficiently similar for communication. We do not therefore need to assume that people literally share mental entities if we want to explain language or communication or a common wealth of ideas.

We now can see the theory does not depend on an enigmatic entity the 'associative rule', which somehow performs reference - or maybe it does, but then it does no harm. The point of the theory is not to explain how any individual does 'reference'. That, in my opinion, is ultimately the realm of psycholinguistics. The point of this theory is to show we do not need to assume people have a shared mental entity, a concept or Sinn, to explain communication and a common wealth of ideas. The nature of the associative rule is not relevant, and can - as far as the validity of this theory is concerned - any mechanism that does or will do the job.

Another consequence in quite another domain is that we cannot expect computers to truly 'understand' language unless they also can perceive the world, or that part of it which is relevant to the part of language they are supposed to understand. According to the theory sketched above, the ability to relate an associative rule to a social extension, and to infer an individual extension from it, are crucial to understanding the meaning of a word, and we cannot do this without perceiving the world. More particular, when this theory is correct, we would expect associative rules to be based to a large degree on perceptual qualities of perceivable objects, and not just on conceptual properties. I.e., it is important to know what a tiger looks like if we want to compare our associative rule to the social extension, not just to know it is a big striped carnivore. I do not want to deny conceptual properties play a role in fixing an individual reference, however I do want to stress that purely a bundle of conceptual properties cannot be sufficient in most cases. When we assume humanity needs shared mental entities, concepts, to communicate, it is more natural to assume these concepts are defined in terms of other concepts, as has been done in different degrees by Frege, Russell, Strawson, Katz and Fodor. When we do not assume there are shared mental entities, any mechanism will do as an associative rule, and in such a case a mechanism at least partly grounded on perceptual qualities seems more natural. Contemporary psycholinguistics also supports such a model, as the focus there has shifted from semantic theories based on concepts to theories centring around prototypes and similarities between objects and those prototypical objects. (However, from psycholinguistics it is also clear that concepts do play a part of their own. I would not want to be seen as propagating the view that it is only perceptual qualities that matter.)

A third consequence is that we may fear too little attention in logic and philosophy of language has been dedicated to the study of reference other than reference of individuals (as in research into uniquely identifying descriptions and proper names, which have received plenty of attention). When we assume concepts to be shared among humans, they are either concepts of individuals (uniquely identifying descriptions), or primitive predicates (as in 'Ta' meaning 'a is a tiger' in predicate logic) or bundles of properties (which can be analysed as: 'Tigers are striped carnivores', which could be ∀x( Tx → Cx & Sx) in a predicate logic). If we do not assume concepts to be shared, things are not that easy to analyse purely in terms of predicate logic anymore. More specific, it becomes more important to know how we actually fix the reference of a predicate, not just what happens when we postulate it corresponds to a set for all speakers of a language. This all of course would not mean predicate logic is false or not useful, just that a part of the puzzle is still missing.

A final consequence of the theory of reference as social interaction would be that potentially a set of testable consequences can be derived from it once it has a suitable formal representation. For instance, the theory would probably predict that individual extensions can be slightly different even when efficient communication is possible. Also it might be possible to relate the rate of change of the meaning of words to variables as the size of the language community or the frequency of use of words. A formal model might predict the occurrence of 'semantic drift' as noted above. Also the theory might predict different characteristics of the social extension for different kinds of words. For instance, it would be natural to expect words which denote things which fall in neatly divided categories such as animals or musical instruments to have social extensions which are clearly delimited: either objects are 'in' with a height weight (most speakers agree that this object is 'in') or objects would be 'out' or 'in' with a very low weight (most speakers agree this object is 'out'). Words which denote things which do not fall into such neatly divided categories such as utensils or jobs might be expected to have a social extension with 'fuzzier borders', to borrow a phrase from Wittgenstein. Issues such as those have been investigated in psycholinguistics, and it would be very interesting to see if findings there agree with a formalisation of this theory. However, without a formalisation, it is not possible to derive testable consequences, and all that can be shown at this point is that such testable consequences might very well appear once a formalisation is made.

Appendix: Draft of a Formal Account

To develop a formal theory of interactive reference, we need a model which greatly simplifies the 'real world' with its many languages, differentiated and changing language communities and very diverse types of possible referents. In any science, scientists make a model starting from some assumptions which simplify reality. Those assumptions may be untrue in reality, but they may very well provide a model which quite accurately describes some processes taking place in the real world.

First, we will assume there is only one language, with a fixed, isolated and homogeneous community of language speakers. With 'fixed' I mean the speakers remain the same through time, with 'isolated' I mean that there are no outside linguistic influences on those speakers. I.e. there is no immigration or emigration, there is no contact with speakers of another language, there are no births or deaths. With 'homogeneous' I mean that all speakers are equivalent in terms of language capability and authority. I.e. there are no young children or foreigners learning the language, and there are also no experts whose opinion on specific subjects weighs heavier than the opinion of others.

Thus we may postulate: a population P consists of j speakers (of a language).

P = {S1, S2, S3, S4, ... Sj }

We will further simplify and assume the language consists purely of nouns which can refer only to tangible objects: the vocabulary V of the language consists of m nouns.

V = {N1, N2, N3, N4, ... Nm}

The world or universe in which those speakers live, is filled with atomic objects (i.e. there are no objects which are made of other component objects). A universe (of discourse) U consists of k objects (or things).

U = {T1, T2, T3, T4, ... Tk}

The speakers of L all have associative rules for each noun in L. An associative rule is a function which can decide whether this noun applies to an object. I.e. the associative rule for the noun 'cat' would enable a speaker to tell whether any object in U is a cat or not. The associative rule is thus a function which yields a truth-value: an object either is a cat or not. If the object is a cat according to the associative rule, the function will yield 'true', if not, 'false'. (We can also say the associative rule is an 'IS_A' function as is often done in information technology.) There are many such functions: one for each word for each speaker. I will therefore superscript those functions to indicate a speaker and noun. I.e. aS1,N4, aS5,N2, aS7,N4 et cetera are all names of functions, namely associative rules of certain speakers and certain nouns.

An associative rule as,n for speaker s and noun n is a function which yields a truth-value for an object t:


For the time being, the associative rule will remain a 'black box': it is nor relevant how a speaker decides whether a noun applies to an object, as long as the speaker is able to decide this.

The individual extension IEs,n of speaker s for noun n is the collection of objects in the universe for which the associative rule of speaker s and noun n yields 'true'.

IEs,n = {t | t ∈ U & as,n(t)}

There is an individual extension for each noun for each speaker. In reality the individual extension of a word, i.e. 'cat' often cannot be established: there are simply to many cats to list them all. However, in theory, given a universe with a finite number of cats and enough time, an individual extension could be made. So, in the real world too, an individual extension is a concept of theoretical relevance.

Now we can postulate the social extension: the social extension SEn of noun n is a set of tuples with objects of the universe and associated weights.

SEn = {< T1, W1 >, < T2, W2 >, < T3, W3 >, < T4, W4 >... < Tk, Wk >}

The weight W of a particular object t in SEn is the sum of all occurrences in individual extensions of all speakers (or, the number of speakers which will say 'yes' when asked whether noun n applies to object t), divided by the number of speakers. For brevity, it is better to include only those things with an associated weight greater than zero, i.e. only those objects for which at least one speaker believes they are categorised under this word, so weights will be in the interval < 0, 1].

For a more formal definition of the social extension, we can postulate a set CEn (the collective extension of n) as the set of all individual extensions of all speakers for noun n:

CEn = {IES1,n, IES2,n, IES3,n, IES4,n.... IEj,n}

Then we can derive the object collection OCt,n of object t and noun n as follows:

OCt,n = {x | x ∈ CEn & t ∈ x}

The object collection OCt,n of noun n and object t is the set of all individual extensions which contain this particular object t. Next we can define it,n as the cardinality of OCt,n:

it,n = |OCt,n|

The weight of object t in the social extension SEn will then be the number of elements in OCt,n divided by the number of speakers j, and the social extension can be defined as:

SEn = {< t, w> | t ∈ U & w = it,n / j }

Now we have a fixed, isolated and homogeneous community of language speakers living in a universe filled with atomic objects. Their language consists of a finite collection of nouns, and every speaker is able to decide for any object and any noun whether this object fits this noun. We have also formally introduced the notion of individual extension and social extension. There is however no mechanism for meaning change or interaction yet.

Learning is a process whereby the individual extension is adjusted through comparison with the social extension. When speaking to each other, the speakers will often use their nouns ostensively: they will see or even point to an object, and refer to that object with their noun. Ostensive use is any use where it is clear for the listening language user which object is referred to. In this simplified model, non-ostensive use is not relevant.

Through interaction, speakers will get to know (parts of) each other's individual extensions. Then it is possible that speakers find out their own use of a noun is not consistent with the way other speakers use it. When confronted with this phenomenon, a speaker will try to adjust her use of the noun so that it better fits the way other speakers use this noun. It does not make sense to have 'private meanings' for words. Language is a social instrument, intended to communicate with others speakers in a sensible, understandable way. (This simplified model of course does not cover poetic or metaphoric use of language where this might not apply so rigorously.)

In general, we may assume that after a number of ostensive communications with other speakers, a speaker will adapt his associative rule so that his individual extension better fits the individual extensions of other speakers.


The new, accented, function a's,n(t) will reflect the learning. This can of course be modelled as a function itself. r is a learning function, which takes as its arguments an associative rule and a social extension, and produces a new associative rule. The social extension is written WE here, to reflect that any weighted extension will do, for reasons shown later. Here: WE = SEn.

a's,n(t) = r( as,n(t), WE )

A border case can be called 'perfect learning', where the whole language community is totally transparent to all users and the associative rule can be adapted in any way desired. In such a border scenario, all speakers will know the social extension and try to adapt their individual extensions to the social extension. For instance, they could decide to change their associative rule so that any object t for which 50% or more of all speakers say that noun n applies to this object, their new associative rule a' will incorporate this object t, and any object u for which less than 50% of all speakers say that noun n applies to this object, their new associative rule a' will not incorporate this object u. If all speakers do this, then meaning change will stop and the system will remain in a state of equilibrium. However, the more interesting cases are those where the language community is not totally transparent, i.e. speakers do not know the exact way every other speaker uses the language. Either they do not know all other speakers, or they do not know for all objects whether a noun applies, or both. Maybe the speakers are also not able to adapt their associative rule in each desired way.

A model which reflects these imperfections would be a model where not every speaker is in contact with every other speaker, and in communication, only a limited number of objects are considered in ostensive language use. After a time period, a speaker can then make a partial social extension PSEs,n of a noun:

PSEs,n = {< T1, W1>, < T2, W2>, < T3, W3>, < T4, W4>... < Tn, Wn>}

The superscript s in PSEs,n reflects the fact that this partial social extension belongs to a particular speaker. Another speaker, having spoken to other partners about other objects will probably have a different partial social extension. The weights in a partial social extension are the number of speakers who have asserted that noun n applies to object t, divided by the number of speakers with whom object t has been spoken about in ostensive communication. In this way we will again yield weights in the interval < 0, 1] reflecting the consensus of the subpopulation involved in establishing the partial social extension. Note that the learning function r will take PSEs,n as an argument.

The actual learning function chosen will of course influence the behaviour of the model. An algorithm could be:

If the weight of an object in the (partial) social extension is above a threshold hmax and this object does not occur in the individual extension, the associative rule is adjusted so that this object will be incorporated in the individual extension.

If the weight of an object in the (partial) social extension is below a threshold hmin and this object does occur in the individual extension, the associative rule is adjusted so that this object will not be incorporated in the individual extension anymore.

This algorithm assumes that the associative rule can be changed in any desirable way.

Obviously more work could be done on the formal account.


Gottlob Frege 1892, Über Sinn und Bedeutung, in: Frege 1980

Gottlob Frege 1980, Funktion, Begriff, Bedeutung, herausg. Patzig, Göttingen

Labov 1973, The boundaries of words and their meanings. in Bailey & Shuy (eds), New ways of analysing variations in English (pp. 340-373) Washington DC: Georgetown University Press