**Introduction:**

Yesterday I wrote about the crisis (<- link to 152 Nederlogs on the subject since
September 1, 2008); today about one of the most fundamental human
capacities, namely the capacity of - symbolical - **representing**
arbitrary real, imaginary and impossible
things, that is
in the present Nederlog represented by four entries in my Philosphical
Dictionary.

I believe that the capacity of representing and its adequate
understanding are fundamental for understanding what it is to be human,
and indeed for understanding what human understanding is
(representation), and I believe that I present something that is in
part original, while I know the formalities will not make it more
popular, but these does get explicated in the remarks that follow, and
in my
Philosophical Dictionary.

The reasons to quote this today are that my eyes are not too well, nor
am I, and I wanted to change the subject, while this is at hand and
important.

Three good books that deal with aspects of representing, of which at
least two are not as well known as they ought to be are Ogden &
Richard's "*The meaning of meaning*"; Engell's "*The creative
imagination*" and Tough's "*The development of meaning*". [1]

Most of what follows should be mostly self-explanatory if you follow
the links, that are all to the Philosphical
Dictionary. And there is an endnote.

**1. On representing**** **** **

**Representing**: Something A
represents something B if and only if the properties, relations and elements of A are
systematically correlated with the properties, relations and elements
of B in such a way that - some of - the latter can be inferred from
the former for those who know the correlation.
This seems to be a uniquely
human abillity in so far as it depends on the human ability to reason with symbols. In logic and mathematics
relations that represent are isomorphisms or morphisms.

The idea that something A **represents**
something B is very fundamental and occurs in many forms.

An often useful instance of
representing is a map
that represents some territory.

See: Formalizations
of representing, Map .

**2.** **On maps**

**Map:**
Representation
of some features and relations in
some territory; in mathematics: function with
specified domain and range. A.k.a. **mapping**.

The ideas of a **map**
and the closely related **mapping** are very fundamental, and are
somehow involved in much or all of human cognition and understanding -
which after all is based on the making of mental maps or models of
things.

The first definition that
is given is from the use of "map" in cartography and the second from
mathematics, but both are related, and mappings can be seen as
mathematical abstractions from maps.

**1. maps: **It is
important to understand that one of the important points of maps (that
also applies to mappings) is that they leave out - abstract from, do
not depict - many things that are in the territory (or set) it
represents. More generally, the following points about maps are
important:

· the map is usually
not the territory (even if it is part of it)

·
the map does
usually not represent all of the territory but only certain kinds of
things occurring in the territory, in certain kinds of relations

·
the map
usually contains legenda and other instructions to interpret it

·
the map
usually contains a lot of what is effectively interpunction

·
maps are on
carriers (paper, screen, rock, sand)

·
the map
embodies one of several different possible ways of representing the
things it does

·
the map
usually is partial, incomplete and dated - *and *

· having a map is usually better than having no map at all to understand
the territory the map is about
(supposing the map represents some truth)

·
maps may
represent non-existing territories and include guesses and declarations
to the effect "this is uncharted territory"

It may be well to add some
brief comments and explanations to these points

**Maps and territories**:
In the case of paper maps, the general point of having a map is that it
charts aspects of some territory (which can be seen as a set of things with properties in relations, but
that is not relevant in the present context).

Thus, generally a map only
represents certain aspects of the territory it charts, and usually
contains helpful material on the map to assist a user to relate it
properly to what it charts.

And maps may be partially
mistaken or may be outdated and still be helpful to find one's way
around the territory it charts, while it also is often helpful if the
map explicitly shows what is guessed or unknown in it.

**2. mappings:** In
mathematics, the usage of the terms "map" and "function" is not
precisely regulated, but one useful way to relate them and keep them
apart is to stipulate that a function is a
set of pairs of which each first member is paired to just one second
member, and a map is a function of which also the sets from which the
first and second members are selected are specified. (These sets are
known respectively as domain and range, or source and target. See: Function.)

Note that for both
functions and maps the rule or rules by which the first members in the
pairs in the functions and maps need not be known or, if it is known,
need not be explicitly given. Of course, if such a rule is known it may
be very useful and all that may need to be listed to describe the
function or map.

Here are some useful
notations and definitions, that presume to some extent standard set
theory. It is assumed that the relations, functions and maps spoken of
are binary or two-termed (which is no principal restriction, since a
relation involving n terms can be seen as pair of n-1 terms and the
n-term). In what follows "e" = "is a member of":

A relation R is a
set of pairs.

A function f
is a relation such that

(x)(y)(z)((x,y) e f & (x,z) e f --> y=z).

A **map** m is a function f such that

(EA)(EB)(x)(y)((x,y) e f --> xeA & yeB).

That m is a map from A to B is also written as:

"m : A |-> B" which is in words: "m maps A to B".

There are several ways in
which such mappings can hold, and I state some with the usual wordings:

m is a **partial**
map of A to B:

m : A |-> B and not all xeA are mapped to some
yeB.

m is a **full **map of A to B:

m is a map of A to B and not partial.

m is a map of A **into** B:

m : A |-> B and not all yeB are mapped to some xeA.

m is a map of A **onto **B:

m : A |-> B and not into.

One reason to have partial
maps (and functions: the same terminology given for maps holds for
functions) is that there may well be exceptional cases for some items
in A. Thus, if m maps numbers to numbers using 1/n the case n=0 must be
excluded.

**3.**** On meaning**

**Meaning:**
What a term, statement, symbol, gesture or sign refers to or represents.

This is a very important
notion, and not
easy to explain
well. Two good explanations in book form are **Ogden & Richard**'s
"**The meaning of meaning**" and **Lyons**' "**Semantics**".

One important **ambiguity**
about the
term "meaning" should be noticed to start with, since it vitiated quite
a lot of analytical philosophy. It is this: By the **meaning** of a
term of statement, such as "elephant", one may refer to either some of
the ideas or concepts people
may have about elephants, such as mental pictures or criterions by
which to recognize elephants, or some real elephant(s). The same goes
for statements, like "I saw an elephant in the zoo".

A way of keeping these
apart is to write
"elephant" for the term, 'elephant' for the idea or concept, and
elephant - without quotation marks of any kind - for the thing one calls by
the term "elephant" and may remember by the idea 'elephant'.

Unless I say so explicitly,
in general I
will use the term "**meaning**" to refer to **ideas or concepts** one
may have, rather than the entities one's ideas or concepts represent in
some reality,
if only because often one does not know whether there
really is something as one means by the term, and because one must be
able to understand the meaning of a term or statement before one can
sensibly make up one's mind whether one believes it to represent
anything real (in the sense of "real" one uses).

**4. On formalization of
representing. **

**Formalizations of r****epresenting**: There are
many formalizations of the fundamental idea of ** representing**,
that may be defined as
Something A represents
something B if and only if the properties, relations and elements of A are
systematically correlated with the properties, relations and elements
of B in such a way that - some of - the latter can be inferred from the
former for those who know the correlation.

Now let's try to make this
more precise and general with the help of set theory:

Suppose $ is a society, I is a
set of ideas, D
is a domain,
and L is a language. If S is a set, S* indicates its powerset. A
language is here identified with its set of terms, and it is
presumed the language contains predicates and subjects.

Also, a domain or universe
of discourse consists of anything one may have ideas about, whether
real or unreal, true or false, or containing much or little.

Then I define four variants of "In society $ function i helps to
**represent** domain D by
ideas I" using a language L, where the function is the above
correlation. I first list them with minimal explanations, and then give
some comments.

**Representing domains by
ideas:**

r($, i, D, I) IFF i : D*
|-> I* &

(ae$)(I inc a) &

(deD)(D_{j} inc D) (deD_{j} iff i(d) e i(D_{j}))

In words:

In society $ function i
helps to **represent** domain D by ideas I iff i maps the powerset
of D onto the powerset of I and I is included in every member a of $
and for everything d in D and every subset D_{j} of D, d is an
element of D_{j} iff the i of d is an element of the i of D_{j}.

Clearly, the fundamental
point of the definition is the equivalence in the last conjunct, that
relates statements about the domain with statements about ideas about
the domain.

Note that here and in the
later definitions set theory is used to define the notion of
representing in various forms, so that in effect the notion of
representing is represented set-theoretically, and that this involves
an assumption to the effect that the domain D and the ideas I are
fairly considered as sets
or classes of
things.

Also note that the notion
of ** powerset**
is used to make sure that all the possible distinctions that can be
made set-theoretically can be rendered in the presumed equivalence that
is the kernel of representing. [2]

**Representing ideas by
language:**

r($, j, I, L) IFF j : I*
|-> L &

(ae$)(L inc a & I inc a) &

(xeI)(I_{k} inc I) (xeI_{k} iff (EPeL)(EseL)(j(x)=s
& j(I_{k})=P & Ps) )

The translation is similar
to the one given above, and so is the main point of the definition.

The difference with the
previous definition is that here ideas are correlated with the terms of
a language, that is supposed to have predicates and subjects.

**Representing language by
ideas:**

r($, m, L, I) IFF m : L
|-> I* &

(ae$)(L inc a & I inc a) &

(PeL)(seL)(Ps iff (ExeI)(EI_{k} inc I)(m(s)=x & m(P)=I_{k}
& xeI_{k} ) )

This is the converse of the
previous definition, and may be taken to involve or explicate the
notions of meaning
and linguistic truth: The statement that something called s has a
property called P is - linguistically - true, in effect, if whatever is
meant by s belongs to the set of whatever is meant by P.

The reason to insert
"linguistically " is that even if it is, say, a linguistical and ideal
truth that whales are fishes, this may be false in the domain of facts.
To establish that one needs the converse of the first definition:

**Representing ideas by
domains:**

r($, d, I, D) IFF d : I*
|-> D* &

(ae$)(I inc a) &

(xeI)(I_{k} inc I) ( xeI_{k} iff d(x) e d(I_{k}))

As I remarked, this is the
converse of the first definition and may be taken to involve or
explicate the notions of denotation and
factual truth:
The idea that something x is an I_{k} is true if and only if
whatever x stands for belongs to the set of whatever I_{k}
stands for.

Before extending these
definitions by including the notion of probability, it may be well to
make a few remarks on the terms I introduced.

- Every human being (rare
exceptions excluded, who tend not to acquire a language at all) is
educated in some human society in which
he or she learns some language, and
- every human being may be
credited with having quite a few ideas that are much
like those of other human beings.

This accounts for the
references to a society $ and a language L, about which I will say a
little more below.

- In each definition,
the language L and the ideas I are, where appropriate, asserted to be a
subset of every member of the society $ - which means that all of these
definitions are somewhat idealizing.
- The four functions
introduced - i, j, m, and d - refer to correlations one learns when
learning a language L and what its terms are used to
refer to. It is much harder to explicate precisely what this must be
like than to assume it exists and has the property claimed by the
equivalences.

Finally, one part of the
reason to explicitly refer to a society is that learning a human
language happens in a human society, and another part of the reason to
do so is that then one can make a number of distinctions, assumptions
and definitions that cannot be made without it.

All of the above can be
taken **probabilistically** which then generalizes the above. To do
so it is convenient to introduce the notion of approximate equality as
in "(p(xeX_{i}) ≈ p(f(x)ef(X_{i}))" where "≈" is taken
as "differs no more than e from" i.e. "0 <= | p(xeX_{i}) -
p(f(x)ef(X_{i})) | <= e", and where e is some convenient
small number.

Here e is clearly itself
between 0 and 1, and if it is 1 the asserted approximate identity is
useless since it conveys no information (as the difference between two
probabilities is never larger than 1). However, an advantage of
introducing probabilities is that probabilified propositions
admit of more subtle analyses and distinctions.

The above four definitions
using probability
on the plan just sketched are as follows:

r($, i, D, I,) IFF i : D*
|-> I* &

(ae$)(I inc a) &

(deD)(D_{j} inc D) (p(deD_{j}) ≈ p(i(d) e i(D_{j}))
)

r($, j, I, L) IFF j : I*
|-> L &

(ae$)(L inc a & I inc a) &

(xeI)(I_{k} inc I)(EPeL)(EseL) (p(xeI_{k}) ≈ p(Ps)
& j(x)=s & j(I_{k})=P)

r($, m, L, I) IFF m : L
|-> I* &

(ae$)(L inc a & I inc a) &

(PeL)(seL)(ExeI)(EI_{k} inc I) (p(Ps) ≈ p(xeI_{k})
& m(s)=x & m(P)=I_{k}))

r($, d, I, D) IFF d : I*
|-> D* &

(ae$)(I inc a) &

(xeI)(Ik inc I) ( p(xeI_{k}) ≈ p(d(x) e d(I_{k})) )

Most of what needs to be
said about these definitions has been said when presenting their
non-probabilistic form, and one can see that even if e is rather large,
say 1/2 or 1/4, one may have ideas about the real probabilities of
facts and things that are adequate enough
to help one guide one's decisions.

We can now also use LPA
and express and consider:

(*) (ae$) aB (be$) bB
(Ei)(Ej)(Em)(Ed)(EI)(ED)(EL)

(r($, i, D, I) & r($, j, I, L) & r($, m, L, I) & r($, d, I,
D))

that is:

All members of society
believe that all members of society somehow share ideas about the
representation of domains, ideas and language, and about the meanings and denotations
of terms.

Here the "somehow" refers
to the fact that most members of society would find it hard to specify
by what functions they relate ideas, domains and expressions, even if
they know quite well how to do it - and indeed (*) claims no more than
that the required functions exist.

Also, it may be observed
that in fact these functions are stipulative and symbolic and what
matters are the defining properties that insist on certain kinds of
equivalences (or approximate identities when probabilities are used).

Furthermore, it should be
noted that (*) expresses - even while it attributes to all members of
society $ a belief about all members of society $ - a minimalistic
idea, in that (*) does not imply any agreement on the ideas, language,
domain and functions used to correlate their items between a and b or
any other members of $.

What in fact seems to be
attributed by members of a society, which may be taken as small as a
family in which a toddler is learning a language and trying to reach
some general assumptions about doing so, is the following rather
stronger assumption:

(**) (ae$) aB
(Ei)(Ej)(Em)(Ed)(EI)(ED)(EL) (be$) bB

(r($, i, D, I) & r($, j, I, L) & r($, m, L, I) & r($, d, I,
D))

that is - and here it may
help to think of the society as a family:

All members of society
believe that there is a language, a domain, and a set of ideas with
appropriate functions
such that all members of society somehow
share ideas about the representation of the domain, the ideas and the
language, and about the meanings and denotations
of terms.

For at least to those who
learn the language, it will thus be represented, and indeed in any
society there are many ideas
and experiences
that are - it would naturally
seem - shared by all members of the society.

Also, if one takes the
society $ small enough - say, one's family and friends - both (*) and
(**) will be true, and indeed there will be quite a lot of shared
beliefs that count as presumptive knowledge.

If one takes the society
larger, what one shares with all others in it in terms of beliefs will
be less, but even so all speakers of the same language share many ideas about that
language and what its terms
mean,
while all humans that know some language may be taken also to share
quite a few ideas, namely at least about language in general, and about
human beings in general, and the things all humans must do and know in
order to survive and function as a member of some society.

See also: Fantasy, Imagination, Rationality, Realism,
Reason,
Representing,
Science

**Endnote:**
These and related subjects - representing, meaning, logic and such - I
would like to write (much) more about, but often don't because I need
more
energy to do it well, while there also seem to be very few who are **really
**interested in these subjects, especially if they are
(also) treated with formal methods.

And while I agree it can be done without set theory or logic, I think
one or the other is needed to do it well, and not to get caught by
subtleties.

I spent a lot of time reading about representation and meaning, but
found it is mostly not treated well, according to my understanding -
and perhaps I will be so bold as to try to write out something like
"Basic formal logic and formal philosophy" in Nederlog.

If I do this at all, this is the only way to do it - piecemeal, if and
when I can and feel like it - but I make no promises, and I also have
no illusions this will make Nederlog more popular.

It is something I would like to do if I can, if only to have written
out these fundamental things as clearly as I can. [3]

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