Formalizations of representing:
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 settheoretically, 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 settheoretically can be rendered in
the presumed equivalence that is the kernel of representing.
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 nonprobabilistic 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.
