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.

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.