I defined truth as follows:

(*) A statement is true iff what the statement means represents a fact. Accordingly, the truth is whatever exists in reality, whatever is real.

This lemma is given to explaining this formula formally, presupposing some basic logic. The main point of the whole exercise is not its formality, but that using some formalities one can give a fairly full and precise explanation and listing of precisely which assumptions enter here.

What we want to explain are subject-predicate statements of this form

(1) Fa

in which something named a (with the term "a") is asserted to have a property named F (with the term "F").

The present explanation of "truth" differs from standard ones like that on Tarskian lines in insisting that an adequate definition of truth refers to both ideas and reality, but conforms in this respect to Aristotle:

"Spoken words are the symbols of mental experience and written words are the symbols of spoken words. Just as all men have not the same writing, so all men have not the same speech sounds, but the mental experiences, which these directly symbolize, are the same for all, as also are those things of which our experiences are the images.
(..)
As there are in the mind thoughts which do not involve truth and falsity, and also those which must be true or false, so it is in speech. For truth and falsity imply combination and separation. Nouns and verbs, providing nothing is added, are like thoughts without combination or separation; 'man' and 'white', as isolated terms, are not yet either true or false." (p. 7-8, "Aristotle - selections", W.D. Ross Ed.)

These Aristotelian intuitions will be followed below, except that the the mental experiences of different men or of the same man at different times in response to hearing or reading the same term of which they know the meaning are supposed to be similar.

The answer to the question how to define "is true" for a statement like (1) in brief is this:

(2) "Fa" in language L is true in domain D IFF r(d,m,L,D) and d(m("a")) e d(m("F"))

which is in words:

The sequence of terms "Fa" that is a statement in language L is true in domain D precisely if language L represents domain D using functions d (denotation) and m (meaning) - abbreviated: r(d,m,L,D) - and the denotation of the meaning of the term "a" belongs to the denotation of the meaning of the term "F".

Here the quoted part in red is a statement in the language L for which we provide the truth-definition, and the rest of claim (2) and indeed all of it is part of natural language (here: English) enriched with variables and enriched with set theory, which occurs in (2) in green.

The main task now is to explain this notion of representing. Here is a definition - fairly lengthy, but explained below:

r($,d,m,L,I,D) IFF (ae$)(Iea)(T_ieL)(T_jeL)(D_ii inc D)(D_j inc D)
                     (  $ is a society of speakers of language L                &
                        a is a speaker of L                                             &
                        I is the set of ideas of a                                     &
                        T_i is a term of L                                                &
                        T_j is a term of L                                                & 
                        D_i is a subset of D                                             &
                        D_j is a subset of D                                             & 
                        d : Terms of L |-> powerset of I                          &
                        m : powerset of I |-> powerset of D                     &
                        m(~T_i)=powerset of I-m(T_i)                                &
                        m(T_i)=m(T_i & T_j) U m(T_i & ~T_j)                            &
                        d(-D_i)=powerset of D-m(D_i)                                 &
                        d(D_i)=m(D_i O D_j) U m(D_i O -D_j)  ) 

In words this comes to the following, where the powerset of the set is the set of all subsets of the set, and the domain is some set to the elements and subsets whereof the terms of language L refer:

We say that language L used in society $ represents domain D for the speakers of L precisely if

• for every speaker a of L and a's ideas I
• for every term T_i and every term T_j of L
• for every subset D_i and every subset D_j of the domain
• there is a function m (for meaning) that maps the terms of L to the powerset of ideas of a
• there is a function d (for denotation) that maps the powerset of ideas of a to the domain D
  
  and such that
   
• the meaning of the negation of a term is the powerset of I except that part that is the meaning of the term
• the meaning of a term T_i is the union of the meaning of the conjunction of Ti and T_j and the meaning of Ti and the negation of T
• the denotation of the complement of a set is the powerset of D except that part that is the denotation of the term
• the denotation of a set Di is the union of the denotation of the intersection of D_i and D_j and the denotation of the intersection of D_i and the complement of D_j

The first five of these points list the wherewithall for the last four assumptions, which are mostly what matters.

What is to be noted here is especially that - as I define it - truth requires both a domain of  ideas of speakers of language L, that can be rendered as a set, and a domain of things that the terms of L denote, that also can be rendered as a set.

The speakers of L are those who have learned the meanings of the terms of L, and have learned many of these meanings by learning also the denotations of these meanings. Thus, they may know that the term "elephant" represents a large mammal with a trunk, and they may have seen some real elephants, but they also may know that the term "mermaid" represents a creature with the upper body of a woman and the lower body of a fish or dolphin, which they may have seen pictures of, but no such real things in the real world, because mermaids do not exist.

The last four of the above points attribute properties to the functions that were earlier assumed, and these properties are in fact such that negations of terms and complements of sets behave as one intuitively expects (what is not X in domain D is everything in D except what happens to be X in D, for example, and the same for I) and such that conjunctions of terms and intersections of sets also behave as one intuitively expects (what is X in domain D is made up of what is both X and Y in D plus what is both X and not Y in D, and the same for I).

And now we can see the point of the whole notion of representing:

Supposing that for speakers of L domain D is represented by functions m and d, then the claim that "Babbar is an elephant" is true in D precisely if the denotation of the meaning of "Babbar" is an element of the denotation of the meaning of "is an elephant".

Note that as Aristotle insisted in the above quotation, the terms "Babbar" and "is an elephant" both do have a meaning (if they can be understood at all) but by themselves no truth, and may have a denotation (if what they mean represents something real) and thus may be said then to refer to something that exists, yet still without being truths, for truth and falsity arises from the combinations of terms into statements.

And we can now also start giving an explanation why the Dutch statement "Babbar is een olifant" would mean the same as the English statement "Babbar is an elephant": Namely, because "Babbar" in either case is taken to refer to the same thing, and "is een olifant" is taken to refer to precisely the same set as "is an elephant" - and this regardless of whether there really are any elephants. (If Babbar dies and all elephants also are exterminated, still the two statements in the two languages mean the same provided their constituent terms mean the same, as they well may do regardless of the existence of Babbar or of elephants.)

Thus, the present explanation of the notion of truth that involves both ideas and domains undoes a confusion of ideas and domains (e.g. as propounded by Davidson) that derives from a confused understanding of model theory.

Also, it should be mentioned here that the domain is arbitrary as long as the speakers of the language L agree what set of things - what universe of discourse - the language they presently use is supposed to be about.

It makes sometimes sense to say that "In Conan Doyle's fictional England, Sherlock Holmes lived in London" - supposing from the start that what one chooses to speak about is Conan Doyle's fictional England. Thus, one may also insist "In Conan Doyle's fictional England King Arthur did not live in London" - and indeed, both claims in this paragraph gives some information about what Conan Doyle's fiction is about.

Apart from that, usually and mostly natural language (if not used postmodernistically) is supposed to refer to such things as are in the everyday natural reality every human being lives in, and that is best if not fully charted by science.

Finally, it is not difficult to extend the above definition of represents in (3) to one that also allows reckoning with numbers and probabilities. This is sketched in some detail in The measurement of reality by truth and probability, and also in representing formalized.