The main feature that a language must possess is its extensibility. When a human being learns a language, after knowing a set of sentences D together with their meanings, he is able to understand a lot of new sentences that he never has read before, that is to say he extends the set D to a super set D*. Without this extension possibility, human progress hardly would be increased.
Now, let us analyze the structure of this feature. Suppose that two human beings, say John and Thom, have learned a set of sentences D of some language L. Then John writes a sentence S which does not belong to D, therefore it is in an extension D* of D. Since Thom has never read before the sentence S, in order to understand it, he must extend D to a super set D** containing S. Now, we have two extensions D* and D** of the sentence set D. If both D* and D** are not equivalent, then the dialog between John and Thom is not possible, because each of them assigns different meanings to the same sentences.
Language extensions only can be useful if they are equivalent. From an algebraic view point this fact means that both sets D* and D** are generated by D.
Under what conditions D* and D** are equivalent? The answer is in categorical algebra: Under those conditions in which extension is unique up to isomorphisms. These kind of extensions are called universal and are obtained by means of monads. A monad is a triple (F, h,k) where F is an endofunctor sending D into D*, and h and k are natural transformations satisfying some well-kwon conditions. The term "monad" is used, because its paradigm is the concept of free monoid, and languages are partial free-monoids.
In short: To be extensible, languages must be structured as partial free monoids and their extensions must be unique up to isomorphisms.
Identity functor is always a universal extension, but it is of no use, since adds nothing. The optimal extensions are those that being universal are generated by minimal sets of sentences. Thus, a language is optimal if its generation system is minimal. Accordingly, an optimal language can be learned by knowing a minimal set of sentences.
In addition, if the extension is built by means of a monad, it is unique up isomorphisms, that is to say, it is ambiguity free. The problem is easy to understand for those who know categorical algebra. Unfortunately, to build a universal language algebraically requires a lot of work, but its structure it is very easy to be understood. The existence is obvious, since identity functor is a trivial example. It is only a matter of optimization to find some useful structure.
Finally, consider that human thought only can run by means of some language. You are free of equipping your thought with a speedy horse or on an slow donkey.
To build an algebraicaly universal language is a matter of human progress.
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