A cognitive system that can be completely described by a formal theory would contain only a fixed and finite amount of information or knowledge. However, a cognitive system can be extensible, i.e. it can reprogram itself, if it is able to take up new information from some source, like the environment, and then use this information in the processing of other information (see Two Conditions of Extensibility). The new information becomes part of the systems active information. In this way, the system can change the way in which it is working.
I suppose that the cognition of human beings is extensible in this way. This insight has consequences for the way language and other sign systems we are using are working. There is a tradition in analytic philosophy, in formal linguistics, in cognitive science and artificial intelligence research to view “natural language” as something that is governed by rules that can be described completely in terms of a formal theory. But if any description of cognitive processes is incomplete because they can be extended by new information, this project cannot be feasible. Any description of how language influences other cognitive processes and how language is influenced by other cognitive processes would be incomplete because both the cognitive processes belonging to the understanding and generation of language itself and the cognitive processes outside of language which interact with the language part of the system would be capable of change.
“Natural” language could turn out to be less natural than this formalistic tradition of linguistics and analytic philosophy of language normally presupposes. It might, to a very high degree, be an entirely cultural phenomenon that does not have much of a specific, innate, genetically programmed basis. There might be some innate structures to start with, but our language structures and our use of language could develop beyond them or leave them behind altogether. Moreover, if language can develop out of the confines of such innate structures then such innate structures would not be required for language to work or to evolve in the first place.
Therefore, I suppose that the formalistic tradition in cognitive science, linguists, artificial intelligence and analytical philosophy, attempting to find a general formal semantics of natural language, does not work. Any formal theory of how language works, any formal semantics of natural language, any attempt to completely define the meaning of “meaning”, is incomplete.
If this is so, the specific formalisms proposed by researchers in this tradition are less fundamental than they seemed to be. Any individual mind might contain structures that can be modeled by such formal systems, but these are always incomplete and they do not form the most fundamental level by which knowledge is represented in the mind. They might differ from person to person, from culture to culture and between different stages of development of the same person. The static formalisms and knowledge representation languages proposed in this tradition might simply be less important scientifically than many people thought them to be. New formalisms or extensions of existing ones can be developed because new information not derivable from the existing formalism can be incorporated into the system. As a result, the semantics of our languages and sign systems are open.
And it is this openness, we could also say “plasticity” or “creativity”, that enables us to adapt our language to new conditions and to develop our culture beyond what is there at any time.
(The picture is from http://commons.wikimedia.org/wiki/File:Discussion.png)
 Just to give a hint what, in my opinion, replaces these special formalisms: Just like AI researchers can use the plasticity or reprogrammability of programming languages to implement new or modified formalisms, the plasticity or reprogrammability of the brain can be used to change the way our brain is working. And one result of this is that we can invent new syntactical devices and that the semantics of the languages and sign systems we are using is open. The approach of the AI-researchers to find “the” structure of cognition and semantics is what is blocking their efforts. The openness, creativity or plasticity must be programmed into the systems themselves. This, however, is a topic beyond the scope of the current article.