EDIT : I have accepted an answer for the time being
I have accepted an answer to this question. The answer and subsequent comments exchange has given me some more insight into my question. It does not, however, answer it from the perspective of formal logic. I am still curious about this aspect of my question. Stated more directly: How could Derrida's argument (or the interpretation of it given in the quoted passage below) be analyzed in terms of a logical system?
Ultimately, I am trying to understand why this kind of thinking was so vehemently opposed by Searle and others and would be interested in a response from a different methodology and discipline than Poststructuralism itself.
Since I am unfamiliar with Postmodern philosophy I have been reading Deconstruction: A Reader. In the introduction the author makes a claim that I find to be somewhat dubious. He states the following:
Consider the following universal system:
Imagine a system in which all the system of the world (A to Z) are related. In order for this system to be a system (for it to be systematic) it must be closed. A to Z must be related in clear and predictable ways in order for their relation to be systematic and this means that there must be a discernible limit to the action of the system. Therefore the system must be closed. So, let us call this system of system the universal system and its aim is to relate all known system to one another. There are two consequences from this description. First, if there is indeed a limit to the action of the system, something must exist outside the system. Whenever we draw a limit we are defining what is inside the limit by presupposing an outside of the limit. Thus, if there is something outside the limit of the universal system (and it must have a limit in order to be a system) then the system cannot be 'universal' because it leaves 'something' unaccounted for. Secondly, what is it that the universal system does not account for? Which system lies outside the closed field of systematic relations described by the 'universal system'? The answer is the system itself. The universal system cannot account for itself as a system. To do so would be to recognize the impossibility of maintaing the limit to the universal system and so to admit that it is neither systematic nor univeral. In this way the universal system, far from regulating all systems, would have to admit that no system can be truly systematic because it is not possible to maintain the rigorous purity of a limit. The universal system demonstrates the impossibility of the system per se, and so undoes the logic of a 'properly' maintained inside and outside.
It would seem to me the author is trying to state that a universal system, by definition, has to exclude something in order to maintain its completeness. But why? If it is a universal system (a set of all sets?) then why must there be a "discernible limit to the action of the system". And if the system "is closed" how can this be the case? If the system is universal then it includes everything there is to include.
Further it seems the author's usage of the words limit is more like a boundaries between two zones (the system and not the system) rather than actual limit. So yes, "something must exist outside the system" and "whenever we draw a limit we are defining what is inside the limit by presupposing an outside of the limit". But if this is actually true then by what right do we have to call this system of systems a universal system in the first place?
Furthermore, why can the universal system not account for itself? If this is the missing piece (Derrida's "supplement") then it seems the author is implying that what lies beyond the limits of the universal system is itself. This make no sense to me. In fact, this appears to be a paradox.
It seems to me that the author is making some kind of informal reference to Godel's Incompletness Theorem. I do not have the proper background in the mathematics behind this (or formal logic for that matter) but as I understood it, Godel's theorem applies to the arithmetic portion of axiomatic systems. How does that apply in this case?
And the bigger problem to me is that how does any of this extend to discussions on sociology, language, consciousness, and so forth? Ultimately, the author is going to use this logic to define the relationship between many different kinds of systems (social, political, scientific etc) and do not see how this is possible given the description above.
I would be most interested in a response that answers this question from the perspective of formal logic.