The key point, however, is that even in our discussion of the Laws of Thought themselves, we are constantly making assertions - statements that are not themselves provable by means of reason - no matter how we proceed, as in order to do so reason would have to turn and act on itself. We might well be able to construct a ``rational system'' on top of different Laws of Thought. Even if we do make the somewhat arbitrary decision to use the Laws of Thought that do include both the laws of contradiction and excluded middle, the result, expressed in human language, is not unambiguous. There are multiple ways they can be mapped into e.g. set theory or number theory with different (equally ``valid'' and even useful) interpretations of the terms. Some mappings, especially self-referential ones, may well get us into trouble with consistency and completeness, as we'll see later. Ultimately, choosing this particular pair of assertions and defining just what the English words used mean has the effect of defining a particular kind of ``Universe'' and restricting our symbolic reasoning process to that Universe (and possibly to certain classes of super-set-Universes that contain it), for better or worse.
There is one last point we absolutely must make before proceeding to examine set theory per se and explore these ideas in more detail. To each of our human Selves (presuming for a moment that you are not a space alien or very smart canine reading this text) the thing we call ``being'' in English is strictly an experiential state. To each of us things ``are'' only to the extent to which our Self individually and instantaneously is experiencing them.
Our linguistic symbolic reasoning process most often refers, however, to objects that are assumed to have an independent external reality that lives outside of Self. Yet the reasoning process itself manipulates ``things'' that in fact do not exist in that we are not directly experiencing them by means of our external senses. Our assumption of the ``reality'' of these objects and the validity of the relationships we wish to assert between them is based on memory and inference, where memory itself is an experential state in the now that references sementic objects from a presumed past!
Time therefore comes into the rules of logic in a very, very dangerous way, dragged in by the existence of implicit tenses of the verb ``to be'' that is used2.37. Objects have being in English when they possess properties such as temporal persistence. Suddenly we find that our Laws of Thought beg many questions by de facto imposing an implied spatiotemporal geometry (where we might as well grab space as well as time to have a place to put ``Everything'').
It is interesting to note, before we conclude, that the Laws of Thought are the basis for the fundamentally dualistic Western philosophies, ``to be or not to be'' is indeed ``the question'' when we expect that every proposition we might make to be definitely either true or false. Einstein himself wanted that dad-blamed electron to really be at one and only one position, regardless of whether or not we could know what that position was. Yet when we pass from the relatively sterile realm of symbolic reasoning, through the semantic intermediary process of mapping out relationships between symbol and object, to the point where the results ultimately apply to the semiotic conceptual relationships in human affairs expressed in real languages (where we certain wish to be able to use language as a tool for or as the basis of reasoning), we learn that according to the Rules of Real Life contradictory things and paradoxes exist all the time as perfect understandable expressions in language.
For example, we almost certainly know some ``wise fools'' (and might even be one ourselves, or be married to one). It is interesting to note that oxymoronic constructions play a critical role in the koans of Zen, and that Zen logic (such as it is) has recognized the non-dualistic interpretation of ``Nothing'' as ``Not in the Universal Set'' since long, long before the development of formal set theory. These will be quite important in the poetical (not logical) development of the themes (not theorems) of this book.
The existence of wise fools is doubtless related to how absurdly easy it is to argue about nearly anything and sound perfectly reasonable on both sides while doing so (a favorite game of philosophers from the time of Socrates through to the present). We'll have a lot of harmless self-referential fun with oxymorons later.
At this point we're far from done with picking on reason, and will eventually take a look at e.g. ``General Semantics''2.38 and the fundamental problems of trying to reference complex time dependent composite objects in terms of single collective symbols in a reasoning systems, the problem with perception vs a presumed reality, and of course axioms. First, however, let's think about sets. Stripped of annoying features of human language and time-perceptions, it seems like the laws of reasoning are in fact the beginning of, or at least could be formulated within, an abstract set theory. Set theory is pretty cool stuff, so maybe we should take a look at it in more detail...