The Laws of Thought above are essentially classical laws. They describe objects in an essentially unitary way, and embody a perfectly dualistic classification of all the objects to which they can be referred. Things that are, are. Things either are or are not, and must be one or the other (whatever a Thing is and whatever all that ``means'').
For a moment, let us forget the ``game'' of logic and use our senses instead of our sense2.23. Is nature (as an inferred collection of of real Things) really like that? In classical physics the answer seems to be yes, things have a definite state and that state can be (in principle) completely known by means of measurements. I can therefore speak of (say) an electron as an object of definite charge, angular momentum and mass that has a definite existence at a definite position in space at a definite time with a definite velocity. According to the Laws of Thought, a unitary object cannot be at more than one place at one time, and most definitely cannot be ``created'' out of nothingness - in fact this was quite literally one of the first conclusions arrived at by Parmenides and embraced by Aristotle.
Nature, on the other hand, says otherwise. In quantum theory (which I occasionally teach) real ``things'' such as electrons can ``be'' at once a particle with particle-like properties and a wave with wave-like properties. By coupling macroscopic systems to the peculiar way things like electrons are necessarily described, Schrödinger's famous cat in its infernal box can in principle be in a state that must be described as both dead and alive, at least until we open the box2.24 .
Even if the idea of building a box containing an apparatus and cat that is sufficiently decoupled from the random state of the rest of the universe to make this actually work is a bit dicey (I personally do have a few issues with this, as do/did many physicists including for that matter Schrödinger himself) the basic point is still valid at the microscopic level: if one makes any given partitioning into a dualistic set of states that are described by classical coordinates of the electron to whatever degree that you like, one must place the electron into a state of mixed being/not-being in another set of states.2.25
We find that even though the electron is a point particle that can be measured (in principle) to be at a single point in space as accurately as we like2.26, in general we cannot think of the electron as an object that is either at point or at point where , or (perhaps more importantly) reason as if it were. The point-like electron is also wave-like, and can be thought of as ``being'' at both points until one performs a special kind of measurement that localizes it, which makes some other coordinate uncertain. There are two-slit experiments that have been done with actual electrons that perfectly illustrate this point, as well as the even more mysterious Aharonov-Bohm effect2.27 .
In quantum field theory things are even worse, with empty space being perfectly capable of turning into an electron and positron pair for a sufficiently brief period of time to ``polarize the vacuum''. Certain experiments (the same ones that often try to localize an electron by for example bouncing other charged particles off of it) can knock pairs out of the polarized vacuum, creating electrons out of one kind of ``nothing'', as it were. Existence itself (of electrons) turns out to be a wee bit ambiguous from the point of view of the laws of thought, yet these things are all readily observable so that we know that in some sense this ambiguous picture is ``true''.
For many years some of the brightest of physicists rejected the formulation of quantum mechanics even though it worked and nothing else did simply because it appeared to be a description of a system's state that permitted things to ``be and not be'' and hence seemed as if it might lead to the possibility of real paradoxes in our experimental view of nature. However, it was gradually determined that quantum mechanics has its own Laws of Thought that map into the classical Laws in a way that precisely prevents the occurence of real paradoxes, even as it forces us to alter the way we view small objects such as nuclei, atoms and molecules. The quantum Laws simply lack the classical ``sharp'' dualism implicit in the Law of Contradiction and the Law of the Excluded Middle in certain contexts with a ``fuzzy'' sort of dualism between conjugate variable pairs2.28 . Our brains, however, are evolved to form a classical conceptual map, and have a very hard time understanding how a single particle can pass through two slits at once or have properties that measurably change if we change a potential in a region where the particle never visits.
Quantum theory can therefore easily appear to be illogical to an untrained observer (and of course it is, from a classical point of view) but the lesson we learn from its success and from mathematical investigations of things such as curved space geometry and alternative logical systems is that the Laws of Thought above, however useful in certain contexts, may not in fact be ``universal self-evident truths''.
Don't misunderstand me - the Laws of Thought are rules that work well as the basis of a system of reasonably consistent and complete reasoning that is a posteriori relevant to much of our experience. They might, however, not be the only such set of rules, and (as we'll discuss at great length later in this work) the phrase ``reasonably consistent and complete'' (as opposed to absolutely consistent and complete) was very, very deliberately chosen. Therefore, even if we accept them as a provisional basis for reasoning (as we will throughout most of this work) we need to carefully consider the possibility of there existing alternative systems of reason all the way down to the Laws of Thought themselves, and indeed try to determine an even more abstract way of encapsulating the foundation of a system of reason.
A final thing to carefully note for later before moving on is that these Laws seem only to function in that peculiar sixth realm of our sensory experience, our memory, imagination, and other Self-generated interior monologue. Reason requires an object, where ``things'' of our experience just ``are''. Hmmmm.
The process of critically examining the very basis of how we ``think'' rationally is something to take very seriously. Reason is so much a part of our everyday lives that it cannot hurt to turn reason back upon itself (a process that we should expect to lead to some ``odd'' results and paradoxes, as self-referentiality leads to very strange results) and try to understand both how it works and its limitations. Human conflict is all about disagreements and somehow we think that if only we used Spock-like logic2.29 there could be no disagreement, no contention, no violence, no war. We expect reason to be able to provide answers to those SUW-class questions that arise in every human heart, and we expect philosophers to ``do the job we've paid them to do'' for a few thousand years and come up with those answers and communicate them in a form we can understand and agree on. After all, we have plenty of folks selling various forms of snake oil, be it religious dogma or political tripe, who are all too happy to fill in the missing pieces to our great collective misfortune.
A significant part of this book is devoted to trying to convince you2.30 that pure reason is a subtly flawed tool, especially when applied uncritically to the ``big questions'' of the last chapter. Reason is great if you are a physicist or computer programmer or mathematician and work from the right set of axioms, postulates, premises, definitions, to desired conditional conclusions that may or may not empirically seem to apply to the real world for reasons that we cannot absolutely prove or fundamentally understand.
Reason, as one can easily see, requires a presumed mapping between ``objects of our sensory experience'' (including meta-objects as symbols in our imagination) and ``meta-objects as symbols in our imagination'' that can never be justified by reason itself and is inevitably self-referential. However rigorous and powerful a structure reason erects on whatever foundation of assumptions one (literally) dreams up, reason itself can never provide or justify that foundation (or do the dreaming!), and self-referentiality leads to its own set of problems as we shall soon see in considerable detail.
With this as a motivation, let's take a closer look at the classical Laws of Thought listed above and think about the details of what they say and try to make sense of them in English2.31 as all too often we have discovered that statements in human language are not sufficiently precise to serve as a basis for the development of either math or science. Note well that I used the phrase semantic assertions to describe at least two of these laws. Without wishing to get drawn into the morass of semantic terminology2.32 2.33 - semantics is about meaning, and is more often than not associated with ``what symbolic objects represent'' distinct from the class relationships that are asserted to exist between the symbols as representatives of those objects.
With this in mind, the Laws of Thought appear to be used as the basis for establishing relationships across all symbolic mappings (symbol into meaning with associated functional relationship) in some Universe. This needs to be clearly understood. They serve as presumed constraints on both the (semantic) symbolic mappings and their ``classification'' by means of rules, and yet appear themselves to be rules applying to the union of all symbols and presumably to the ``objects'' to which they are referenced in the dictionary defining the mapping.
Classification in the abstract can be viewed as grouping into sets. This leads us to ask: is there is a way of viewing the Laws of Thought themselves as statements associated with a symbolic set theory stripped of (almost all) their semantic content? This is the topic of the next chapter, but first let us try to deconstruct the English and abstract the meaning as best we can.
As sentences, the assertions are very simple. They contain nouns: ``Whatever'', ``Nothing'', and ``Everything''. These are clearly class or set delimiters, but of the most dangerous kind. ``Everything'' in particular suggests that the laws of thought apply to objects2.34 drawn from a Universal Set. ``Whatever'' then refers to any object drawn from that set, which is straightforward enough.
``Nothing'' is even more dangerous a concept than ``Everything'', however, as it has two possible meanings! One meaning might be ``The complement of the Universal Set'', which is usually taken to be the empty set even though, for a set theory to be truly closed, the empty set is itself an object of sorts in the set theory (which is a problem that can lead to certain famous paradoxes unless the set theory is carefully axiomatized to avoid them). The other meaning might be ``Not a set at all, including even the empty set'', effectively creating a ``complement'' to the Universal set that is not the empty set.
In set theory and mathematics, ``Universal Sets'' are very, very tricky2.35 . They become even trickier when the Universal Set in question can be embedded inside a larger set as a subset. For example, the set of all integers can be viewed as a Universal Set for the purposes of discussions in e.g. number theory, but is a particular subset of the set of all rational numbers, which in turn is a subset of all real numbers, which in turn has an interesting relationship to geometric algebras and manifolds - e.g. complex or quaternionic numbers, two, three, or N dimensional spaces. Oh, dear! It turns out that there are an infinite number of ways ``the integers'' can be viewed as particular subsets of ``larger'' sets (whatever the latter might mean).
If the Universal Set in question is just the integers, what exactly is the complement? It might be the empty set of integers (a list of integers with no members), all non-integer real numbers, all complex numbers that are not, in fact, a real integer, or it might be not a number at all, it might be ``Nothing'' (in the set theory). Which of these is used depends on what one wishes to do with one's reasoning process.
Our difficulties are not over. In addition to these apparent implicit references to a Universal Set, the laws of thought contain various present tense forms of a single verb construct: ``(to be able) to be''. Leaving aside the notion of being ``able'' to be something per se as irrelevant2.36, ``being'' itself is the core relational concept in both of the non-tautological assertions. This is rather shocking, really, as the English notion of being is a very subtle concept with all sorts of baggage brought over from inferences made on the basis of experience. In mathematics there is no such concept, really, as everything one manipulates is fundamentally ``imaginary'', not in the sense of complex numbers but rather in the sense that the ``objects'' being manipulated are symbols in our imagination. There is no such thing as ``the number one''; it is only an idea, it has no being except as an idea. There is therefore a significant disconnect between the term ``being'' in English (or for that matter, in Greek!) and any sort of concept in e.g. set theory.
In my opinion, the only way this word can make any sort of sense in both worlds is if a set object is considered to ``exist'' if it is ``something'' - an object drawn from some presumed Universal Set. The most important kind of existence will be experiential, existential existence - the real existence of real ``things'' whatever they might turn out to be, and which include the symbolic objects of our experiential imaginations. It will be vigorously asserted in the next chapter that this mapping (to produce a consistent logical system at the end of it all) requires that the complement to the Universal Set be not a set at all (including the empty set) and that both ``Nothing'' and ``non-being'' refer to this sort of complement, not the ``mere'' empty set within the Universal set.