Set Theory and the Laws of Thought

We should now have an interesting, if static, perspective on the set of all things in the real world. All (say) $N$ objects in the ``existential Universe'' $E$ can be grouped into sets by permutation, forming $\Pi(E)$ with cardinality $2^N$. These permutations can in turn be permuted into sets of sets $\Pi^2(E)$ with cardinality $2^{2^N}$. However, nature selects only a small subset of $\Pi^2(E)$ - particular groupings of objects according to certain rules. We simply don't see any of the vast, the good-friends-with-the-infinite, other possible set groupings. We therefore for many purposes define these objects to be a basic existential set, e.g. the set of all atoms, and form its power sets instead of including all of the non-observed sets from the cosmic all.

Every possible grouping of objects into sets, though, is contained in the $\Pi^n(E)$ recursion. Predicate logic and set theory can only be judged to be a ``theory'' or ``valid'' according to whether or not any given predicate, constructed according to any presumed set of rules, successfully identifies objects in the $\Pi^n(E)$ hierarchy. This construction is so far very nearly axiom free. We have really assumed very little about $E$ except that it exists, that it has a finite cardinality, and that it contains discrete identifiable (in the formal sense) objects, objects that can each be mentally permuted.

We are ignoring for the moment many questions of interest to mathematicians - such as what we need to do if the cardinality of $E$ is truly infinite or if $E$ is a continuous set. Dealing with infinity and continuity is irrelevant to our descriptive process, because even if $E$ is infinite and continuous we can at least imagine a similar continuous permutive process (which leads instantly to infinitely infinite infinities) to generate the analog of $\Pi^n(E)$, causing us to throw away infinitely more unrealized possibilities as we do not see either the infinity or the continuity, only the finity and immediacy of a single slice of the possibly infinite possible. So to speak.

To speak strictly metaphorically, even though the Universe may live in a meta-Universe of possible set groupings analogous to the real line, infinitely divisible and infinitely permutable in every tiny segment no matter how small, the Universe itself is just one of those groupings. It may well be like unto an irrational number - infinitely unlikely in a set with uncountably infinite cardinality - but it is what it is. Furthermore, we can always renormalize this imagined real line so that the Universe is the integer number one. There may be lots of other possibilities out there, but if we can't see them they really don't matter. If we can see them, they stop being ``other possibilities''; our Universe $E$ and the sets $\Pi^n(E)$ just turns out to be larger and more complex than we thought but still is Unitary. This process of conceptually expanding the Universal set $E$ occurs all the time in physics, as we extend into the microcosm.

Accepting $\Pi^n(E)$ as the extremely naive existential set Universe of thought that our brains co-evolved structured wetware and language to cope with, we can at last consider the laws of thought and see what they mean in terms of this fundamental set-theoretic Universe.

The first of the Laws of Thought, the law of identity, states that any thing that is, is (itself). However, English (and doubtless Greek or Sanskrit or other languages in which the law is or has been formulated) is strongly multivalent and thought is an important thing to get right. We had best proceed extremely carefully and not assume that we actually understand what this means. We will begin by defining a ``thing'' as ``any object in the $\Pi^2(E)$ hierarchy''.

We also have to be careful to define the word ``is'' (and all various forms of the verb ``to be'' and - in a moment - the concept of ``not to be'', or negation of being). Among other things we cannot help but associate different tenses with this verb. We will therefore have to agree to mentally ignore all concepts such as ``was'', ``will be'', and so on. Our laws of thought are formulated as static statements associated with a static description of sets, not with a dynamic conceptualization of predicates that permits us to convert one set into another. This is actually remarkably consistent with physics and relativity theory, where time is just another dimension like space and one can imagine stepping ``outside'' the set of all space-time events and considering the whole ball of wax to be $E$.

With that carefully established, the law of identity becomes a beautiful, tautological existential statement. Any ``thing'' is an object selected from $\Pi^n(E)$, and as this set hierarchy was itself imagined (not ``constructed'', as it a priori existed the instant $E$ itself was established) by a process of identification, this law is the law of identification. Our set Universe is precisely that which can be identified, drawn from the set of all permutations of the existential set $E$ that can be identified. If we can identify, that is, if a statement selects an object from $\Pi^2(E)$, then that statement is valid; otherwise it is not.

Mathematicians and dreamers may object that this definition is cold and heartless - it excludes all sorts of reasoning about non-existential Universes, things we might imagine, things we might dream up. Basically all abstract thought. Not so - it merely acknowledges that those subjects contain an infinity of traps for the unwary mind that will require axioms to deal with, as it is absolutely trivial to conceive of imaginary universes in which six impossible things happen before breakfast^3.16. Abstract thought will turn out to be a simply lovely game and all sorts of fun, but we need to remember that it is a game where we can easily twist the rules back onto themselves into impossibilities, inconsistencies, paradoxes, and worse. Not so with $\Pi^n(E)$. It is the very definition of mundane.

What now of the difficult laws, the ones involving nothing and non-being, the negation of the two ideas that we had to work so hard to clearly and unambiguously define above so that the law of identity could be viewed (literally) as a Universal Truth?

Note well that negation is a very subtle and difficult concept, so much so that positive set theory^3.17 excludes it and manages to get along amazingly well without it.

Nevertheless, in the English statement of the laws of thought (and in Aristotle's and Parmenides' Greek statements as well) negation is very much present, and of course negation is a key part of logic, which either proceeds from the laws of thought or the laws of thought proceed from logic (depending on who you happen to be speaking to at the time) so we have to at least figure out what we are going to do with it in our set theoretic expression of those Laws. Let us start with the law of contradictions (as I wrote it a couple of sections ago: No thing can both be and not be. This was a somewhat clumsy way of writing it, but now that clumsiness will serve us well as we have at last defined what a ``thing'' is and what ``being'' means, which gives us at least a chance at defining what ``no thing'' and ``non being'' are.

Even so, we will discover that there are many distinct linguistic meanings of negation of ``thingness'' and ``being'' with regard to the existential $\Pi^n(S)$ Universe. Let us list a few of them. A ``thing'' in the existential set $S$ is an object in its associated $\Pi^n(S)$ Universe, so ``no thing'' might be:

• An object in another, disjoint Universe. For example, if $S$ is the set of natural numbers (without worrying in the least about whether or not this set closes at infinity - we will adopt the point of view that infinity is something that we reach by a limiting process and that existential truth is derived from truths associated with finite sets of arbitrary (variable) size), and we introduce an irrational number, that number is ``no thing'' in $\Pi^n(S)$ - it can be identified with no nonnegative integer or set of sets of sets of nonnegative integers. If we consider the set of all apples, it might be an orange.
• An imaginary (or if you prefer, hypothetical object. This is really a special class of the first case, because we can define an existential set Universe associated with the imagination and awareness, self and otherwise, itself. It is an important special class because for each one of us, it is the only set Universe that we directly experience. It is also where e.g. all abstract sets such as those of mathematics live, as it is otherwise remarkably difficult to find a piece of $\pi$ in your refrigerator.
• This is the really tough one - no object at all, in any set theory. The negation of all set theories. The utter lack of an existential set Universe of any sort. If this sounds scary, it should. The utter negation of thingness of any sort is almost inconceivable to a conscious mind, and this is the source of many religious beliefs that basically assert that this kind of negation cannot exist.

The first two can be associated with predicates in various ways, as can the third one, but they are very different ways and can lead to considerable confusion when one attempts to develop ``logic'' based on one of these forms when somebody else wishes to use another.

This is the easy part. What about ``being''? We defined being in the law of identity as ``being identifiable'', where being identifiable itself basically meant being a set in the $\Pi^n(S)$ hierarchy. Not being is then pretty straightforward. It means not being in the existential set Universe, period. Of course this is now an existential tautology and every thing is in the existential set Universe, unless we somehow embed that Universe in a larger one as one might embed the natural numbers in the complex plane. Which is cheating in so many ways, especially if the Universe one is trying to embed is the actual existential physical Universe in which we live^3.18 No thing is not in this Universe.

However, what about the inheritance from the other two forms for a ``thing''? And more important, what about predicates? People tend to use the laws of thought to decide propositions or the set equality of predicate expressions. Let us consider these separately. Being is now well defined for our existential set Universe and indeed is a tautological extension of the law of identity for that Universe. We don't really need a law of contradiction in this approach, only a criterion for establishing identity, which is doubtless the observation that led to the development of positive set theory.

We are still left with what one might call ``strong'' nonbeing - not being in any possible set Universe, nonbeing in the absolute sense - and ``weak'' nonbeing where a set exists, but just not in the right set Universe, where $z = 1.0 + i\pi$ exists but where attempts to reason about it as a natural number involve either some sort of restriction/projection from the complex plane to the natural numbers or extension/embedding of the natural numbers in the complex plane. To do either one requires axioms, many axioms, and many theorems derived from those axioms besides and hence is far beyond our analysis of the laws of thought.

There is one more sense in which nonbeing is used, however - and for better or worse it is one of the most common forms of usage and is completely different from the ones associated with identity and existence in the set Universe. This ambiguity is one major source of paradox and antimony. In many cases, not being means not being in the same identity set. That is, the law of identity can be interpreted as saying ``A thing (identifiable in the set Universe $\Pi^n(S)$ for some existential set S) is either in set $A$ from somewhere in that hierarchy or it is in $\bar{A}$, the complement of $A$ (from the entire hierarchy).'' This is of course a useful thing to have around when trying to decide if an object ``belongs to'' set $A$ or if it doesn't, when trying to define an axiom of equality^3.19 . It is also the source of much dark evil when nonbeing in the strong or weak tautological sense are confused with nonbeing in this sense.

The possibility of antimony is apparent when one considers how differently the empty set is treated by the two meanings. The empty set is is always a set within any set theory, existential and permutative or not. Its existence is an axiom in positive set theory, but one can also just ``observe'' it as the outcome of evaluating a false formula. In our permutative approach, it is just the set of all set objects selected zero at a time, one of the possible permutations of objects that exists even for five year olds seeking the various ways of grouping a small pile of pennies on a table. It is an explicit but often invisible member of the subsets of $\Pi^n(S)$ - I like to think of the empty set at any level of the hierarchy as being the ``set brackets'' of the hierarchical set itself, so that $\{ \framebox{$\heartsuit$}\} = \{\} \bigcup \framebox{$\heartsuit$}$ in a manner of speaking, since of course we technically cannot speak of $\framebox{$\heartsuit$}$ all by itself outside of a set container. If sets are metaphorically objects in a box, the empty set is the box, which can be empty but always is there. Its presence is required to that operations like intersection close within the set theory where a full set theory allows set objects to be manipulated with the operations of union and intersection as part of its basic definition.

If not being is used in the sense of not being equal, or not being in the set of true statements as an essential part of predicate evaluation, it cannot also be used in the sense of not in the set Universe. Russell worked far too hard to define his paradox (which we will discuss in some detail later). He might just as well have tried to create ``a set of all things that are not in any set including the empty set''. Say what? Clearly this kind of ``paradox'' isn't paradoxical at all in an existential set Universe, it's just a meaningless statement.

The final law of thought, the law of the excluded middle, tells us that every thing must either be or not be. Once again, now that we know what a thing is, we can see that this is a tautology of the law of identity for the strong or the weak formulation of nonbeing - all things (objects in the $\Pi^n(S)$ hierarchy) are, so sure, they are or they are not - not. Once again we see how a wide range of problems can come from extending over the existential set Universe boundary and allowing ``things'' to exist (sort of) in sets ``outside'' the existential set Universe $S$, in which case we can talk about things that are not, like pink hippogriffs dancing the tango or irrational numbers in a natural number Universe. We observe that for the strong version of nonbeing this law sounds rather odd - nothing doesn't exist where anything or everything do, quite literally.

Finally, we observe that as before this statement has a different meaning altogether when used inside a set Universe as another form of disjunction. In this context we can interpret the law as saying that every object in $\Pi^n(S)$ is either a member of any particular set in $\Pi^2(S)$ or it isn't. There isn't anyplace else for it to be, after all, because we exclude imaginary Universes or embeddings of the set $S$ and because no Universe at all cannot exist in the presense of a Universe that does^3.20.

If we mix up these different interpretations, and use contradiction or excluded middle on the one hand to refer to actual impossibilities and on the other hand to partitionings of actualities into disjoint sets (be those sets sub or super to the set Universe in question) then we are bound to get ourselves into trouble.

Still, this was a generally successful effort. We note that the law of identity is a pure tautology when expressed in terms of an existential set theory, and that the laws of contradiction and excluded middle are irrelevant restatements of the same tautology - just another way of stating the principle of identification that defines the set Universe as the $\Pi^n(S)$ hierarchy in the first place. We also see that there are at least two or three other ways these laws can be interpreted with greater or lesser meaning and utility. The difference is that these interpretations require axioms where I would argue that the law of identity itself and its two strong-form statements of negation are tautologies of a naive theory that is nearly axiom free after the assumption ``suppose one has a set $S$ of objects that actually exist''.

It seems worthwhile to see how our new strong definition of nonbeing enters into existential set theory as the absence of any set including the empty set as the empty set is very much a part of $\Pi^n(S)$. To make up a formal theory that manages to sound like a set theory extended to ``include nothing'' let us now introduce a new concept (at least in western thought) - the null set.