In the previous chapters we have seen how the Laws of Thought, as one of the basic tenets of Logic, are in fact quite troublesome and not at all ``obvious truths''. Since they are generally used to determine truth in an extensible manner, this is a problem. We also saw that to make the way they are usually framed in English (at least) precise and to not admit existential paradoxes, we had to try to formulate them in terms of existential set theory (where set theory in general is ``the mother of all reasoning systems''). Only in existential set theory (with a well-defined set Universe) could the abstractions of ``being'' or ``non-being'' be found, only by extending this set theory with the null Not a Set (while leaving the empty set as an element of set theory within the set-theoretic Universe in question) could we deal with certain paradoxical statements that appear to be perfectly understandable, well-formed English sentences that parse out logically to be nonsense, contradiction, to things that violate our intuitive ideas of true and false, existence or non-existence, or that simply fail to actually specify any set in our set Universe including the empty set.
In this chapter we'll home in on Logic per se, especially in the context of symbolic reasoning systems that are do not strictly reference an external existential reality that is what it is, but instead can be used (sometimes in enormously subtle ways!) to make self-referential assertions. The purpose of this book is in no sense to denigrate the power, the beauty, the simplicity of Logic (or its cousin Mathematics); it is to point out that it is a sterile kind of beauty that cannot in and of itself give rise to a single absolute truth relevant to the physical world we live in, and that can all too easily implode, making all theorems in the system essentially unprovable. There is a fundamental disconnect between experience and reason that cannot be filled in by reason, and reason applied to itself proves to be unreason if one is not reasonably careful!
The easiest way to accomplish our goals will be by presenting a very few examples of logical arguments (famous logical arguments at that) to illustrate the different parts of a system of logic. We will see that systems of logical inference, without exception, can be described in terms of actions taken on sets (no surprise), that rules of inference are in some sense set-theoretical definitions or operations, and that to go beyond the elementary list-making and categorization operations of a raw set theory we have to dress the sets up with a mix of definitions and axioms.
A language is often viewed as a system of definitions, a dictionary. All dictionaries4.1 share a fundamental self-referentiality that makes them far from trivial logical objects. Tarzan aside, it isn't at all obvious that real human beings are capable of taking a dictionary for a strange language alone and learning the language thus represented4.2. Indeed, there is considerable evidence to the contrary - without a Rosetta stone, without a context or pre-existing relationship in terms of which a decryption algorithm can seek information compressing patterns, the dictionary is arbitrary and can even continuously change. Modern cryptography is based on this premise - it constructs a highly nontrivial ``dictionary'' so that statements are (ideally) indistinguishable from random noise, at which point there is no informational compression at all. A further problem is that even within a single language with a fixed dictionary, all dictionary definitions in that language are circular - they are written in words in the dictionary, whose definitions are written in terms of other words, which are written in terms of other words, until eventually you find that the dictionary is nothing but a set of equivalence connections with a certain pattern. Yessir, Tarzan's accomplishment puts John von Neumann, Shannon, and the rest of them4.3 right into the shade.
If we know something about the Universal set to which the dictionary applies, we can sometimes guess a consistent mapping between the ``real'' patterns of the subsets of that set and ``virtual'' patterns of the dictionary terms, possibly aided by visual cues such as a ``picture of a tree'' next to its definition that help us establish at least some provisional mappings. In essence, the dictionary represents a code, and to break the code we have to determine a homologous set of linkages between the dictionary and the system to which it refers. Ultimately this task is made extraordinarily difficult because there is no guarantee that any homology will be unique. Given the high degree of degeneracy (redundancy) of human language it will almost certainly not be unique4.4.
Dictionaries do not intrinsically specify a system of logic, however, and a language is not simply the set of homologies represented within the dictionary and some reference system. Dictionaries (real ones, not idealized ones) are only rarely complete - perhaps when they reference some ``simple'' closed system that is capable of being well-defined (literally) such as the ``dictionary'' of a computer or mathematical language.
Because of their intrinsic incompleteness - a complete dictionary for something like a real world would require the moral equivalent of a word for every event in space-time that completely specifies the homology between that event and all other events, plus the ability to represent all higher order homologies built on top of the raw physical homology - the ``language'' of human experience, of poetry, of illogic and paradox and contradiction - a dictionary is most generally an approximate, or coarse grained set of homologies, and requires something more to aid in the abstraction of relationships before anything like a system of logic ensues.
We've encountered just the tip of this particular iceberg in our discussions of sets, where ultimately the dictionary is what is required to identify each object and sort it into its own identity set when confronted with the Universal set. It is not enough to identify an object as a ``tree'', it has to be able to identify an object as this particular tree, with its own unitary and unique existence, as of this particular moment in its existence. Where in fact the tree is made up of a dynamically changing set of molecules, the molecules are made up of atoms, the atoms are made up of electrons and nuclei, the nuclei are made up of protons and neutrons, and the protons and neutrons are made up of quarks - ultimately a complete definition of this tree extends to the subatomic level, to the fundamental level, and extends through space and through time as a set of intertwining relationships.
This in turn doesn't necessarily recognize or encode the relationships and structures that emerge at the higher degrees of complexity. It is not at all easy to understand this tree's particular role as a home for nesting birds and eventual source of firewood based on an understanding, however complete, of its subatomic structure4.5. Specifying relational operations is like specifying the syntax of the language. We can define an apple quite precisely (if we try hard enough) as a concatenation of specific molecules that underwent a particular process of development in natural history without ever mentioning that apples are good to eat, that an apple a day keeps the doctor away, that a thrown apple can be used to bean someone on the head, that deer are attracted to apple trees in the back yard at certain times of year because they are good to eat except those yards of healthy people who bean deer on the noggins with apples any time they dare to show their furry little faces!
When we come to reason, we find that in addition to a set of definitions (that are fundamentally arbitrary and certainly not ``obvious truths'' or ``provable'') we need to specify relationships in order to be able to operate on the objects that are appropriately defined within the theory. I leave this term deliberately vague - operating on an object might (for example) be an action that ``transforms'' (in a sequential reasoning sense, not a temporal sense) a defined object from one state to another. Or it might be viewed as a sorting or categorization operation, one that takes an object or subset from one set and places it in another. Or it might assert a more abstract relationship between objects or collective subsets that we discover we need as we proceed. Ultimately such relationships function as rules of our system of reason. There are two primary kinds of rules involved in formal logic. One is the so-called rules of inference which are (as their name suggests) a set of rules that permit one to ``infer'' provisional truth relationships. The other is the set of axioms of the theory.
These two things are differentiated primarily by rules of inference being presumably self-evident statements - in fact, the Laws of Thought in disguise. They presumably ``come with the territory'' of set theory, universes and mutually exclusive partitionings of identity relationships, although I'm hoping that the previous chapters were enough to make you a bit skeptical that this is in fact the case. Axioms per se are simply unprovable assumptions, the hypotheses that lead one to this system of reason (or this set theory, this branch of mathematics, this hypothesized universe, this computer's microarchitecture) and not that.
It is a fairly recent discovery that it is possible to choose different hypotheses and reason validly to different conclusions even in that most precise and self-evidently obvious of mathematical systems - ordinary geometry. It is worth repeating like a mantra that while the sum of the angles in a triangle in plane geometry is radians, in an infinity of other two-dimensional geometries it is not. If we change the assumption that the two-dimensional surface is ``flat'', the result goes away and is replaced by new, different results.
Eventually, I'm going to compress rules of inference into a very limited set based on the set theory above, which does not require things like the Law of Contradiction and the Law of Excluded Middle to universally operate within the Universal set of the theory but rather to differentiate that which is (in an external Universal set, including the empty set) and that which is not (is ). ``True'' and ``False'' will be particular sets that may or may not be exclusive and exhaustive within set Universes that contain the system of reason being used (making it self-referential) distinct from ``Being'' and ``Non-Being''. The particular extension that permits it to be applied to True and False categories will then become an axiom of a particular system of reason applied to something else that is moderately concrete
This is a very good thing. It uses these two rules only to state the truly obvious - ``Contradictions cannot occur'' - without specifying precisely what a contradiction is (which requires both definitions and other axioms and a set of objects that might or might not be contradictory). In fact, perhaps it is better to think of it as being ``Contradictions do not occur'' as an assertion or constraint on possible sets of symbolic reason that we wish to consider in case they prove useful in particular contexts. The null set (the impossible) is not in the Universal set (the context in question), regardless of how objects are parsed into nonempty and empty or true and false sets within the context, the set Universe of the problem.
We have to do this. One apparently ignored consequence of Gödel's theorem (which we will cover, sooner or later) is that the existence of a single undecideable statement in a theory, together with the usual rules for inference, makes all statements within the theory undecidable just as surely (and using the exact same mechanism) as the existence of a self-contradiction in the theory makes all statements in the theory self-contradictory. In fact, the Law of Contradiction and Law of Excluded Middle can easily be shown to be false in any system of symbolic reasoning that admits Gödellian knots, which is pretty much any system of symbolic reasoning that matters, in particular, English (or other human languages).
This doesn't (ultimately) mean that they have to be abandoned. What it means is that they are not self-evident truths but are rather axioms, assumptions that can be used, in a constrained or limited way, to build a system of contingent relationships. If one can introduce an axiom or axioms that are capable of specifying truth relationships in the theory - a very big if indeed, since this is categorically impossible in nearly all interesting cases - one can build pretty little systems of classical Boolean logic with its Venn-diagrammatic disjunctive truth relationships. Whether or not these systems are useful is a different matter entirely and will depend strongly on what our goals are!
Let's see what the problem is, though, as people have grown remarkably attached to classical Boolean logic because it is a limiting case of the way our brains are more or less hardwired to think, and is ``built in'' to normal language and validated by all sorts of quotidian experience. It is a box that it is very, very difficult to think outside of, because most of what we think is implicitly derived using classical Boolean logic and one has to work hard to either clearly demonstrate its self-destructive nature in human language or other nontrivial symbolic systems or to bootstrap from it to a more general system of reason.