Logical Complexity

For reasons unclear even to me, I have always been deeply fascinated by logic. I studied logic and set theory on my own in High School, and was exposed to Boolean algebra and logic design in college. I poured over Elements of Symbolic Logic by Hans Reichenbach and Introduction to Symbolic Logic and its Applications by Rudolf Carnap. I discovered Language, Truth and Logic by Alfred Jules Ayer, Philosophy of Logic by W.V. Quine, and Philosophy of Logics by Susan Haack. [I highly recommend each of these books.] My logic library today includes many other books, some good and others not so good, some in the old Aristotelian tradition, others in the modern symbolic mode, and still others in the Boolean mode peculiar to engineers..

Two books, however, generated a deep disquietude in me and their topics continue to do so to this day. The first, Gödel's Proof by Nagel and Newman, I discovered while in High School. I was amazed that such a result could be proved and shaken by its implications for mathematics, as is nearly everyone when they first encounter it. [Though perhaps the best known, most popular exposition of Gödel's Theorems, I do not recommend this book. It simplifies the topic so much that its derivations are defective and misleading, but perhaps later editions have improved - I refer here to the 1960 edition. I instead recommend reading Gödel in the original, On Formally Undecidable Propositions of Principia Mathematica and Related Systems, though it is much more difficult to follow.] Gödel's Theorems are the most famous and disquieting results of 20^th Century logic, and like the uncertainty principle and relativity in physics, they are often cited as justification for some position in fields far outside their applicability.

The second book was the contentious Laws of Form (LoF) by G. Spencer Brown, another book I do not recommend. Opinions concerning Brown are decidably mixed and tend to the two extremes - either he is a visionary or a charlatan (see some of the reader comments in the reviews at the link). The book is short and obscure, with an eclectic notation not conducive to a thorough and clear exposition of logic. Brown's style is difficult and egotistical, but there was an idea behind it which I found fascinating, viz., the introduction of imaginary values to logic. Brown argued:

"The fact that imaginary values can be used to reason towards a real and certain answer, coupled with the fact that they are not so used in mathematical reasoning today, and also coupled with the fact that certain equations plainly can not be solved without the use of imaginary values, means that there must be mathematical statements (whose truth or untruth is in fact perfectly decidable) which can not be decided by the methods of reasoning to which we have hitherto restricted ourselves."

which is a viewpoint having great resonance with mathematicians and engineers who routinely employ the powerful methods of complex analysis to arrive at useful results. Unfortunately, Brown's development was defective. Though attempting to formulate an imaginary logic, Brown limited the possible logical values to only two formal constants, making it effectively a propositional logic; what is worse, one of these constants is not indicated by any symbol, but simply by a blank space; worse still, by identifying imaginary logical values with variations in time (an understandable error for an engineer familiar with logic circuits), Brown confused a possible interpretation with the formalism.

My first reading of LoF left me completely bewildered - it was unlike any logic I had ever read. There seemed to be something of value, if only I could grasp it. I was frustrated for years by my inability to understand, but kept returning. I only began making progress when I abandoned Brown's peculiar notation and translated his text into a more conventional notation. This enabled me to see the errors in his development, and to attempt to correct them, resulting in a complex logic which I detail in my manuscript Systema Logica [a .pdf summary of the Systema is available at the Yahoo Laws of Form Group files area, here]. Brown's simple exposition was the starting point – I only found it necessary to standardize the notation, reorganize and formalize it, and add the appropriate definition of the formal imaginary logical constant, V. Part of Brown's insight was that imaginary values are most easily introduced in a formalism with two primitive operations, only one of which, complementation, is needed to completely define the fundamental property of the imaginary constant. The resulting formal system is a multi-value, extensional, boolean equivalence algebra and an associated arithmetic, requiring two primitive logical operations, complementation, indicated by a postfix tilde "~", and concatenation of distinction symbols, representing disjunction, and two primitive relations, logical equality (not material equivalence) and derivability (instead of implication). Those familiar with Laws of Form may be disappointed, for this complex logic bears scant resemblance to Brown's extravagant and eccentric work – it is instead a retreat from his mystical visions and psychological speculations to more conventional and concrete realms. Lest I seem unduly critical of Brown, I wish to here acknowledge my debt and thanks for clarifying many concepts, including the clear distinction between axioms, postulates and initials, substitution and replacement, theorems and consequences, proof and demonstration, but most of all for pointing out a direction and providing a map which, however flawed, could be followed.

I already used some of these results in my solution of Dodgson's eight hitherto unsolved sorites [see here]. In articles that follow, I will use these methods to resolve the Liar Paradox, provide a different interpretation of Gödel's Theorems, and discuss their implications in the re-entrant forms Brown introduced in LoF.

What is most curious to me is that the two books most influential on my views are two which I do not encourage anyone else to read.