Noesis 76 - December 1992
Chris Langan to Rick Rosner
by Chris Langan
October 09, 1992
Dear Rick:
Since I'm sending you my letter to Ron. I might as well take the opportunity to rectify what you modestly call your "misunderstanding of Godel", involving the supposed inability of tautological systems to generate "interesting results". Read closely; if you want to get to the bottom of the controversy over "metaphysics", it doesn't get any clearer than this.
First, let's take a look at the word tautology. Its meaning in the vernacular involves needless repetition or redundancy. But in logic, its meaning is more precise and more benign. It describes a statement which is analytic, or true solely by virtue of its logical form. This reaches its limit in 2-valued prepositional logic: e.g., A v ~A (law of the excluded middle); ~(A & ~A) (law of noncontradiction). In this notation, variables are sentential; "A" stands for any complete formula or "predicate". Such tautologies are self-referential; we can let "A" stand for the whole tautology in which it appears (e.g., A v ~A --> (A v ~A) v ~(A v ~A)). Since logic is entirely developed by deductive substitution from initial tautologies - as is a geometry from its axioms - these tautologies form what you'd call a "reflexively true tautological framework". They are "highly resistant to outside contradiction" because, in order to be comprehensible, any such contradiction must be formulated in terms of propositional logic and therefore submit to the very tautological rules it purports to "contradict".
It is possible to take an outside perspective on 2-valued prepositional logic by extending the set of truthvalues on which it relies. This perspective is that of many-valued logic. However, if you want to be able to regard statements as being either true or false but not both at once, you cannot take this perspective. Even if we were to take an MVL perspective for theoretical purposes (as sometimes we must), we would have to "translate" our results into 2VL in order to make them consciously comprehensible.
So we have three definitions of "tautology". In order of strength:
1. The self-referential sentential tautologies of 2VL;
2. Less general analytic statements like "daisies are flowers";
3. Any statement that is repetitive or redundant.The extreme generality of propositional logic usually makes it inadequate as a theoretical formalism. Most scientific theories are sets of objectively-interpreted predicates making qualitative and quantitative attributions with respect to objectively-interpreted object variables. It would thus be useful to "relax" the prepositional definition of tautology so as to extend its applicability to predicate logic. This can be done in a self-referential way through the well-known distinction between autologous ("the word short is short") and heterologous ("the word illegible is not illegible") predicates. Unfortunately, this distinction involves a nongeneral assumption that we cannot usually make: that predicates are being typographically interpreted, or that predicate logic is being used only in reference to how it is written. So we must suspend the self-reference criterion. This, of course, leaves us with definition (2) above.
The self-referentiality of sentential tautology owes to the fact that these tautologies can only be expressed as things of the kind to which they refer, i.e., logical formulae. But this is rarely the case. For example, we sometimes make general statements about contexts which contain neither the statements themselves nor their objective images. These statements then comprise a "metalanguage" formulated in a context which properly includes that to which they refer...i.e., in a semantically context including a theoretical object language and its object universe, or referent context. Examples of statements requiring metalinguistic formulation are those attributing truth or falsity to sets of semantically interpreted object-level expressions. In these and other such cases, we will be using the term "tautology" in reference to any universal generalization over the referent context ... something "repeated" for everything in that context, but not necessarily for itself.
But first, a preliminary note. A non-self-referential tautology always implies a restriction of its referent context with respect to reality as a whole. Otherwise, there would be no place left to formulate it in which it could avoid referring to itself. There is just one obvious context which cannot be restricted in this way: reality as a whole. This, of course, is the universe of any theory of metaphysics. Like propositional logic, a metaphysical theory must be formulated within the context to which it refers. So, given our "relaxed" definition of tautology, it will be understood that tautology becomes self-referential by necessity at the "metaphysical limit" of predicate-logical theorization.
Note also that the cognitive syntax of the human mind - the time-invariant aspect of human mental functionabi1ity - qualifies as a tautology in the same self-referential sense as does metaphysics. Whatever it considers - itself and everything else - it must consider within its own definitive constraints. In other words, it can consider its own structure and operation only within its own structure and by its own operation, and everything else (all which is outside or beyond it) only as an externalized potentialization of itself (i.e., as that which can be considered within it). If the phrase "itself and everything else" seems suspiciously close to the phrase "reality as a whole" - the "universe of metaphysics" mentioned above - then you already glimpse what must follow.
Any nominal tautology (or "tautological theoretical framework") is of one of two kinds. Either it is analytic over the entire domain of definition of its argument, or it isn't; either it covers its entire universe, or it doesn't. In the former case, it is a valid tautology with respect to the given application. In the latter it is not, and if it is nevertheless tautologically applied, we call it an artificial tautology or a pseudotautology. Artificial tautology is the worst bane of inductive and empirical reasoning; it pretends to yield a kind of information it cannot yield, and to describe things completely which it actually describes partially, not at all, sometimes, or never. Most supposed "metaphysical" theories are of this variety (e.g., the Pepper theory of metaphysics, whose root concept is tautological only with respect to behavioral psychology and not reality in general).
Artificial tautology is especially insidious when, in indeterminate contexts with undescribed contents, it becomes "self-implying" in a manner which parodies true logical analycity...e.g., when the rules of inference of the theory in which it is misplaced ignore the ordinal distinction between its "antecedent" and "consequent". As widespread examples, take such notorious prejudices as "those defendants who most cleverly deny guilt are always guilty". In any court holding this belief, no hapless innocent can be clever in his own defense without "proving himself guilty"! This statement's claim to generality resides in the supposed exhaustivity of the domain of definition of its antecedent (the universally-quantified set of defendants from which "the cleverest" are taken) and the universal quantification ("always") of its synthetic consequent. In empirical contexts, this is a blueprint for disaster.
Another example, paraphrased from Noesis 73: "Those theorists most certain of their theories are necessarily dogmatic (and insane and stupid to boot)." Even though demagogues, who prey on the ignorance and prejudice of their audiences, are marked by just this kind of cynical reliance on artificial tautology, the wider definition of "demagogue" - which involves "leadership" - prohibits us from saying that the author of this particular one is "necessarily a demagogue". All we can say on its basis alone is that he's doing a transparent and rather sorry impression of one.
However, it's just as clear that any valid tautology, by virtue of its applicability over specific distinctions within its universe, must be general in a sense often confused with uninformative. This confusion is only natural for those preoccupied with seeking various kinds of specific information. If we want to cure an ill, it isn't enough to know that we need "a medicine"; we must know the specific kind of medicine we need.
If the story ended here, we'd be in big trouble. To get specific information, we need to use deductive reasoning. But we can only do so by starting with generalities and "working inwards". This means that without general info, there can be no specific info. In other words, we can't call the fact that we need "a medicine" "worthless information", since without it, we can't even begin to find the specific medicine we need. Generalities - and the inductive reasoning which produces them - are absolute prerequisites of "interesting" deductive theories.
Generalities reflect a general truth: not all of them are created equal. Tautology, as the very broadest kind of generality, is the most necessary prerequisite for informative theories. Thus, if Godel had ever said anything like "tautological systems cannot generate interesting results", he'd either have to draw some fast qualifications, or we'd have to rip the officer's stripes from his "genius" uniform and bust him down to privatdocent. Systems consisting only of tautologies may be informationally impoverished, but that's only because we haven't yet developed their primary advantages: their tautological structures relative to their data-universes .
All informative systems must have tautological bases relative to their universes...i.e., must come from premises (or axiomatic sets of premises) that are true for all things under all circumstances describable within them. Any system which does not is founded on premises which exclude some aspect of its universe, and is useless for arguments involving it. Where this excluded aspect is unknown, we cannot identify the arguments for which the system is or is not useful. This, of course, eviscerates the entire system from an informational standpoint, on the other hand, if the excluded aspect is known, then adjoining this info to the system in a way allowing it to interact with info already there extends the system to cover it. and there must now exist a tautological basis of the system with respect to its whole universe.
Notice what this says about the plight of pre-CTMU theories. The validity of any of them must be relativized to those aspects of the universe for which its basic premises are tautological; whatever information it contains exists only for them. Wasn't it too bad that the info in particular pre-CTMU theories was inapplicable to the contexts of other such theories...i.e., that all specific-theories couldn't be combined to amplify information about the contexts of each of them in least within logical constraints up-plying to relationships among the universes themselves?
Information is not an absolute quantity. It exists relative to the contexts in which it is applicable. If you know that apples are edible, but you have no apples, then you have no useful information on how to feed yourself, hut if you have an endless supply of apples, you have quite a bit of info indeed. On the other hand, no number of apples can alone make "apples are edible" yield info on how to fix your TV. Unfortunately, standard information theory just wasn't equipped to deal with these and other aspects of its titular subject matter. While Shannon-style information was a sufficiently powerful concept to promote the development of modern communication and computation systems, it had its limitations. The CTMU was invented partially to rescue the world from these limitations by redefining information in a more powerful way.
As readers of Noesis will recall, this crucial redefinition begins with a mutual, recursive interdefinition of information and cognition within a "reified tautology" called a quantum transducer. The quantum transducer, being paradoxiform by direct analogy with tautologically-based inference, models the way subjectively-tautological cognitive syntaxes transduce information in time. The universality of this model allows reality to be reduced to it, and thus to (cognitive) information. "Information" is the objective aspect of the quantum transducer for itself and for all others; it is cognition-for-cognition, equating generalistically to a cognitive identity relation on that part of reality to which it corresponds (i.e., the part containing all the transducers playing active and passive roles in it).
As you suggested in Noesis 73, my "certitude" regarding the CTMU rests on its tautological structure relative to all humanly-comprehensible reality (I seem to recall mentioning something to this effect during one of our two conversations)...and a few related "tricks" like paradox-distributivity. Formulating reality as a tautology was an obvious move. The reason no one succeeded before me is that doing so required a basic (and rather counterintuitive) restructuring of our perceptions and conceptions of reality.
A primary effect of this restructuring was to eliminate certain barriers existing among various submetaphysicaI disciplines. Every field of human inquiry contains valuable information, but it has always been difficult to transfer this information across interdisciplinary boundaries. Thus, the elimination of these boundaries - the construction of a "universal formalism" - opens various realms of inquiry to relevant but otherwise-inaccessible information once "hidden" in the alien formalisms of other realms. The "liberated" information is then free to combine in synergistic (or even chaotic) ways to reveal new insights.
I know you remain skeptical of certain implications of the CTMU, largely because you're unfamiliar with the logical and model-theoretic criteria for proof. Hut you must at least know that they involve conjunctions like type theory and probability theory, and the theories of physics, computation and decision. Furthermore, these conjunctions are used to solve problems which cannot otherwise be solved, at least with any amount of ease. Your skepticism notwithstanding, it is obvious that this kind of "informational chain-reaction" can he a powerful generator of insight.
There is one problem in particular that cannot be solved without a CTMU-style tautology and its attendant informational explosion: that of providing a general, logically consistent picture of the universe. This owes to the fact that the basis (root concept) of any correct theory of metaphysics must be tautological relative to all conceivable aspects of reality. Because the "metaphysical universe" is so all-encompassing that it exceeds the set of all self-inclusive sets, where "self-inclusion" is synonymous with the kind of self-description on which (prepositional) tautology is defined, it must reduce (or regress inductively) to the broadest and most powerful tautology the human mind can formulate.
There is only one such "universal tautology", and therefore only one correct basis for metaphysical theorization. To convince you of this, I offer the following informal and highly simplified "proof". For the purposes of this proof, think of "information" as that by which transducers distinguish among objects or ideas. The phrase "T excludes d" means that the theory T contains neither the info d nor a deductively heritable generalization of it. The point of exclusion is to excuse us from differentiating between two theories, one of which is either a notational variant or deductive evolution of the other. Such theories pass as virtually identical; "different" theories have different tautological bases.
SHORT FORM: Say that there are two true but different theories of metaphysics M and M', one or each of which contains information inferentially excluded by the other. Call all such info "d". Since M, M' are both true, and the distinction between two truths is itself a truth, d is true. Since metaphysics is comprehensive over reality by definition, it can exclude no real truth. But at least one of the pair M, M' excludes at least a part of d. So at least one of the pair is not a theory of metaphysics, and the assumption that two such theories exist is self-contradictory. This implies that there is at most one true theory of metaphysics.
Could there be no true theory of metaphysics? According to the above discussion, metaphysics reduces ultimately to the human cognitive syntax (or more accurately, its symmetric self-expansion). So "no true theory of metaphysics" would imply that human beings lack a cognitive syntax. If this were so, human cognition would he random and patternless. But it isn't. So there is one true theory of metaphysics, and this is by definition the CTMU.
It might be objected that the CTMU, being based by definition on the human cognitive syntax, already resides in each of our minds and thus represents no informational gain. But this syntax is not so easily formulated within itself, and equating metaphysical reality to it is neither obvious nor simple. As explained above, a net informational gain comes from freeing information once "locked up" (artificially isolated) within U*-pseudotautologies and the scientific and mathematical theories implicitly based on them.
Now that we have the essential picture, let's try for some detail. Let Ui, be that part of a generalized universe U* to which we refer as the physical universe, or the set of all things directly observable by Ui-observers. This is a recursive definition in which Ui is defined on Ui-observers and vice versa, and varies with choice of subscript. Subscripts correspond to cognitive equivalency classes within U*, or sets of observers sharing the same information-transductive syntax. Ui consists of that part of U* specifically decidable to Ui-observers, and is mathematically equivalent to the cognitive class itself. Assume that the class Ui is human.
The term "metaphysics" is variously construed. In certain usages it encompasses alternate (or "parallel", or independent) realities with no physical meaning. In the Aristotelian sense - and ours it is the totality of theoretical potential relative to the physical universe. While there is nothing mutually antithetical about these constructions, metaphysics relates to physics only as an exhaustive domain of ultimately Ui-effective "hidden causality" undecidable by conventional scientific means. The real universe U* is an extension of Ui by adjunction of this domain.
U* is related to the physical universe by a form of connectedness loosely characterizable as "relevancy"; i.e., it is an extension of Ui generated by causal regression. From Ui, it appears as "causal potential" manifesting itself in Ui as "physical effects". For Ui, U* is unique. For suppose that Ui were contained in many realities corresponding to many Ui-distinguishable metaphysical tautologies. For the differences among them to "register" in the minds of Ui observers, they must be specific relative to the Ui cognitive syntax. As relatively specific tautologies are of lower order than the "tautological" Ui cognitive syntax itself, the universes to which they apply - i.e., the realms of Ui potential and Ui-relevant "alternate reality" they represent - must be partial and therefore properly included in U* (which is complete by definition and theoretically infinite). It follows that U* is unique up to indiscernability: if "other versions" of U* exist, they must be within it, inductively homomorphic to it and indistinguishable from it.
It would be easy at this juncture to point out that by "reifying" information as the quantum transducer, and distributing the quantum transducer over reality, we have removed the major distinction between U* and any theory describing it. Whereas only the latter was formerly regarded as "informational", so now is U*. The U*-decscriptive theory is now merely a sort of endomorphic "self-equivalency" of U* as perceived by Ui. We could conclude our proof on these grounds alone; if U* is informational and "unique" for Ui, then so is the metaphysical information to which Ui regards it as "equivalent". But we can make this even clearer.
A theory of metaphysics is formulated by inhabitants of the real universe it describes. Relative to (Ui , U*), it is a description of U* by the observational subsystem Ui of U*, or a U*-self-description based on a Ui-formulated U*-quantified tautology applying to the "metaphysical" extension U* of the jointly-observable reality (Ui) of the Ui cognitive equivalency class of U*. The circularity of this description reflects the necessary self-referentiality of tautology at the metaphysical level.
Suppose that there exist Ui-discernible theories of metaphysics M and M' on {Ui, U*}. The Ui-discernability" of M, M' implies that they are Ui-informationally disjoint: (M ∪ M') - (M ∩ M') = [illegible] ∅. The "infometrical" form of this relationship is graphically expressed as
M---------(d)---------M',
where the edge (dotted line d) represents syndiffeonesis (difference within a cognitive class)...i.e., information in the sense given above.Now, the disjunctive information represented by the edged exists in M ∪ M', which, by the self-referentiality of metaphysical tautology, implies that it exists in their common universe U*. So the edge d represents real information that must be included in the real universe U*. By our initial assumption that M and M' are both theories of metaphysics and therefore tautological on U*, d must be included in both of them. But since d is defined as disjoint information - whence the way it disjunctively separates M and M' - this leads to a contradiction. I.e., the nonuniqueness of M and M' violates the universality criterion of metaphysics.
Now let's see if we can recap all of this.
Aristotelian metaphysics is universal, containing in principle all Ui-relevant information (Ui-potential) U*. A theory of metaphysics M is an open inferential system which, because necessarily universal, reduces to a Ui-recognizable tautology T on U* heritable in M via generalized rules of inference (where "generalized inference" is just logical substitution). As specific information equates inductively to ancestral generalisms, and U* is both unique and Ui-indiscernible from T, the identification M = T = U* is practically unconditional. Now suppose that there exist two Ui-distinguishable true metaphysical theories M and M'; i.e., two Ui-distinguishable Ui-tautologies T and T'. These can only be Ui-distinguishable by virtue of a nonempty Ui-informationa1 disjunction: i.e., disjoint information d = (T ∪ T') - (T ∩ T') > ∅ recognizable in/by Ui (where the information in T or T' equals the scope (image) of its universal quantifier, and ∅ is the null set). This information d, being the distinction between two Ui-perceptible truths, exists in Ui and thus U*. But as it is disjoint information, one member of the pair (T, T') does not contain it. So this member does not cover U*, is not a U* tautology, and thus is not a theory of metaphysics. On the other hand, M = Uj = 1, 2... Mj, where the jointly U*-exhaustive Mj are all "true", Ui-distinct, and M-nonexluded, does and is.
So the assumption fails, and there can be only one correct theory of metaphysics at the tautological level. This, by definition, is the CTMU. I.e., the CTMU takes this existential proof of metaphysical uniqueness and uses the implied system as the identity of a transductive algebra meeting the conditions for human cognition by its homomorphic relationship to the human cognitive syntax. So for the human cognitive equivalency-class, the universe is generalistically identical to the CTMU tautology.
Soi-disant "metaphysicians" have been debating the merits of so-called metaphysical theories for centuries, usually claiming to argue from "logical" standpoints. The only accord they have been able to reach is an "agreement to disagree". Sadly, this has left the uncloistered masses with a level of metaphysical understanding not far above that which guided them through the last Ice Age, and science without a clue as to the meaning of what it is doing. If this is not a monumental injustice to humanity, then humanity has vastly overestimated its own importance.
Fortunately, mankind does have a protector against the abuses of time and energy being perpetrated upon it even now by mainstream philosophy. With the coming of the CTMU, time has run out forever on this conspiracy of the blind: the blind, sighted at last, can newly behold reality through tears of shame and gratitude; and the rest of us, freed from the rotting conceptual bonds of traditional "wisdom", can finally anticipate the fulfillment of our collective intellectual identity.
As a start down that road, the information in this letter alone exceeds that of a standard Ph.D in "philosophy". Think of it as a primary gateway into logical self-awareness.
Regards, Chris
COPYRIGHT 1992 BY C.M. LANGAN. ALL RIGHTS RESERVED
Copyright (c) 1992 by the Mega Society. All rights reserved. Copyright for each individual contribution is retained by the author unless otherwise indicated.