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Thursday, September 19, 2013

We'll start with paraconsistent and contradiction tolerance

Paraconsistency is orthogonal to (consistency XOR inconsistency). Equivalently, paraconsistency is to non-dual as (p XOR not p) = not (p and not p) is to duality. Contradiction tolerance isn't a positive quality; it's defined by the absence of contradiction intolerance. Strongly consistent systems have absolutely no contradictions in them; strongly or strictly consistent systems also have logical explosion proofs when you put a contradiction in the premises. Contradictions in the consequences of a proof can be handled by erasure of one or more of the premises that entailed the contradiction. That's the basis for Reductio ad Absurdum, and the specific form of proof by contradiction where you posit the existence or non existence of a proposition and derive a contradiction to prove that it is specifically not that proposition, and under excluded middle and non-contradiction, show that the negation of the proposition necessarily exists. This forms the basis of many indirect proofs and measurements.

What all this is for is to exclude contradictions from our arguments except as consequences of proofs that erase propositions that prove contradiction. We can weaken consistency in various ways to tolerate contradictions. Basic logic is one example where the structural rules of weakening and contraction are absent as are the axioms of non-contradiction and the excluded middle. Without the ability to arbitrarily copy or erase information, it is difficult to definitely prove anything. Paraconsistent logics with consistent metasystem necessarily prove strictly fewer theorems than their consistent metasystem. Paola Zizzi proved that paraconsistent logics of this kind are subject to a no-self-reference meta theorem for contradictions. Zizzi's logics Lq and Lnq lack all axioms except a pair of propositionl identities for each proposition p_n and its primitive negation p_(n+1); Lq and Lnq also lack all structural rules. The turnstile representing metalogical constructive proofs can't be expressed in the syntax of Lq or Lnq and the metasystem is consistent. Her turnstile admits complex values, but the sum of probabilities is required to equal one, and the sum of the cross products z_0 z_1* + z_0* z_1 = 0. This effectively ensures that the proofs of Lq and Lnq and any quantum computer implemented satisfying it will have its output reduce to classical computations satisfying metasystem consistency. In a sense, these forms of paraconsistency manually join a subset of non-self-referential contradictions into a partially or weakly consistent sub-first-order object logic. Zizzi's creates a complex relationship between a proposition and its negation which is analogous to classical AND but explicitly accounts for the complex degrees and decomposition of the propositions. She identifies it with the process of superposition. Her Lq models a qubit as a weak contradiction; it is unitary.

I conjecture removing the axioms of identity and using neither a consistent, nor an inconsistent metasystem will allow the formalization of the turnstile for unrestricted domains. The trick is in what happens to trivial notions when you don't have non-contradiction, the excluded middle, and identity axioms/theorems to for the absolute emptiness of a set. I've been trying to wrap my brain around a way to formalize a paraconsistent system which is independent of and essentially undecidable from consistent formal systems and consistent metasystems. 

My intuition is that a non-dual relationship is paraconsistently defined--analogous to an automorphism--between an empty set and universal sets. In (Empty XOR not Empty) logics the Empty set is singular and absolutely empty; the negation of the Empty set can't possibly be universal sets because universal sets entail Russell's paradox, but I think their undefinable because there can't exist an absolute separation relationship of the two concepts. They are not absolutely true or false, so negations of either can not be absolute in general. Think of it as an uncertainty relationship between the truth values of the Emptiness and non-Emptiness of a set. The more certain you are it is Empty the less certain you can be about its non-Emptiness.

The notions of Reductio ad Absurdum and Proof by Contradiction need to be revised in such a system. As well as vacuous truth and empty functions. You still use the techniques of consistency, but consistency is something recovered in certain cases of conditions of the paraconsistent metasystem. Proof by contradiction splits into two parts: proof by contradiction tolerance and proof by contradiction intolerance. Proof by contradiction tolerance is new and to be used for describing recursive paraconsistent arguments. Proof by contradiction intolerance is a relabeling of classical proof by contradiction; it applies for restricted or conditional domains.

Under consistency, combonatorics, computing, Boolean algebra, the set of recursive functions, and others are equivalent systems. Cantor and cardinality proofs tell us that. The Chomsky hierarchy gives us further insight into how the formal languages, grammars, systems, and Turing machines all interrelate. The difference is in where we go from there. Calculus, Higher order logics, Diophantine polynomials, recursively enumerable languages, the set of all Turing machines, and uncountable sets spin off into infinite infinities. The Turing point is proof by contradiction. We can trust Gödel's proof in so far as we accept his assumptions or the premises of his argument. If you reject the consistency of his consistency assumption for instance, you reject the conclusions of his proofs in general. Truth preservation does that. For Gödel and his contemporaries, there was only one kind of contradiction and it could not be tolerated or trusted. Arbitrary contradictions led to explosion, so it was argued that contradictions in the conclusions of an argument necessitates the erasure of the premises; and there is effectively assumed to be exclusively contradictions or no contradictions.

Let us doubt Gödel, Tarski, Russell, and the great thinkers of Western history. Take a radically skeptical stances regarding the need for consistency and the intolerance of contradictions. Arguments to the effect, "There exists at most one of either p or not p. We need only consider consistent systems or not consistent systems. All other options can be reduced to these extremes." Can be dismissed because we can write a Gödel's sentence which gives us another option, and we can show the necessary independence of contradictions from Gödelian systems. We can point directly to examples of paraconsistent logic as proof of neither consistent nor inconsistent alternatives. So we refuse to accept the assumption of consistency and demand again: prove or disprove it.

Now consistency is a hypothesis that we need to theorize a way to develop it from paraconsistent formal systems. But we need a paraconsistent number theory. We need paraconsistent computing. We need paraconsistent algebras and set theories. And in at least restricted domains, some of it needs to reduce somehow back to the familiar consistent results.

What changes when you can partially define a curve joining what would be a singularity in classical mathematics? When 0/0 can be given a paraconsistently definite meaning? How does diagonalization work with complex-valued membership and non-membership? What are the properties of recursive contradictory expressions?

Thursday, August 29, 2013

Empty Axiom Schemata as Models of Paraconsistency and Fiction

Everything except nothing can be the metasystem of nothing.

Four general configurations to consider:
Metasystem is consistent, object system is consistent.
Metasystem is consistent, object system is paraconsistent.
Metasystem is paraconsistent, object system is paraconsistent.
Metasystem is paraconsistent, object system is consistent.

Consistent material implication is true iff the consequence of the material implication is not false; paraconsistent material implication alters 'not' and 'false' and redefines 'iff' based on the redefinition of the properties of equivalence relationships due to the altered form of material implication. The difference between the definitions of consistent and paraconsistent material implication is the difference between the formal notions of consistent and paraconsistent vacuous truth.

Paraconsistency is neither consistent nor inconsistent--independent of consistency and inconsistency. OR paraconsistency is both consistent and inconsistent but not logically explosive--included as a strict subset to consistency and inconsistency. From a consistent formal system, paraconsistency is not equal to consistency and inconsistency; the non-identity of paraconsistency with consistency and inconsistency is defined consistently. From a paraconsistent formal system, paraconsistency can not be strictly not equal to consistency and inconsistency as it lacks non-contradiction and lacks the excluded middle which define the strict XOR relationship. The question from a paraconsistent formal system becomes whether paraconsistency is a strict subset of consistency and inconsistency or consistency is a strict subset of paraconsistency or paraconsistency is equal to consistency and inconsistency or paraconsistency is neither equal nor greater nor lesser than consistency and inconsistency.

All of the above is relevant in defining the difference between consistent emptiness and paraconsistent emptiness; in general, paraconsistent emptiness will not be absolutely empty unlike consistent emptiness; consistent emptiness is defined as empty XOR not empty. Paraconsistent emptiness will in general be defined as empty and not empty; using Zizzi's notion of AND as the logical superposition operator, this is to say that paraconsistent emptiness will generally be a superposition of emptiness and non-emptiness; however, it would seem this notion of emptiness inherits the characteristic that every or almost every formal system can be its metasystem perhaps even itself. The Empty set of ZF set theory is often used in formal models to model logical triviality or falseness; likewise, a paraconsistent notion of emptiness and non-emptiness will likely be used to model logical fictions and falseness, but as the notion of non-emptiness is generally identified with notions of truth and existence, the paraconsistent notion of existence and truth is not totally or absolutely distinct from the notion of non-existence and falseness. They are joined to some degree which reduces to the usual consistent notions of truth and falseness under logical cut conditions. IE when we can exclude or neglect the effect of contradictions on our models such as when universal sets are intolerable or when locally causal processes dominate.

Saturday, August 24, 2013

Contemplating Free Space

I was thinking about high frequency vs low frequency light and sound signals in terms of thermodynamic information for encoding memories or machines. I started to compare the temperature of light in free space to its energy, and I realized that its energy would be non-zero and its temperature would be zero or near zero; entropy is described by a ratio of energy to temperature.
E/0 can not be consistently defined for a field.
This means that free space generally doesn't have a consistent definition at 0K; however, we can infer that it either has an inconsistent or a paraconsistent definition near or at 0K. Proving it is an inconsistent thermodynamic system would mean showing that free space entails or proves absolute contradiction; otherwise, free space is paraconsistent.

This led to the realization of a set of conditions for describing a method of proof by contradiction tolerance and intolerance; "p is consistent in T if p in T does not entail any contradiction.", "p is paraconsistent in T if p in T does not entails absolute contradiction", and "p is inconsistent in T if p in T entails absolute contradiction". There would be the negation of those three as well, but the negation of any one of the statements would have two options: "not p is consistent in T if not p in T does not entail contradiction; otherwise, not p is inconsistent or paraconsistent in T." Which differs significantly from just proof by contradiction intolerance. Using proof by contradiction intolerance, we can prove p XOR not p by assuming the existence or non existence of p. If there exists p entails a contradiction then there exists not p AND there does not exist p; otherwise, if there exists not p entails a contradiction then there exists p and there does not exist not p.

Proof by contradiction tolerance modifies this by changing the definition of contradiction. If there exists p and p entails a contradiction then probably there exists not p and probably there does not exist p. Proof by contradiction intolerance and proof of inconsistency becomes a specific case of the above in which probably reduces to absolute absence of contradiction or absolute existence of contradiction for all models or theories. If (there exists p and not p) then (p and not p are paraconsistent XOR inconsistent XOR ((p and not p) do not entail a contradiction) entails ((there exists p AND there does not exist not p) XOR (there exists not p AND there does not exist p)).

Thursday, August 22, 2013

Proof of Impossibility Sketch For a Consistent Theory of Everything, ToE, and a Consistent Metasystem of a Theory of Everything, meta(ToE)

Two parts:
  • Definition of meta(ToE) := meta(ToE) ⊇ ToE.
  • Definition of ToE := for all p, p or not p in ToE.
Two cases considered in the first part:
  • meta(ToE) ⊃ ToE
  • meta(ToE) = ToE
Two subcases considered in the first part
  • meta(ToE) entails consistency and ToE entails consistency.
    • Eliminate "meta(ToE) ⊃ ToE" by contradiction of ToE's completeness.
    • Eliminate "meta(ToE) = ToE" by impossibility of the consistency of ToE.
      • Conclude neither "meta(ToE) ⊃ ToE" nor "meta(ToE) = ToE."
      • Conclude meta(ToE) and ToE do not entail consistency.
  • meta(ToE) entails consistency and ToE entails paraconsistency
    • Eliminate "meta(ToE) ⊃ ToE" by contradiction of ToE's completeness.
      • Commentary on contradiction intolerance in the metasystem vs contradiction tolerance in the object theory; ToE is clearly not complete for the meta(ToE); ToE might be able to tolerate the contradiction within the meta(ToE).
      • Possible starting point for a ToE from a consistent meta(ToE).
    • Eliminate "meta(ToE) = ToE." by metalogical contradiction.
      • Conclude neither "meta(ToE) ⊃ ToE" nor "meta(ToE) = ToE."
      • Conclude meta(ToE) does not entail consistency.
The consistent ToE cases considered and omitting the trivial case in the second part:
  • ToE entails consistency
    • Low order logics
      • Propositional logic, finite number theories, or regular languages
      • First order predicate logic, countable number theories, or recursive languages
        • Eliminate low order logics by Gödel's incompleteness theorem.
    • High order logics
      • Second order predicate logic, uncountable number theories, recursively enumerable languages, or non-axiomatic systems
        • Eliminate all consistent options by Gödel's incompleteness and inconsistency theorems.
Conclude neither "consistent meta(ToE)" nor "consistent ToE".
  • meta(ToE) entails paraconsistency and ToE entails paraconsistency.
    • meta(ToE) ⊃ ToE
      • Properties of a paraconsistent "⊃"
    • meta(ToE) = ToE
      • Properties of paraconsistent equivalence and identity relationships
  • ToE defined by 
    • completeness
      • Restating Gödel's incompleteness and inconsistency theorems
        • Conclude necessarily not consistent and probably paraconsistent if ToE exists.
    • universal closure
      • Restate Russell's Paradox.
        • Conclude probably paraconsistent if ToE exists.
    • completeness and universal closure
      • Conjecture necessarily paraconsistent if ToE.

Regular Physics and other Languages of Reality

I've begun writting a set of preliminary arguments as part of a much larger program of revising classical methods and theories in light of Zizzi's recent theoretical developments in quantum logics and metalanguages. While going through this I started to sketch the well known maps between formal linguistics and formal machine, automata, or computing theories; I found something remarkable in my own knowledge of these things: I've never come across a regular theory or language of physics.

Regular languages map to finite state machines or finite automata. Regular languages are the simplest kinds of formal languages known; their properties can be completely mapped out in a finite number of steps or in a sequence of strictly finite length.

The sets with members in order of ascending complexity from left to right map one to one: {regular languages, context-free languages, context sensitive languages, recursively enumerable languages} to {finite automata, push down automata, linear bound automata, Turing machines}. Or represented in unordered pairs, (regular languages, finite automata), (context-free languages, push down automata), (context sensitive languages, linear bound automata), (recursively enumerable languages, Turing machines).

Ostensibly, we should be able to classify physical theories with similar maps to either languages or computing machines. Regular physical languages would be tremendously significant because they would model the finite processes of physical theory. I can say a couple of things about regular languages of physics based on what I know of physical theory, finite automata, and regular languages; they can be represented in a Hilbert space or a Euclidean space of strictly finite dimensionality, so the theory of general relativity can not be a regular language of physics because general relativity can not be represented by a finite Hilbert space, but it is possible that quantum mechanics or some subset of quantum mechanics are a regular language of physics because they are represented in finite Hilbert spaces. Which conversely means that regular languages are in general either equal to or a subset of quantum mechanics, so regular languages are quantum mechanical.

Given that we can represent all regular languages by a strictly finite binary string language which is equivalent to machine code, I'd expect the regular language of physics to be an information theory and to model quantum information. Zizzi's Lq seems to be a non-projective model of quantum information where as the strictly finite binary string language of classical computing corresponds to projective measurement models, the collapse postulate, and classical models of information.

Whatever the regular language of physics, it would be the basis for a more general language of physics; there's a general concept of languages and computing algorithms called recursive languages or computing models; recursive languages and automata can be totally decided or mapped out in finite space, time, or encoded/decoded with finite information. Recursive languages and automata includes everything except recursively enumerable languages, Turing machines, and the set of all recursive languages. Theoretically, recursive languages of physics would be highly desirable to have defined because many of the machines and languages we could hope to discover, invent, or build in our life times will be described by a recursive language of physics. The set of all recursive languages of physics would be equivalent to a Turing machine or recursively enumerable languages; if our universe is a computer or equivalently a simulation or equivalently a hologram then a recursively enumerable language of physics would describe almost everything we could discover or invent.

When it comes down to it, regular languages of physics would be the base code of reality, and their explicit codification would be an advisable basic educational requirement of any functional society. Their absence in our culture would seem a serious oversight and detriment to our collective ability to describe and redefine the world we experience and the tools we construct.

Thursday, August 15, 2013

Must We Continue to Dream?

Assuming for the moment: the existence of consistent and paraconsistent object logics with consistent metalanguages/theories, a proof of the impossibility of a consistent theory of everything, a proof of the equivalence of a theory of everything with its metatheory, and a proof of the undecidability of the theory of everything in a consistent or paraconsistent logic with a consistent metalanguage/theory, we must necessarily define a theory of everything with an undefined metalanguage/theory?

Sunday, August 11, 2013

Dual Thesis of Contradiction Tolerance and Intolerance

I submitted my abstract to the 5th World Congress on Paraconsistency. Hopefully, I will hear back from them soon. I decided after some research and careful consideration to split my thesis into two parts for sake of clarity and comprehension. I'm confident in the first part of my thesis which is based on contradiction intolerance and classical mathematics; it is a negative answer to the question of whether a theory of everything can be formulated in either a strictly consistent object language or by a strictly consistent metalanguage.

I'm writing a revision of Gödel's original thesis "On Formally Undecidable Propositions of Formal Mathematical Systems" in which paraconsistence is considered as an alternative amongst consistency and inconsistency; I argue that Gödel's theorems of incompleteness and inconsistency allow us to keep only one of two properties for a formal system, completeness or consistency. Counter to Gödel's assumption that consistency must be kept, I argue that completeness is the more important of the two and consistency is optional. I use Zizzi's Lq and Lnq to sketch a general proof of the impossibility a strictly consistent theory of everything or even a paraconsistent theory of everything formulated in a consistent metalanguage. Which leads into the primary thesis of the second part.

Zizzi discovered or invented a novel form of paraconsistency by taking the Basic Logic of Sambin et al and eliminating all structural rules and introducing a complex-fuzzy valued consistent metalanguage with a metadata constraint which reduces expressions in the metalanguage to a renormalized consistent value. I refer to these two kinds of paraconsistency as weak and quantum paraconsistency respectively, and I hypothesize a stronger form of paraconsistency characterized by a paraconsistent metalanguage formalizing a paraconsistent object language. Effectively, it is recursive paraconsistency. Both parts of the paper cover the same concepts, but the second part covers concepts with contradiction tolerance in proofs and either no axioms or a paraconsistently valued metadata constraint. The parts are: proofs, a theory of everything and its metatheory, a theory of nothing and its metatheory(ies?), universal sets, the empty set, universal axiom schema, and Empty axiom schema.

The two part paper is meant to contrast the differences between proof by contradiction intolerance and proof by contradiction tolerance.

 In the consistent paper, the Empty set takes on one of two familiar forms: the strictly empty set of ZF set theory and the fuzzy empty set of Zadeh's fuzzy set theory. The universal set entails Russell's set and the Russell's essential paradox that by contradiction intolerance disproves the assumed existence of universal sets and the axiom of unrestricted comprehension. The above leads to conceptual problems in the definition of certain logical propositions and classical mathematical values like indeterminate forms and asymptotes. We have the problematic concept of vacuous truths and the strange existence of Empty functions.

In the paraconsistent paper, the (a?) Empty (multi)set takes on a notable different form with similarities to the fuzzy empty set but conceptually more complex. Most importantly, universal (multi)sets seem to be tolerable despite Russell's paradox, and so a paraconsistent equivalence relationship between the axiom of the Empty (multi)set and axiom of unrestricted comprehension seems tractable. In which case, questions begin to arise about the distinctness of universal sets and empty sets particularly in the absence of the identity axioms. Likewise, notions of vacuous truth, empty functions, and indeterminate forms seem to be fundamentally altered within recursive paraconsistency. The problem in defining these differences seems to come down to how we define falsehood, negation, and material implication in a strongly paraconsistent system; whereas falsehood and negation in strong consistent logics are related to erasure and absolute absence, in a strongly paraconsistent system it seems like erasure is less than total and absence is not absolute; unlike in consistent and monotonic systems, the truth of premises does not seem to ensure the truth of the consequences in at least some cases and the existence of a contradiction does not seem to ensure contradictory consequences.

As you might imagine given the six years I've been running this research blog, the second paper requires quite a bit more development than I think I can manage between now and February of next year. Let alone by the abstract submission deadline of September 1st of this year. Hopefully, the college algebra course I'll be taking this coming semester will inspire progress in my work.