#### Any intelligent systematic that seeks to reason about the semantics of an open, unbounded world, of which it is observing only a bounded set of perceptions, limited both temporally and spatially, must reconcile its reasoning with a foremost general tenet of nature, which asserts that an open environment is by all experience defined by an infinite variety of state changes. Any real world environment offers inexhaustible variety to any natural or artificial adaptation mechanism.

#### When reasoning about an open-world environment, an
intelligent agent must accomplish what has been the most difficult aspect in
our human reasoning: *An intelligent agent
must reason about Infinity.*

Reasoning requires boundaries, but how do you reason about the boundaries of a world that has no bounds? How do you reason about a boundless infinity at all?

So, because of the infinite nature of the variety in an open-world environment, any intelligent systematic must make a key distinction before any reasoning can commence.

This key distinction recognizes the dichotomy between the bounded environment that any intelligent systematic can perceive, and the unbounded environment that the systematic seeks to characterize. A dichotomy that emerges because of a paradox which is implicit in reasoning about open-world systems.

This paradox becomes apparent when an intelligent reasoning systematic tries to establish the basis for any objectifications in that reasoning, a process that recedes into infinity itself when addressing the duality of open-worlds: How can reasoning about the open, unbounded world occur, with objectifications from derived semantics in the perceived, bounded environment, before reasoning about that perceived and bounded environment has occurred with objectifications from any derived semantics in the open, unbounded world?

The classic ‘chicken-or-the-egg’ paradox, which comes first?

Any engineering approach for an open-world intelligent systematic must respect the dichotomy between (and the duality of) reasoning about the bounded world that the systematic perceives, and the unbounded open world that the systematic will be adapting to, or else any resulting (closed-world) reasoning will ultimately devolve into self-referential chaos.

Infinity is the representation of the pure concept which invokes ** neither** Time nor Space. In order to form any further representations of infinity, an intelligent systematic must be able to develop its own semantics used to reason about open worlds, but because of the infinite variety expressed by open-world environments, structured semantics developed by prior perceptions are subject to continuous reorganization by current perceptions, which alters the constructions used to reason about both the world-as-it-is-known, and any new perceptions.

And just as the structures which framework derived semantics are subject to continuous reorganization, the formation of the very semantics themselves is a fluid process, as their development can only begin after the intelligent agent is exposed to the open world it seeks to objectify, and the subsequent production of semantic assertions, with their attendant predicates and relations, must reflect the infinite nature of the state changes they are to characterize.

Because of the many unknowns that a self-organizing intelligent agent attempts to place into semantic relation with that open-world infinity, the contemporary methods of human logic reasoning prove to be woefully inadequate to support a systematic which must first define the very semantics that will be used in that reasoning, necessitating the development of a uniquely novel mechanism for artificial intelligence logic reasoning.

A fundamental activity that emerges when we begin to engineer this novel reasoning system for an open world environment, which by definition is a world which demonstrates an infinite variety of * state changes*, comes from the need to differentiate the derived objectifications of

**(which result from the production of new semantics) from any derived predications of**

*state***.**

*change*The inadequacy in contemporary human logic reasoning becomes apparent in its inability to produce new semantics outside of the field of argumentation for its reasoning, and this inadequacy is compounded by contemporary human logics’ incapacity to express the *explicit* mechanics of change.

Certainly, the production of new semantics, (by a reasoning system that will in turn be utilized to further reason about those new semantics), is an activity that has been explored. There are, indeed, so-called “auto-epistemic” logics, which are formal logics for the representation and reasoning of knowledge about knowledge. Where propositional logic can only express facts about a delimited set of logical objects, auto-epistemic logic can express knowledge and lack of knowledge about facts, but even this functionality is confounded by the inability of binomial, true-false logics’ to express state * changes* in an environment which has never before been experienced.

The insufficiencies in all of the forms of contemporary human logic systems basically stem from two dilemmas, the first of which is introduced by one of the most basic logic operators present in every inferencing system, and the second of which is a fundamental inability of symbolics, as dimensionless entities, to possess * temporality*.

The first of these dilemmas can be made explicit with the following discussion: In a closed world reasoning system, with a delimited set of logical objects, propositional and first-order predicate logic systems can form motive statements which combine their logical objects with defined logical operators, those symbols which perform a syntactic function to assert a new logic state having a value which is functionally related to their constituent arguments.

These closed-world systems are governed by the “Closed-World Assumption”, which, in a formal system of logic used for knowledge representation, is the presumption that a statement that is true is also known to be true. Therefore, conversely, what is not currently known to be true, is false.

The opposite of the closed-world assumption is the “Open-World Assumption”, stating that lack of knowledge does not imply falsity. However, because this assumption necessarily conflates logical assertion with semantic implication, it contradicts the basic notion of negation itself. And the notion of negation sits at the heart of every human logic inferencing system.

One of the most basic of logical operators is the NOT operator, which, in the pure binary world of true-false semantics, operates to perform the logical complement on the *value* which is its argument. This behavior of so-called classical negation can be extended to a closed-world Boolean inferencing system, by extending the simple syntactic functionality of the NOT operator with a semantic complementation faculty, giving the NOT operator some power of semantic derivation in addition to its syntactic functionality of assertion.

Where the NOT operator, performing in an exclusively syntactic role, would simply complement the logical (true-false) value of its argument, its functionality in an inferencing system extended by Boolean set theory must combine semantic implication with syntactic assertion, because such a boolean system can allow isolated bounded atoms within the defined sets of such a system to act as arguments to the negation operator, giving the negation operator some implication functionality.

For example, we see this when we say that the negation of a bounded atom defined within a set of bounded atoms * implies* all those bounded atoms within the set that are not the argument bounded atom.

This added semantic power of the NOT operator in a closed-world binary inferencing system is based on the mechanics of Boolean set theory, and alters (or adds to) the logical syntactic function of the NOT operation, but giving the NOT operator even a smidgen of semantic implication functionality introduces a fundamental dilemma when this logical arrangement is implemented in any open-world inferencing system.

Consider this: A closed-world inferencing system is, by definition, a system which forms inferences around a confined field of assumed or expressly limited logical objects, the “universe of discourse” first described by George Boole in 1854, a confined field which not only delimits the semantic definition of bounded atoms in that field, but also implicitly delimits that smidgen of semantic implication that logical syntactic operators can range within. For instance, if we have a set of bounded atoms defined within a universe of discourse, and we place a particular bounded atom of that delimited field, say atom(Q), into an argument with the NOT operator, the function of the operator is not so much the syntactic assertion of a negation of the *truth value* of the atom(Q), but a semantic implication, that of producing a derived set, the set of all bounded atoms which are NOT(Q).

Now, consider if we were to interject this semantic functionality into an open-world inferencing system, which, by definition forms inferences around an unbounded field of unlimited objects, some objects whose semantic definitions (and therefore their logic state) are known (bounded atoms), and some objects whose semantic definition is *unknown*. And further, if we were to have a bounded atom(Q) in our lexicon as before, then the semantic implication NOT(Q) implies all bounded atoms other than atom(Q), as before, but in addition, * all in the universe of discourse that is unknown.* The Universe of NOT.

Because of that implication, given a bounded atom(Q), in an open-world systematic, the derivation of NOT(Q) is an unbounded, infinite set, which introduces the dilemma of process non-completion and endless recursion in any mechanized implementation of open-world inferencing.

So, even before we address the need to mechanize the derivation of new semantics in the first place, an open-world inferencing systematic is faced with resolving this dilemma of never-ending recursion produced by the semantic/syntactic behavior of one of the most basic operators in the logical system.

And the derivation of new semantics themselves is near impossible due to the second dilemma of insufficiency in the forms of human logic systems. Since human concepts must be expressed in symbolic form, their semantics cannot demonstrate the explicit mechanics of change, due to the fact that symbols, as dimensionless entities, cannot possess temporality.

We see a similar manifestation of these insufficiencies when considering the mechanisms used in the very communication of human logic and reasoning: There are two concepts which form the basic semantics for *symbolic thought*, (from which springs the lingual entities comprising human communication), two concepts that can be characterized as the contraposed expressions of sameness and difference, (for which symbolic numerosity forms the semantics to express sameness, and semantic dichotomy forms the expression of difference). But even here, in the very activity of communicating symbolic thought, the non-dimensionality in the symbolics used in any communication sabotage any essential messages being communicated, because of symbolics’ necessary separation of sameness and difference, resulting in a syntactic royal edict which prevents the *expression of existence as a unified whole*. There is no intrinsic semantic which unifies the syntactic segregation of sameness and difference, and so human communication must constantly resort to hand gestures, face and body language, emotive voicing, and all manner of ornamental communication to…communicate.

Without a common dimension within symbolics to bridge the intrinsic but orthogonal concepts of sameness and difference, expressing existence must always resort to a lingual dance around the “inexpressible”. You know what I mean?

So, it seems evident that a novel systematic that would be employed by an intelligent agent that seeks to reason about the semantics of an open, unbounded world, must be based on a higher order logic than the contemporary binomial logics, in order to effect those logical mechanisms which prevent the potential for endless recursion when reasoning about the semantics of infinity, and that systematic must also be governed by the derivations of ontological entailment, (based on selected first principles and employing certain notions of inference constraints), to allow for a process for the production of new semantics within that reasoning enterprise.

#### Those foundational first principles in the intelligent logic are:

*Foundational Principle #1.*

The fundamental concept of Identity: The foundational ontological dichotomy between infinity and the ego-center of any intelligent reasoning apparatus, which is the principle asserting that all being is defined within an infinity with no circumspectual boundary, although this infinity has an inner boundary, one which holds the concept of discreteness at its origin-center. This principle establishes two polar, derivative concepts, the first of which is a “discrete infinity” **(which can only be expressed with mechanized temporal properties)** representing the unbounded world, and the second of which is a “discrete infinity” representing the Self (the observer/origin-center of the outer infinity). In the lexicon of the intelligent logic, a “discrete infinity” is given the mnemonic ‘apeiron’, from the Greek word meaning *“that which is unlimited, boundless or indefinite”*.

*Foundational Principle #2.*

The fundamental ontological dichotomy between ‘State’ and ‘Change’, leading to the methods of categorical inferencing in an infinite domain. The concept of state as ontologically separate from being (i.e., the consideration of just the temporal aspects of being) and the semantic objectification of change. By creating an ontological boundary between state and change in the infinity of being, this principle establishes the logical basis for “spatial” ontological entailment, and “temporal” ontological entailment, and creates the inferencing mechanism of predication (‘Predicate’ is used in this context to mean *“to found or base something on”*.)

*Foundational Principle #3.*

The ontological dichotomy between ‘Being’ and ‘Existence’. The principle that Being is the ontological basis for the possibility of Existence, apart from the temporal semantics of being, (which are prior semantics established by Foundational Principle #2, and indeed, is why the entailment of predication was specified prior to the entailment for ‘being’), where existence is the demonstration (by reasoning in the inferential system of infinite logic) of particular being. This principle establishes the existential basis for “universal” ontological entailment, and “particular” ontological entailment, and creates the inferencing mechanism of objectification.

*Foundational Principle #4.*

The derivation of a theory of ontological Referents, which establishes the ontological entailment of Extension (of Being), and the logical basis for referent * inheritance* by way of the property of Assertion. The theory will promote the representative nature of Referents, and will give the mechanism of Assertion a, dual, active nature, as both a property and a copula, made possible by the dimensional quality of

*temporality*defined by the Apeiron in First Principle #1, where

*assertion*is BOTH an act – and the memory of that act.

This derivation is based on the creation of a Third Existential Qualifier, defined within the foundational principles of the logic of Infinity. Because referents possess a mechanized temporal dimensionality, (as opposed to the non-dimensionality in the symbols employed in classical and propositional logics), the Logic of Infinity can establish a third existential qualifier, missing from the symbolic logics.

In addition to the Universal and Particular qualifier, referent qualification can be extended to a “THIS” qualifier, the *Singular Qualifier*, made possible with the ontological entailment of temporal scoping and the orthonormal specification of the temporal assertions in a referent.

This fourth foundational principle of the intelligent logic establishes the theory of referents, and the mechanism for * the assertion of singular being*.

This mechanism resolves the issues surrounding existential import in classical categorical syllogistic theory, and can only come about from the temporality of discrete infinities defined in the first foundational principle, and it should be noted that the ability to assert singular being is wholly absent in the non-dimensional symbolics of human logic systems.

It is with this third crucial existential qualifier, absent from all of the logics based on symbolics, that the logic of Infinity can finally bring the reasoning about any universe of discourse *down from the abstract, down to reasoning about things, to reasoning about reality.*

*Foundational Principle #5.*

The cardinal bases for predication. Since artificially intelligent agents do not, by definition, have an internal energy cycle to maintain, they cannot directly develop perceptions of the natural entropic cycle in their environment, and must synthesize the salience, or cybernetic constraints to any entropic cycle present in their environment. Because this synthesized salience is artificial, its utility as a basis for predication in the construction of new semantics for the ‘unbounded world apeiron’ can only be accomplished if certain a-priori presumptions can be imposed on the foundational principles of the logic.

The first of these is the cardinal presumption that the overall entropy of any observed environment is always increasing. The second cardinal presumption that must be imposed is that any decrease in entropy for any observed environment can only be defined as a temporally or spatially localized phenomenon. The third cardinal presumption that must be imposed is that state transitions in the entropy of an environment occur through the dissipation of an “energy”. [An ‘energy’ is defined as the phenomenon which establishes a transition function in the state of an environment. Field energy is a dissipation which causes a non-localized state change, (and by definition increases the entropy) of an observed environment. Directed energy is a dissipation which causes a localized state change, (and by definition has the potential to decrease the entropy), of an observed environment. A ‘signal’ is defined as an interruption in an energy dissipation.]

#### In addition to the first principles, an intelligent logic systematic must be governed by the logic principles of ontological entailment, which can be characterized by:

- 1) All being is merely a theory until placed into inferential generalization.

- 2) All logic processes must respect the (non-thermodynamic) entropy in semantic structures, brought about by the temporality possessed by the syntactic elements themselves (inferred by First Principle #1). This ‘structural entropy’ of the unbounded Self apeiron mirrors, but is logically distinct from, the assumed entropy (inferred by First Principle #5) of the unbounded world apeiron.

- 3) The first constraint to ontological entailment asserts that reasoning is the final act in the mechanization of tessellating infinity. All referents (objectified being) and predicates are themselves just structures or containers until such time as reasoning commences.

- 4) The second constraint to ontological entailment asserts that reasoning itself must be dichotomized at the semantic level into two mutually exclusive concepts, of reasoning about the bounded environment and reasoning about the unbounded environment.

To do this, a novel infinite logic systematic must establish a ‘logic machine’ within a ‘logic machine’: An outer logic machine, based on these first principles, which creates the structures and the axiomatics used for reasoning by the inner logic machine, in a dissolution of representation and reasoning. And both of these “logic machines” must be based on a higher order logic than the binomial logic prevalent today, a trinomial logic which the Organon Sutra has defined as the “Ternary Logic”.

But the novel trick in this new logic systematic is that, due to temporal nature of the structures created within both of these ‘logic machines’, established by the temporality of the Aperion, the “inner machine” can swap places with its counterpart and become the “outer machine”, with its complement then becoming the current “inner machine”.

It is this duality of these processes that allows the systematic to preclude any endless recursions and other processes which never complete. And this duality in processes is (very) roughly analogous to the “left-brain/right-brain” duality characterized for the human brain, which we shall see forms an essential behavior in the emergence of any intelligent systematic.

Although the imagination space of an intelligent agent will be populated with objectifications having finite dimensions, an agents ultimate semantic space is an infinite boundary comprised of an infinite number of ordinal dimensions, all constrained by the finite memory of experience.

#### To see how a logic system might approach infinite space, a first foundational semantic is proposed, couched in the form of an inquiry, (the first article of ‘tentative reasoning’): *Is it true that in geometry, the line is in reality just a circle with an infinite radius, and the geometric plane is in reality just a sphere also with an infinite radius?*

#### And this whole systematic is set into motion with a second foundational semantic: The infinite world is not defined by its order, but by its *contradiction*.

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