Published in the Proceedings of the First International Conference on Neutrosophy, Neutrosophic Logic, Set, Probability and Statistics, 1-3 December 2001, University of New Mexico, pp. 121-138. ISBN 1-931233-55-1
Abstract
Our world is run by a logic that has no room for values, by a scientific methodology that disdains the very notion. In this paper we try to redress the balance, extracting many modern scientific findings and forms of philosophical reasoning from the field of complex systems, to show that values can and should be made part of an enhanced normative logic derived from Neutrosophy. This can then be employed to quantitatively evaluate our beliefs based on their dynamic effects on a full set of human values.
Keywords and Phrases: Complex Systems, Axiology, Neutrosophic Logic, Intrinsic Values, Synergy, Dynamical Fitness, Attractors, Connectivity, Holarchy, Teleology, Agents
As we move into the 21st Century it is opportune to take a look at where we are as humans and where we are going. The achievements of the 20th Century in material matters are clear, yet this period has done little if anything to improve humanity as a species, we still spend inordinate amounts of time on war and similar destructive practices. The cry from all quarters seems to be that our lifestyles are becoming unsustainable, that we have lost our values. Yet our world is run by a logic that has no room for values, by a scientific methodology that disdains the very notion. Here we will try to redress the balance, extracting many modern scientific findings and forms of philosophical reasoning from the field of complex systems, to show that values can and should be made part of an enhanced normative logic derived from Neutrosophy, which can then be employed to quantitatively evaluate our beliefs based on their dynamic effects on a full set of human values.
We start by looking in Section 2 at how we define values, and outline in Section 3 an existing science of values. Section 4 introduces complex systems science and we look in Section 5 at the need for a logic of wholes rather than parts. Section 6 looks at non-Aristotelian logics and what a logic of values would require, whilst Section 7 relates paradoxism to complexity notions. In Section 8 we look at the ideas of Neutrosophic logic and relate them in Section 9 to synergy and unpredictability. Finally Section 10 looks at what is still needed to fulfil the goals laid out in this introductory paper. An Appendix relates existing logics to the teleological fitness focus that we adopt.
Like most scientists growing up in a 'value-free zone' it has taken much time for me to realise the self-deception involved in this stance. A recent paper [Lucas2000c] looks at this in detail, and outlines a metascience that includes explicit values to complement the implicit ones already included within the 'scientific method'.
Following on from an heavy involvement in the pursuit of complex systems science I realised that what we called 'fitness' in our experiments (the same 'reproductive' concept that is used in evolutionary biology) was actually a simplified overall value - survival. From this it seemed clear that this could be broken down into many related values (water, food, warmth etc.) all of which were necessary for survival or reproduction and all ignored in the one-dimensional biological practice of 'population genetics'. These form a complex of what I shall call 'primal' values and necessitate a teleological (agent driven) form of science that bridges the objective/subjective divide. In this view these 'values' are simply ends derived from our evolutionary past, and what we can call 'needs' are our genetic predispositions to actions (means) that can meet these ends, by enabling us to generate fitness enhancing trajectories through life. Thus we can regard 'values' as static descriptions of an organism's goals and 'needs' as their dynamic equivalents.
From this insight two others followed, firstly that the number of our needs or values increases with the complexity of the organism and with experience (adding flexibility to better cope with environmental change) - successive stages adding higher values - what I've called 'social' and 'abstract' value complexes (inspired by similar work by [Maslow] and others). Each again contains many dimensions or variables. Secondly, that all these needs need to balance dynamically, they do not exist in isolation but interact in nonlinear ways, both amongst themselves and with their environmental contexts (i.e. we cannot simply add their 'fitnesses' or fulfilments, they are epistatic).
Looking around the Internet to identify work on values led me to discover Robert Hartman and his Science of Axiology. Study of this neglected area of science suggests that three types of valuing are possible. Firstly, we have the binary logic (classification) type of valuing which concentrates on existence or being (ontology). Here we use an either/or judgement, for example we ask is there a tree or not; is he an Arab or not ? This is the 'label' style most common to our descriptive science and philosophy, where we classify objects by lists of such traits or attributes. The second type of value is that of quantity, where we define parameters by size or ranking (cardinal or ordinal numbers), e.g. how heavy is it; is he taller ? This is the 'quantification' mode of valuing common to mathematical science and economics, and is an extrinsic or external (relative fitness) approach. For both of these 'objective' dimensions we assume (implicitly) a subject (or group of subjects) that are 'doing' the valuing of something outside themselves (which may, as in psychology, be another person's 'subjective' experience temporarily regarded as an 'object' to be studied).
The third type of valuing is that currently outside most science, and that is the valuing of wholes, or uniqueness. Here we treat all the values as a unique set of self-supporting attributes that contribute to a one-off absolute fitness or 'value-in-itself'. This is the valuing we see in such approaches as love or art, where the subject is not broken down into parts but ideally is accepted as an self-justifying whole. This is an intrinsic or internal perspective and it is this type of overall viewpoint that we will concentrate on here in trying to adapt Neutrosophy to valuation, where we will go beyond 'subject' and 'object' perspectives entirely and identify ourselves with the system from the inside as it were - we 'become' the system and can then look towards 'our' self-development as a contextually situated or 'embodied' entity.
The first thing that strikes about the [Hartman] approach is that it is an attempt to add logic to values by using forms of [Cantor]'s transfinite sets (Aleph0, Aleph1 & Aleph2). This is ingenious but creates a number of problems which recent work by axiologists such as [Edwards], [Forrest] and [Moore] tries to address, the latter by adopting a finite version of axiology based upon quantum theory. One of the focuses of such work concentrates on ranking combinations of the 3 value dimensions, for example where we employ intrinsic valuations of systemic entities (e.g. love of logic), which is denoted as SI, or systemic disvaluations of extrinsic entities (e.g. excluding the poor), denoted ES. There are 18 combinations of these, the positive aspects are called 'compositions' by Hartman, the negative aspects 'transpositions'. Another problem is in differentiating between partial and full valuations, i.e. where we have 3 values met in a system containing only 3 values, versus 3 values met in a system which has 5 (denoted here as 3in3 v 3in5). Whilst many aspects of this tradition will be included in what follows, we shall instead adopt a different perspective based upon recent developments in the sciences of complex systems.
Over the last two decades of the 20th Century considerable work has taken place in bringing together and taking forward the study of complex systems, defined as systems comprising many autonomous parts and interacting in complex ways. The foundations of this transdisciplinary science go back to work in cybernetics (feedback and homeostasis) and general systems theory by people such as [Wiener], [Ashby] and [von Bertalanffy] in the mid 20th Century, and it was later influenced by work on 'the pattern that connects' by [Bateson]. Following the advent of inexpensive computers this has joined with mathematical dynamical systems theory to form a science of complex systems in their own right, with many topologically isomorphic specialisms appearing (e.g. neural networks, cellular automata, artificial life, evolutionary computation, production systems), for an overview see [Lewin] or [Waldrop]. All these areas consider non-equilibrium systems, where their histories and future trajectories through the possibilities open to them become important, and this demands a more dynamic form of time-critical logic, a logic of change.
Analysing such systems makes use of three critical concepts of complex systems, the first is 'state space' which defines all the possible combinations of the system variables, e.g. 4 binary variables can combine in 16 ways, so state space (the possibilities) here comprises 16 points. It can be seen that this space escalates exponentially with both the number of variables and number of states available for each variable. The second concept is that of an 'attractor', and here the connectivity of the system causes a subset of state space to be preferred, the system will self-organize over time (circular causation) to concentrate on a small area of state space which we call the attractor. Within this perspective there are three types of attractor: point, cyclic and strange (or chaotic). Simplifying a little, the first assumes a static ontology (systemic value); the second a changeable one, one dimension plotted against time (extrinsic value); and the third occupies all the dimensions - system wide (intrinsic value).
In highly complex (high dimensional) systems however there will be multiple attractors present and this brings in the third concept, the idea of 'edge-of-chaos'. Here the system is found to move spontaneously away from either stability or chaos towards a dynamically semi-stable intermediate state (equivalent to the phase-boundary in physics) which comprises a power-law (fractal) distribution of both temporal fluctuations and spatial structure. Analysis of this unstable boundary needs some mathematical sophistication, yet mathematics and complex systems enjoy a difficult relationship. In truth there is no sign as yet of a maths of complex systems that can adequately deal with such complications, just a number of partial mappings that treat abstracted aspects of the whole. I look at this in my introduction "Quantifying Complexity Theory" [Lucas2000b]. In these scenarios the specific connectivity proves crucial, and we can adopt many perspectives, from simple on/off (systemic or Boolean), through weighted (extrinsic or neural) to integral (intrinsic or chaotic). In complex systems science we find that a middle connectivity (generating the 'edge-of-chaos') is needed to obtain the maximum fitness from such value combinations [Kauffman].
By equating values and complexity here I suggest that a maths for one may also form a maths for the other, so we can attempt to evolve such a synthesis. From a psychological standpoint we must I think discard infinities, based upon the practical problems for humans in using such concepts computationally (and also in the difficulty in getting people to relate to such mathematical paradoxes). So what have we left ? The Hartman 3 value stages is a good start. Systemic values can be regarded as philosophical abstractions, my third value complex. Extrinsic values could be regarded as socially driven scientific measurements, my second value complex, whilst Intrinsic values could be regarded as primal values - the survival of the 'organism-in-itself'. It is interesting to note that this way of looking at the matter reverses the standard idea in intellectual circles that 'logic' is at the top of the 'pyramid' of our faculties. We see instead that by intellectualising we remove value, firstly by neglecting the whole for the part, then by reifying the categories into an either/or logic which goes on to form the basis of our system of social values embodied in a legalism of dualist 'right' and 'wrong'. We invalidly reduce intrinsic values to the lower systemic type in many aspects of our daily lives, as seen in our one-dimensional prejudices and our behaviours of conflict and competition.
Given that logic is so highly regarded academically, one way out
of this conundrum would be to adopt an intrinsic form of logic,
in other words one that can accept extrinsic and systemic logics
as special cases. But can we find one ? Fuzzy Logic, which previously
many have been inclined to consider, copes well with extrinsic
values, we can specify a half apple logically FL(0.5,0.5), but
it seems to deal all too poorly with value combinations, a half
AND a half = a half (using Zadeh's intersection operator), i.e.
a similar problem to that found with adding infinities. Moore
makes the point that fuzzy logics require time, unlike systemic
logics propositions are not universally 'true', logical values
change over time (a fresh apple FL(1,0) 
eaten apple FL(0,1) ).
I'd add the idea that intrinsic logic requires context, it is
dependent upon all the other values present, thus needs to be
an interactional form of logic, a connectionist logic. In complex
systems the idea of 'uniqueness' relates to our current specific
position on the 'fitness landscape', in other words it is dependent
upon our history. This can be said to comprise a number of systemic
dimensions (axes) and a number of displacements (vectors), but
the 'now' is defined by how these all interact - what I would
here call intrinsic fitness. This network approach has significant
advantages, since the number of extrinsic paths possible through
the network (the set of possible value relations) grows exponentially
with size and connectivity - a true measure of such intrinsic
value would therefore approach infinity (especially in humans),
as desired by intuitive axiological approaches to the value of
a human life.
A further complication however relates to the concept of synergy, the idea that the whole is more than the sum of the parts (I look at this in more detail along with fitness in my introduction "Fitness and Synergy" [Lucas2000a]) - and this is I think precisely what we are really looking for here. It relates to the complex systems notion of emergence, the creating of new global properties (values) by the combination of parts. Thus no matter how many extrinsic items you add, you don't get the emergent next stage unless you go beyond aggregation and create suitable connectivity - a value add step. This idea also allows us to compare intrinsic sets, a more 'developed' person will have more values ('higher' needs or more discriminative ones), avoiding the ant=human problem of equating the value of all lifeforms. But the potential v actual issue is important here also, we must take into account that experience and education can convert potential (as in a baby) into actual (an enlightened adult) and this possibility must also affect valuations.
Making synergy more specific perhaps, we can imagine a 'precision' axis where we can specify a value in terms of bits, from 1 bit (binary) to infinite bits (irrational number). We can also imagine a 'depth' axis where we can specify the number of values in the system (again from 1 bit to infinite). This gives us an integral plane, or map at a single level. But such interacting entities generate new emergent levels, so we have a further 'height' dimension corresponding to these extra layers (visualise, say, atoms, molecules, cells, organisms, societies, ecosystems, planet). We would regard this 'height' as comprising a further 'holarchic' dimension beyond the three Hartman ones (precision, depth, plane - which correspond to extrinsic, systemic and intrinsic values, but all relating to a single holon [Koestler]). The size (volume) of this 'box' perhaps corresponds to the overall value of the hypersystem [Baas] we are considering. Like the other solutions offered by Forrest and Moore however it has a snag, the general idea of network analysis (and specifically emergence) has proved to be mathematically intractable using all the normal techniques - most complexity problems are related to graph theory and mathematically tend to be NP-complete, insoluble in polynomial time e.g.[Crescenzi & Kann].
For over 2000 years we have used and taught almost exclusively the classical logic of Aristotle (and its modern Boolean equivalent), in which all issues must be 100% true or 100% false and there is no other possibility. That this causes immense problems when applied to humans has been known for many decades. [Korsybski] identified this with the confusion of 'map' and 'territory', this means that we try to force a limited model onto an unlimited reality. From the discussion in this paper on values, we can see that this relates to forcing a 1 bit systemic value (one dimensional) onto a multi-bit extrinsic value (two dimensional) to which it cannot relate, and even worst to forcing the same dualist evaluations onto intrinsically valued humans (four dimensional), a 2 level 'category' or 'type' error - showing the need for a 'higher-order logic' of at least 3 levels, a mathematical meta-model [Palmer]. Given our general pre-occupation with Aristotelian logic, it is sobering to discover just how many non-standard logics already exist e.g. [Suber], so there are a number of less familiar possibilities that might be explored to find a suitable method for a logic of values. And what would we want from such a logic ? I'd suggest the following at least (taking into account Moore's criticisms of Hartman):
a) Evaluate to a higher total value the more values (dimensions) exist, i.e. complexity matters
b) Value intrinsic systems more than extrinsic variables, and those more than systemic distinctions
c) Discard 'Law of Excluded Middle' - which prevents us specifying fuzzy truths (extrinsic values)
d) Provide adequate resolution to deal with real variables, the full variety encountered in life
e) Include a method of treating the 'many' as of higher/lower value than the 'one' (aggregation)
f) Allow for synergy, i.e. A + B can generate an emergent higher value C, a value-add step
g) Differentiate between possibility, probability and actuality, i.e. future choice & past history
h) Be context specific, i.e. allow truth to depend on time, space and interactions/connectivity
i) Differentiate positive-sum & negative-sum trajectories, i.e. dynamical fitness effects
j) Give an intuitively adequate rank ordering for SI, IE and the rest of the combinations
k) Add a measure of 'fulfilment' or personal development, i.e. 3in3 v 3in5, actuality v possibility
l) Allow for circular causality, the multiple interconnected paths of real systems thinking
m) Allow for obligations, the idea that we should not degrade the values of others (morality)
This list seems to go beyond most forms of logic and includes aspects of many types of logic whose implications and technicalities are a specialist task to unravel (see the Appendix for a look at how these relate to our teleologically based fitness viewpoint). Only a few attempts have yet been made to try to combine fuzzy thinking and the more teleologically oriented logics, e.g. [Gounder & Esterline], which brings us perhaps to a novel type of paraconsistent logic (for a quick overview of these see: [Priest & Tanaka]). The one I have been looking at especially is Neutrosophic Logic [Smarandache] which is unique in that it has three axes of logical validity. One is 'truth', one 'falsity' and one 'indeterminacy'. Now of course the latter immediately suggests a role for quantum theory, and also allows for those paradoxes and contradictions that troubled Frege, Russell and Gödel. Additionally in this logic we need not have normalised values (i.e. 0 to 1) we can have 1 AND 1 giving 3 (or anything else), thus synergy seems possible, i.e. the generation of new niches, new opportunities or alternatives. This logic was intended to bridge the gap between literature/arts and science, so is already in the same area as we are considering here, and discusses multiple-value sentences and ways of distinguishing between relative and absolute truths.
Before I consider this as a logic, perhaps I'd better say something about Neutrosophy as a philosophy. The creator, mathematician Florentin Smarandache, was something of an anarchist, a Romanian fighting against the repressive communist regime of Ceausescu in the 1980's. Living a 'double-life' (the 'spin' culture of deceit now familiar to us all) helped him to recognise paradox as crucial, so he came up with a philosophy in which one could prove anything - and also disprove it ! He applied it widely to highlight contradictions - combinations of opposites in stress, and founded the literary movement known as 'paradoxism'. Despite the nihilism suggested, this does have much in common with spiritual ideas (the figure/ground or Yin/Yang) and with complexity science (where we balance static conscious 'rational' order and dynamic unconscious 'irrational' chaos), and so realise that as Smarandache said, "constants aren't and variables won't" - the two descriptions are contextual or transient [Lucas1997]). For humans, if we are too static then we stagnate and die, if we are too dynamic then we disintegrate and die, paradoxically we must be both somewhat ordered to survive and somewhat chaotic to grow. To be human is thus to be indeterminate, to live a contradiction. In an insight from Eastern philosophy, we are not 'either' order 'or' chaos, but 'both' and 'neither'.
We could regard these two axes (of chaos and of order) as those of 'generalisation' (artistic scope) - where we encompass everything but make no distinctions (mystical awareness or intrinsic value perhaps), and 'specialisation' (scientific content) - where we make 'cuts' or valuations across infinite reality, this would relate in dynamical systems terms to taking Poincaré sections (low dimensional projections from a higher dimensional whole). Thus both width and depth can be included, but not at the same time again echoing quantum complementarity and granularity [Smith & Brogaard] - we can see either the whole (dynamic wave) or the part (static particle). We can view the move from 'indeterminacy' to 'true/false' as the making of distinctions, the creation of opposite pairs or dualisms, i.e. systemic values (something akin to [Spencer-Brown]'s 'Laws of Form'), but each such division must exclude all the others in either/or logic. Thus our very act of classifying the world generates its own stresses, a problem not unknown even within conventional science e.g. [Kuhn], where new paradigms or syntheses are occasionally necessary to transcend the tensions of suppressed inconsistencies and contradictions.
Neutrosophic logic itself allows <A>, <Not-A>, <Anti-A> and <Neut-A>. The first two are standard Aristotelian, <Anti-A> is Hegelian (included in <Not-A>) whereas <Neut-A> includes all the other possibilities, i.e. the set of distinctions ignored when looking at opposites (e.g. if <A> is 'white', <Anti-A> is 'black', <Neut-A> includes blue, red, yellow etc., <Not-A> is <Neut-A> + <Anti-A>). The values however are neither binary nor fuzzy but are intervals, allowing vagueness (e.g. it could be 30-40% 'white', 10-20% 'black' and 40-60% 'unspecified'). Another idea included is that of Multispace, where a set M of structure S1 is said to contain also many subsets with different structures S2...Sk not included in S1 - a sort of fractal hierarchy similar both to the layers mentioned earlier and to the structure at the 'edge-of-chaos'.
One of the main tenets of this form of logic is that for any combination of the three dimensions NL(T, I, F) a context or 'referential system' can be generated to make the statements valid. Thus 'truth' can not be applied to all possible worlds, and whether any statement is 'true' in our human world becomes an empirical matter and not an issue of logical analysis. This idea allows us I think to effectively distinguish between intrinsic and extrinsic/systemic value schemes, in that the set of worlds in which a value is 'true' changes with complexity, i.e. context. Within any intrinsic system, any extrinsic value or systemic distinction will fail 'truth' in many frames of reference, whilst the intrinsic value of 'existence-in-itself' will still hold true. For example, the exact systemic statement 'I see the clock showing 12:00" would fail to be truth a minute later, the fuzzy extrinsic truth "I see the time" may hold for many hours, while the intrinsic value "I see" should hold true for all my life. Thus we naturally perhaps can justify higher truth values logically both for intrinsic values v extrinsic and for extrinsic v systemic, if we include domain-specific temporal and (state) spatial context.
In this logic a systemic distinction (a division of the world into system/environment or figure/ground) has a value of one bit, no more, no less - either 'in' NL(1,z,0) or 'out' NL(0,z,1), where z relates to all the undifferentiated content of the two halves. We can go on to make more distinctions, more cuts through the whole. In the limit we obtain a binary set, corresponding to the number of distinctions made, infinite if we wish. An extrinsic value however, a one dimensional measurement, has a variable number of bits of precision - a vague value NL(x,z,y,) where x+y is the range and z includes the measurement uncertainty. In a Fuzzy Logic reduction z disappears and x+y normalises to 1. Again we can make more measurements within our whole, we obtain then a set of reals. From this perspective we see that systemic values are simply low resolution extrinsic ones, crude value distinctions that discard the precision dimension. It may perhaps be quite reasonable to equate, say, 8 systemic values with one extrinsic value of 8 bit resolution (within a linear viewpoint).
Now we take a further step, we group distinctions. We make associations between variables, we create an algebraic matrix, a mesh or network of interactions - an intrinsic system. Given that every systemic is a 1 bit extrinsic, this is a matrix of extrinsics in the limit. Again we see a new perspective, in that extrinsics are just crude intrinsics, low resolution views that discard most of the connectivity effects (the two topological dimensions), i.e. how values interrelate to support or oppose each other (the 'higher-order' causal terms usually ignored in science). To evaluate this stage logically we perhaps may usefully employ a fuzzy matrix logic [Yamauchi] but using neutrosophic triples. However due to the nonlinear and nonadditive nature of such value interactions we cannot adopt a simple global mathematics, applying standard matrix operators to the array. Each intersection pair now may require an individual connective or 'transition function', a local 'law' - reminiscent of a spreadsheet mode (because of the circular causality inherent in complex systems this would then 'hunt' for a solution - that attractor representing the output triple or intrinsic value). It is easy to envisage experimental changes to this array in the search for fitter dynamical solutions. This contextual perspective has much in common with the 'constrained generating procedure' form of emergence pioneered by [Holland], which extends our treatment into more general areas by implying that we must formulate a logic that can generate further triples, i.e. be creative.
Thus we add another stage, the matrix in systems terms (if sufficiently complex) gives rise to emergent properties, a higher level of system, so we have a fourth value dimension, the hyperset of systems, which I earlier called an 'holarchic' value level - a nested heterarchy of intrinsics made up of systems or 'holons'. To take an example, in a rainforest the 'systems' of the geologist, botanist, zoologist, artist and mystic will all see (and value) different environmental 'systems'. These may be disjoint (if the experts are too single-minded) or may overlap, some may be more complex than others, they may differ in 'zoom' ratio (scope in space or time). This is the realm of combinatorics, where everything can be permutated from the set of wholes, factorial combinations of intrinsic modules at many levels. Evaluating this whole obviously causes immense practical difficulties, but we can of course treat relevant subsets as necessary (if we can identify them). Thus 'sustainability' in environmental terms would be such an holarchic valuation. In these cases we need to move from a 2D (plane) logical matrix to a 3D one which includes the various levels, a cubic matrix logic of neutrosophic triples seems required
Looking more at the dynamics, we can relate this to tensegrity, the system of balance proposed for collections of interacting elements by [Buckminster Fuller] in Synergetics (700.00). Here a tension between a continuous 'pull' to truth (an 'attractor' in complex systems terms) and a discontinuous 'push' to falsity (a chaotic move or perturbation in those terms) relates to a balance between convergence and divergence - our edge-of-chaos, or semi-stable state. The indeterminacy relates then to the uncertainty as to whether a change will create or destroy value (or have no effect) - the 'butterfly effect' familiar in chaos theory [Gleick] - this also can include stochastic effects and measurement uncertainties. Trajectories can move in two directions therefore, which Smarandache relates to 'underhuman' and 'superhuman' behaviours - what in my terms I'd call 'dysergy' (negative-sum or sub-animal) and 'synergy' (positive-sum or full human), both of course relating to Hartman's idea of transposition and composition. This relates also to the idea of cancellation and reinforcement of waves in quantum and electromagnetic theory, and we can thus also regard values as being potentially 'in-phase' or 'out-of-phase' with each other. In more general value terms we can say that the three neutrosophic axes correspond to values that are 'good' (positive-effects and thus 'true' beliefs), 'bad' (negative-effects and thus 'false' beliefs) and 'groundless' (unpredictable and thus 'careless' beliefs) - each with respect to a particular situation.
The relation of beliefs to values is a subtle one. We each have a worldview in which we believe certain actions to be advantageous to us whilst others are not, and we tend to reify those theories that have in the past proved advantageous in meeting our needs as 'true', whilst those that have failed as 'false'. But we all have different experiences and contexts, so there is always a tension between these two poles, what I see as 'true' you may see as 'false'. This relates to our often limited vision, and to correctly evaluate the trajectories of an action based upon our beliefs we must take into account how our worldview meshes with those of the other entities with whom we interact. Bringing together our common beliefs in 'truth' generates what we call science, a consensus as to which theories have been tested as being generally effective. Yet even here we make errors, we do not take into account the full range of interactions involved, we reduce the intrinsic whole to extrinsic slices - just as we do individually. It is for this reason that we need to formalise a science of values, generating a logic of interactions that can identify where our narrower beliefs (scientific or more general) fail to be true in terms of the whole.
In such cases the values relate to the hypersystem, i.e. subsets of the whole system may have the opposite form, e.g. for a system of 100 people (simplifying each here to just one systemic value), what I believe is good for me NL(1,0,0) may be bad for you and for 8 others NL(0,0,1) and neutral for the remaining 90 NL(0,1,0). So, for the matrixed hypersystem of interacting values, the overall result would evaluate as NL(0.01, 0.9, 0.09), assuming standard arithmetic connectives (summing over a simple diagonal matrix of triples) and normalising. We see that in overall utilitarian terms the result is 8% negative, thus taking everything into account my belief is proved intrinsically false even if it was extrinsically true. We can also see the relative effects of this action, in that 90% of our social group are unaffected, thus we are not inclined to escalate the issue out of all proportion - a major problem in logics based upon only two axes. Note however that this simple example is unidirectional, it takes account only of the effects of my belief on the group, it doesn't include the effect of the beliefs of the other 99 members of the group on me or on each other. Given better knowledge of what all our values are and how they all interact, we can in principle derive a resultant fitness trajectory for the whole. This applies equally if the whole is just me and the parts are 100 different personal values. For more complex nonlinear connectives the same principle holds, although the mathematical difficulties will of course increase considerably.
It is not our intention here to outline a fully working model, simply to establish the feasibility of so doing. In this section we largely follow [Krivov] (to whom the reader is referred for more technical details) in his attempt to generate a logic based general systems theory for multi-agent modelling. We adapt those ideas here to our neutrosophic value focus. We take as our starting point the definition, analogous both with the classical definition of a Model in predicate calculus and von Bertalanffy's definition of a System, of <Values, Connections>, in other words we have a set of values (function or process) plus a set of interconnections or relations (structure). Our 'agents' originate internal states or goals which must be taken into account, indeed this is our definition of needs - our drives to meet a set of internal values. To include such a teleology we add to our Finite Protocol Language modal operators and time, i.e. Operator(ValueComplex, Time, State) where Operator can be such as needs, prefers, believes, acts etc. and State is the status of the value complex (true/false for systemic, variable for extrinsic, set of included extrinsics for intrinsic, set of intrinsics for holarchic). A Model of the system contains a function stating how the structure and needs will evolve over time, i.e. M(t+1) = F(M(t)), this we refer to as a 'logic machine' (a predicate automata) and it incorporates the connectives or quantifiers that relate values to each other dynamically. The Relations thus have the form Relation(Value1, Value2, Affect) where Affect is a connective that relates the affect on Value1 of Value2, and the overall function is our matrix.
To clarify the real world systems that we are modelling, we have the following progression for any 'agent':
Value Need (for change in that value) Preference
(ranking of alternatives) Belief (fulfilment theory)
Action (environmental output) Reaction (feedback)
Update (belief and need changes).
All agents (and values) may of course act simultaneously, so our
model is a multiobjective constraint satisfaction problem. The
ranking of preferences can cause its own problems in the making
of decisions [Ha], especially within interconnected nonlinear
systems where 'ceteris paribus' (all else being equal) rarely
holds. This brings in logical implication in that some values
imply others to some extent (e.g. the ability to 'philosophise'
implies meeting our primal needs), so a fuzzy implication operator
is required. The timescales for the evolution of the various components
(values, beliefs, preferences, needs, actions) vary, so it may
be possible to model these separately if we wish (given computational
resource constraints). Additionally, important aspects of preferences
and beliefs, as well as actions, are determined by social norms,
which brings in higher-level obligations and canalization of state
space (i.e. [Campbell]'s downward causation based constraints
on alternatives, e.g. laws), and also by environmental issues
(resource availability, costs). It is apparent that needs, preferences
and beliefs can all be fuzzy variables and that we can have considerable
uncertainty as to their standing. This aspect relates to the standard
logical notion of "for all x" (
x), where uncertainty
is zero, through "there exists" or "some x"
(
x) which has variable uncertainty, to complete undecidability
(which we may call Ix). Thus the neutrosophic axis I defines the
improbability that our T/F axes are correct, in other words the
truth value of the believability of our assertions is Bel(1-I),
which relates to the approach taken in the k-calculus in qualitative
decision theory [Ha].
Restricting ourselves just to values, there are in neutrosophic logic many (possibly infinite) forms of definable connective which leaves the possibility of finding a definitional set that matches what we wish to achieve with values. This is complicated to do in formalised logical notation (especially given the open ended set of possible systems, values and nonlinear interactions as here), but we can attempt to do so in more general terms (it may be possible, more formally, to use genetic algorithms to search the space of possible connectives to locate the optimum definitions, given an adequately defined set of goals). I'm primarily interested here however in looking at intuitively simple ways of combining multiple vectors, i.e. n-Tuples - extrinsic values of the form Ei(T,I,F) for i in the set 1..k. If we are to get more from less, i.e. synergy (or conversely less from more, i.e. dysergy) then we need it seems a multiplicative form of connective. For a simplified 2-value case a form that appeals is R = S(A AND B) where R is the resultant value and S is a synergy operator which can vary from zero to plus infinity (or take more nonlinear forms). This allows for both cancellation (S = 0) and reinforcement (S = +2), matching Moore's quantum wave theory, but also allows for other S values for more generality. The AND is our normal logical connective, defined in whatever way we choose for fuzzy truth values. This can easily be extended to cover the multivalued case, and if necessary we can have separate synergy operators for each interacting value, e.g. for 3 values: R = (SAA AND SBB AND SCC ) etc. If we assume normal additivity (e.g. 2 values are greater than 1) then we can generate a fuzzy truth table as follows (A and B both assumed to be 1 here).
| S | R | Comment |
| 2 | 4 | Synergy, positive-sum, increase factor 2 (100%) |
 2 | 2.818 | Partial augmentation, 90° in phase |
| 1 | 2 | Aggregation of values, zero-sum, standard maths |
| 1/2 | 1.414 | Diminution, partially out of phase |
1/  Vi | 1 | Normalised disvaluation, classical logic |
| 0 | 0 | Dysergy, negative-sum, 180° phase cancellation |
Two problems immediately spring to mind, firstly how do we generalise this to neutrosophic triples ? There may need to be interchanges between the T, F & I axes as we vary S, since this seems to convert between T and F. Secondly, can we generalise further to allow for the interaction of a number of input triples to result in the generation of a number of new output triples - as needed for the emergence of new values and levels ? Here we may possibly make use of [Stern]'s Matrix Logic, which uses two by two truth table operators comprising the values true, false, both (synergy) and neither (dysergy) spanning the logical levels of scalar, vector and matrix, and capable of generating autopoietic emergent systems. By generalising these values to fuzzy values and adding the indeterminacy axis we naturally seem to end up with a 3 x 3 neutrosophic matrix logic operator.
It will be noted that our approach to logic isn't static, we move through time - either discretely or continuously, and to cover the increasing generality we need at least a 4th-order predicate calculus of triples. Our viewpoint throughout blurs the distinction between logic and mathematics, and sees logic as simply a 1 bit version of mathematics, whilst mathematics is an infinite bit version of logic. Whether the two can be successfully merged dynamically with values, using neutrosophic logic, remains to be seen.
In this paper we have looked at bringing together three spheres of intellectual activity, firstly the idea of values or axiology, secondly the field of complex systems science and thirdly the area of non-Aristotelian logic. We have examined a number of connections between these fields and can conclude that a new form of paraconsistent logic (Neutrosophic Logic) may prove instrumental in welding together the needs of interactive humans whilst getting to grips with contradictory values and unpredictable complexity. There is much to be done in formalising in detail how this would work in practice, and in taking into account the complications added by evolution and contextuality, within a framework of circular causality and emergence. But the indications are promising that this could be achieved, given sufficient time and expertise.
With the availability of inexpensive computers and the growth in the use of multi-agent simulations we are now perhaps in a position where we can instigate an experimental form of normative logic, looking to use agent evolution to develop and evaluate various logical formalisms. In such systems the agents are generally taken to be autonomous (not globally controlled), teleological (having internal goals), contextual (interacting with other agents and their environment), heterogeneous (different from each other) and autopoietic (self-perpetuating). Whilst much work is yet needed to put such approaches on a firm footing, there are considerable commonalties between the stance taken in this paper and recent work within these areas.
By using a predicate calculus approach common to recent work on formalising complex systems, and both generalising this to a higher-order form suitable for treating multiple levels and including neutrosophic triples as primitives, we obtain a methodology of considerable scope, very suitable for use computationally and potentially applicable far beyond the area of values which has been our main concern in this paper. We should not minimise the difficulties however, we subsume here in our generalisation many specialist fields which all have their own share of difficulties and controversies. Nethertheless, the need to better integrate disjoint mathematical technicalities with interconnected real life applications is clear, and to this end normative concepts can form a bridge that links these two worlds. One final observation is that given our susceptibility for error, in the recognition of values, in understanding their interactions, in defining the scope of our systems and in defining suitable connectives, then the adoption of a form of logic that permits uncertainties and supports paradox seems highly appropriate.
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In this appendix we look briefly at the various forms of logic that we think need to be integrated if we are to create a valid logic of values. We relate the main idea of each of these formalisms to our treatment of values in terms of teleological fitness. One of the key differences we should emphasise is that we replace impersonal logical generalisations by personalised contextual ones. Each statement is related to the viewpoint of a living organism (not always human), which must reason out (not always consciously) ways of meeting their needs within ongoing environmental situations. In human terms this external context is largely social, and so we understand the effects of our actions on our own needs and the fitness of our wider societies to be dependent upon the validity of our socially derived beliefs (we make no distinction formally between ethical or other types of value fitness). By bringing in the idea of personal agency we highlight the tension between global rules and local ones. In a specific context it is often the case that global rules are inapplicable (as they were designed for standardised situations which do not always hold) and local ones are in conflict (different values are mutually incompatible when taken together) so we may need to restrict the scope of our logical rules by imposing appropriate boundary conditions.
Natural Deduction: The definition of a logic without a rigid set of axioms. Our system has no axioms for generality (allowing us to use fuzzy truth values), all connectives are defined as local inference rules applied to arbitrary premises (scientific hypotheses). This open (sympoietic) contextual system contrasts with the closed (autopoietic) global systems of most formal treatments, and discards completeness for applicability.
Classical Logic: Truth value is derived from valid syntactic forms of argument. The premises are considered to be fixed truth values, and only Aristotelian truth values (1, 0) are permitted. We generalise this initially to allow fuzzy truth values for intersection (AND), union (OR) and the other connectives, and later add semantic considerations relating to wider values.
Alethic Modal Logic: Qualifies truth by adding possibility, actuality and necessity. The first we regard as encouraging creative alternatives, new paths through state space; the second denotes our current position in state space. The third implies that a value cannot exist or an act take place unless a condition is met (a critical path analogy).
Deontic Logic: Adds obligation, permission and forbidden operators. We regard the first as an historical social norm intended to avoid fitness reduction (which may be empirically invalid), the second as an allowable alternative (within the social structure) and the last as a denial of a freedom to pursue an alternative (due to implied socially unacceptable disvaluations). Each relation is contextual within a cultural worldview.
Epistemic Logic: Adds knowledge and beliefs. We adopt a coherentist approach to beliefs, based on the circular logic of complex systems, in which all beliefs support each other in a consistent worldview (but one grounded by empirical trial and error, so this is neither absolute nor relative). We can have three types of belief, 'true' - that the proposition will have positive fitness effects if acted upon, 'false' in that it will have negative effects if acted upon and 'indeterminate' where we don't know what result will pertain.
Temporal Logic: Adds future and past operators, which relates to values changing over time (e.g. with metabolism, experience and education) and validity being dependent upon the timescales involved. In our treatment time is regarded as a sequence of discrete steps.
Dynamic Doxastic Logic: Adds propositions and actions that implement changes in beliefs. We thus allow for both changes in base values and changes in the effectiveness of our beliefs in satisfying needs.
Proairetic Logic: Adds preferences, which we relate to rankings of values, and the rankings of alternatives in the achieving of them, plus the changes in ranking over time with actions. This logic includes utility measures, which we generalise to matrixed global fitnesses, incorporating nonlinear interactions of preferences.
Quantum Logic: Adds indeterminacy and probability, discarding the law of excluded middle. Measurement here relates to making a decision between alternatives, the act of choice transferring information (knowledge) from possibility to actuality - a trajectory through state space opening up a new set of resultant possibilities.
Nonmonotonic Logic: Allows new rules to restrict the validity of the existing system, permitting evolution of truth contexts. This allows us to reduce or increase the scope of our values and beliefs, depending upon new circumstances, and adds falsification of existing rules.
Fuzzy Logic: Adds partial fulfilment of truths, an infinite valued form of logic. We allow partial fulfilment of our values (e.g. what we called 3in5) and partial beliefs about them, i.e. partial set membership or completeness.
Mereology: Focuses on the relationship between parts and wholes (usually on a reductionist assumption that the whole equals the sum of the parts). Relates in our treatment to the difference between intrinsic systems and their component extrinsic and systemic values.
Non-Adjunctive Logic: Adds non-additivity, i.e. A AND B
A & B. Here this allows contradictions of joined value systems
and the possibility of dysergy and synergy (i.e. emergence - the
whole not being equal to the sum of the parts).
Higher-Order Logic: Add propositions that act upon sets of predicates rather than atomic facts. Here it relates to intrinsic systems being defined as sets of extrinsic values, and to higher levels of value logic incorporating sets of intrinsics. Thus the 4 orders of logic necessary as a minimum are systemic, extrinsic, intrinsic and holarchic (although these are not regarded as disjoint but are overlapping frameworks).
Neutrosophic Logic: Adds independent true and false axes plus an indeterminacy one. We can have values and beliefs that have both true and false aspects (e.g. good and bad interactions with other sets of values), plus uncertainties as to their effects.
It can be seen that each of these logics expresses only a subset of life's possibilities, so it is not surprising that when applied to real human situations that problems occur and important value data is excluded. It is one purpose of our logic of values to highlight such problems, and to formulate a set of connectives for each context that minimises such reductions.
Note that since any analogue value can be expressed as a string of binary digits, it is in principle possible to generate a logic of values using standard Boolean logic. However the added complications of doing this, together with the already difficult nature of the task, suggest instead approaches that adopt more natural human ways of expressing comparative variability and reasoning. Additionally the advantages of having an axis of indeterminacy, with neutrosophic logic, allows us to keep in mind the partial nature (contextual incompleteness) of all logics, and the vast difference between infinite possibility space and finite actuality space.