**RELATIONS OF ORDER AND EQUIVALENCE **

**IN A BODY OF LAW**

**1 – Relation**

A relation is a set of ordered pairs. The first entry in the ordered pair can be called *x*, and the second entry can be called *y*.

For example, {(1, 0), (1, 1), (1, -1), (-1, 0) is an example of a relation.

A function is also an example of a relation. A function has the special property that, for each value of *x*, there is a unique value of *y*. This property does not have to hold true for a relation. The equation of a circle *x²+y²=r²* defines a relation between *x* and *y*, but this relation is not a function because for every value of *x* there are two values of *y *: √*r²-x²* and *-√ r²-x².*

In mathematics an order is a set on which are defined relations of order (which produce a system), or relations of equivalence (which produce an arrangement).

**2 – Order**

“To put in order” certainly is a work that allows to organize anyway, and in a certain sense, a set of objects, and/or concepts.

Mathematics identifies in an “order” all those properties that characterize all relations that can produce the order itself: i.e., all those properties that any relation must possess in order that the connections of the elements of the set, which is generated, constitute an order of the set itself.

The technical definition of this kind of relation is “relation of order”.

**3 – Equivalence**

Two logic sentences are equivalent if they will always have the same truth value. For example, the sentence *“p→q”* (“IF *p* THEN *q*”) is equivalent to the sentence “(NOT *q*)→(NOT *p*).”

**4 – Introduction to Nomology**

**4.1 – Abstract**

Nomology is the study of human lawmaking (theorisation) that controls and verifies the correspondence of human laws to a correct theory, i.e. to the respect of the statement of true premises, and of a valid argument. If so, then any conclusion is true, any theory is true, any law must be true.

With such a conclusion we do not want to side neither with Natural Law, nor Positive Law, nor with the polyvalent Logic of Karl Popper. We just want to assume that any human law can intervene if, and only if, it is sure to produce a benefit or an improvement to the natural order; in any contrary case no law (control brought about by enforcing rules) is needed, or can be admitted.

In this work we have tried to investigate the process of codification referred to the major aspects of society, nature, science, and other acts or facts token into consideration by a body of law.

**4.2 – Definitions**

Nomology can be defined as the process, which deals with the study of theories and laws. The word nomology is a neologism formed by two Greek terms: *logos*, which indicates ‘the study of’, and *nomos*, which indicates a theory, a law, the government, or the administration of something.

The etymological notion underlying *theory *is of ‘looking’; only secondarily did it develop via ‘contemplation’ to ‘mental conception’. It comes via late Latin *theoria *from Greek *theörìa *‘contemplation, speculation, theory.’ This was a derivative of *theöròs* ‘spectator’, which was formed from the base *thea- *(source also of *theàsthai* ‘watch, look at,’ from which English gets *theatre*). Also derived from *theoròs *was *theoreìn *‘look at’, which formed the basis of *theorema* ‘speculation, intuition, confirmed theory,’ acquired by English via late Latin *theorema* as theorem.

A theorem can be defined as a theory that has been proved, as the Pythagorean Theorem.

A law etymologically is that which has been ‘laid’ down. English borrowed the word from Old Norse *lagu *(replacing the native Old English *æ* ‘law’), which was the plural of *lag *‘laying, good order.’ This came ultimately from the prehistoric Germanic base *lag- *‘put’, from which English gets *lay. *It has no etymologic connection with the semantically similar *legal.*

English has three words *lay*. The common verb, ‘cause to lie’ [OE] goes back to the prehistoric Germanic base *lag- *‘put,’ a variant of which produces *lie. *From it was derived *lagjan, *whose modern descendants are German *legen, *Dutch *leggen, *Swedish *lägga*, Danish *lægge, *and English *lay.*

Law comes from the same source, and it is possible that *ledge *may be an offshoot of *lay *(which in Middle English was *legge*). *Ledger *could well be related too. *Lay *‘secular’ comes via Old French *lai *and Latin *laicus* from Greek *laikòs*, a derivative of *laòs* ‘the people’. And *lay* ‘ballad’ comes from Old French *lai*, a word of unknown origin.

The term *legal*, on the contrary, has a Latin source. The Latin term for ‘law’ was *lex*. From its stem form *leg-* come English *legal, legislator* (which goes back to a Latin compound meaning literally ‘one who proposes a law’), and *legitimate*. *Loyal* is a doublet of *legal*, acquired via Old French rather than directly from Latin. Another derivative of *leg-* was the Latin verb *lēgāre* ‘depute, commission, bequeath,’ which has given English *collegue, college, delegate, legacy, *and *legation.*

A ledger, etymologically, is a book that ‘lies’ in one place. The term was used in 15^{th} and 16^{th} century English with various specific applications, including a ‘large copy of the Breviary’ (the Roman catholic service book), and a ‘large register or record book’ – both big volumes that would not have been moved around much – but it finally settled on the ‘main book in the set of books used for keeping accounts.’ It probably comes from Dutch *legger* or *ligger*, agent nouns derived respectively from *leggen* ‘lay’ and *liggen* ‘lie’ (relatives of English *lay* and *lie*).

We said that the Latin term for ‘law’ was *lex*, defined by G. Devoto as an archaic Indo-European term, which defines the ‘religious law’, and that besides Latin survives only in the Indo-Iranian languages.

2.1 Commonly a theory can be:

A set of general principles drawn from any body of facts or abstract thought (as in science).

Principles governing practice (as in a profession of arts, or in an administrative regulation).

A more or less plausible or scientifically acceptable general principle offered to explain observed facts.

Any theory is an argument, i.e. a sequence of sentences (called premises) that leads to a resulting sentence (conclusion).

An argument is a valid argument if the conclusion does follow from the premises. In other words, if an argument is valid, and all its premises are true, then the conclusion must be true.

Any theory is stated through a theorem, which is the logical process by which verity is deducted from the premises of the theory itself by means of mathematical or grammatical rules of logic.

2.2 Commonly a law can be:

A rule or principle stating something that always works in the same way under the same conditions.

A rule of conduct or action established by custom or laid down and enforced by a governing authority.

The science that deals with laws and their interpretation and application.

A statement of the observed regularity of nature.

A revelation of a supreme will (as the revelation of the divine will set forth in the Old Testament).

2.3 Commonly a law tends to degenerate into:

The control brought about by enforcing rules (forces of law and order).

The imposition of a power.

Logic is the activity pertinent to the demonstration process of a statement (Theory or Law), while only the related science is pertinent to demonstrate a premise.

The doctrine of Natural Law, Positive Law, and Epistemology are the disciplines that deal with the process of legislation and codification.

In the Natural Law, any theory (or law) comes from the observation of the regularity of nature: i.e., there exists a natural order and any codification process moves to the comprehension of the truth from its observation.

Positive Law (or Positive Right) comes from the induction of experiments, i.e., reasoning from a part to the whole, or from a particular to a general conclusion.

In Epistemology theories and laws are just hypothesis. Scientific revolution in mathematics and physics in the early ‘900 demonstrated that science progresses through deep crises and rearrangements of its conceptual apparatuses. Therefore, in the contemporary Epistemology the problem of the definition of scientific criteria is continuously re-proposed.

Bertrand Russell and Rudolph Carnap consider as scientific a theory when all its items can be connected through rules of ‘correspondence’ to observable data.

Karl R. Popper, introducing the notion of ‘falsificability’, considers as scientific a theory only if it is possible to identify the events whose ascertainment can prove its falsity. In any contrary case, theory will result undemonstrated and just ‘corroborated’ (opinion supported with certain evidence).

**4.3 – Theory and Law in Science**

In Science, Theory and Law are terms that indicate a same object, i.e. the statement of a more or less plausible scientifically acceptable general principle offered to explain observed facts. Ohm’s law and Ohm’s theory are the same statement of a general principle of electricity. Therefore, in science law and theory have the same value.

General principles of electricity are accepted because their practical effects are visible, and easy demonstrable in the earth reference system. Nobody can state anything different than that.

General instinct to survival, reproduction, freedom, exchange, and knowledge are needs of nature, therefore they are natural laws, and nobody can state that something coming from nature is unreal, or false. Anyway, their effects are not so easy to be demonstrated as the effects of electricity are, therefore there will always exist someone somewhere who will issue human laws to regulate, by enforcing rules, those activities already regulated by the nature.

In Physics Newton’s theory and the Theory of General Relativity move to a similar gravitation theory. That is the principle of correspondence, which indicates the tendency of two physical laws to coincide when they are deducted from at least two different theories.

A few Epistemologists arrive to state that the scientific assumption of any term is nothing else than the complex of empiric operations performed when they are used. Once such a condition is satisfied, then it is possible to state that the theory under examination is scientifically valid.

Therefore, a theory results verified if observable data could be effectively related each other in the same way of the relations of ‘correspondence rules’ terms connected to data. In any contrary case we can say that the theory results counterfeited.

The inconvenient of this theory lays in the presumption that there will always exist rules of correspondence for all the terms of a theory. In the reality that never happens, because it is possible to demonstrate that almost any scientific theory contains terms with no rules of correspondence (the so called ‘theoretic terms’. Epistemologists tried to solve the problem with several modifications to strict Empiricism, looking overall for a shrewdness, which could give scientific sense also to the propositions containing some theoretic terms. Anyway, remains the fact that the verification concept itself is referred to statements with no theoretic term. These doubts are due to the circumstances that no observation, as accurate as possible, will ever allow to verify any authentic scientific law.

In fact, scientific law state the existence of a certain relationship between variable terms in infinite dominions, so that, in order to verify a law, it should be necessary to verify that the same relationship exists between an infinite number of data (corresponding to variables terms), when it is obvious that data effectively reachable by observation are always a finite number.

Such a difficulty moved the ‘falsification doctrine’ of K. Popper, which states that it is not necessary that a theory results verifiable in order to define a theory as scientific. It is necessary, on the contrary, that its falsity can be proved. In fact, in the act of a theory definition one can indicate a few events whose verification could prove its falsity (potential falsifiers). In any contrary case the theory would not be scientific, but merely metaphysic.

Once the potential falsifiers have been pointed out, scientists will be charged to submit the theory to extreme tests to verify whether it resists to the falsification attempt. If positive, scientists can state that the theory has not been ‘demonstrated’, but just ‘corroborated’.

Therefore, a real science shall be constituted by theories seriously corroborated, and we must note that any corroboration, however serious, will never result as definitive, because nobody can exclude on principle that further proofs could lead to exits in antithesis with the proofs till then performed.

The existence of such a plurality of epistemologies confirms the actuality of the problem about the formulation of a criterion of science, which can give a precise sense to the objectivity of this knowledge, and to its effective superiority in comparison with pre-scientific knowledge, without any further appeal to a presumed absolute metaphysical basis of science.

**4.4 – Law and Theory**

How many civil or common bodies of law have currently as same casual links with social theories as law and theory scientifically have?

A civil or common law must at least demonstrate be a theory, in order to be real (proved or just corroborated), therefore to be right. If not, any civil or common law will just be a power imposition.

In nature every living being is authorized by natural programs (instinct to survival, reproduction, freedom, exchange, and knowledge) as nature needs to oppose any power imposition, even if it comes from a process of codification imposed by a Government, an Administration, and/or any presumed or self-styled positive law.

An organism that formulates a lot of codifications and laws is the State, which becomes the highest Institution when it is assumed as a ‘body of law’. ‘Law and Order’ enforce state laws, unlike scientific laws. Any State law is coercive, even when the case that has to be regulated is not by nature.

Now, if a body of law wants at least to be a theory- as any scientific statement- shall contain demonstrability criteria, or the proof that it cannot be forged. Therefore, to become eligible any law must demonstrate that: I) premises are really scientific; ii), the argument is logic; iii) conclusions come from premises, passed through a coherent (valid) argument, and support all its statements.

In order to do that, any law must demonstrate that its statements are natural (logical) and not forged. In any contrary case the law shall result undemonstrated, therefore false.

One of the most frequent objections to the aforementioned definition is that the demonstrability of a State law can take long and hard procedures, while society’s problems need urgent interventions.

That is nonsense, because a law has a sense only when it is able to classify a real order, or when it is able to really improve a natural order.

Human society, as any other living being society, has a recognized and accepted natural order that allows applying all instincts for living. Such a natural condition needs no human manipulation or forges to live on. No civil or common law is needed, but to improve technology. Therefore, if human laws want to reach a real improvement, they shall have at the most engineering skills, which shall be the definition of highest organisation levels. In any case, such a manoeuvre- as in all engineering process- must come through a theorem (that has been defined as the logical process by which verification is deducted from the premise of the theory itself by means of mathematical or grammatical rules of logic).

Now, even if we make any effort, any terrific enormous effort to try to identify the correspondence of human laws with scientific theories, we can see only a few, very few civil or common laws eligible as theory. In all other cases they are just a power imposition.

**4.5 – First Conclusion**

In the Abstract we have defined Nomology as the study of human lawmaking (theorisation) that controls and verifies the correspondence of human laws to a correct theory, i.e. to the respect of the statement of true premises, and of a valid argument.)

Therefore, with a law we can state a relation of equivalence (natural law) when we describe a natural process, identifying a strict equivalence, or arrangement between the statement of the theory and the condition of reality. With a positive law usually we state a relation of order, that is, we try to organize with a certain sense a set of concepts, setting up an artificial order to the elements of the set. That is, with any positive law (relation of order) we tend to innovate to the natural order of a set, because we believe we can improve its efficiency.

Definitively, consciously or unconsciously, each time we issue a positive law we innovate reality, because we believe we can be more efficient than reality itself.

**5 – INNOVATION**

In a state of nature human beings live through consistent patterns or “regularities” in the way ecosophic systems evolve over time. We can articulate these patterns in the form of theories, and sets, as follows:

**5.1 – Theory of Completeness of Parts.**

Ecosophy arises as the result of synthesis of previous separate matters (disciplines) into a single whole. In order to live and to be viable the system includes three basic sets:

Demand Set

Production Set

Non-Rational Set.

Each set is a closed set. If any of these sets is missing or inefficient, to that extent the ecosophic system is unable to survive and prevail against its competitive systems (i.e., those systems, which impose power, e.g., Political, Military, and Violence System.)

**5.2 – Theory of Entropy and Energy Conductivity. **

An Ecosophic System evolves in the direction of increasing efficiency in the transfer of energy from outside to inside. This transfer can take place through a condition or state that can be called entropy, as in Physics (it is the case of using the same term just because it indicates the same phenomenon.) The higher is entropy the higher is conductivity. Therefore, the higher is conductivity, the lower enthalpy.

Entropy can be argued as the thermodynamic quantity that characterizes the trend of closed systems (i.e., those systems, which do not exchange matter or energy with surrounding environment) to evolve to the maximum equilibrium. Entropy is the quantity that signifies the non-reversibility of natural phenomena, as it is the index of energy degradation. Energy and matter degrades while entropy increases, thus resulting inapplicable.

N. Georgescu-Roegen firstly used the theory of entropy in Economics, in order to emphasize as economic processes are not “circular”, and non-reversible, and that the stock of natural resources tends to exhaust itself (this theory is also used by major ecologists.)

In Information Theory, people use the term entropy as the “quantity” of information (the higher entropy, the lower information.)

In Physics any entropy increase indicates the system’s passage to a state of greater disorder.

Imagine, for example, the passage of water from solid state to liquid state: in solid state molecules are tied each other in the ice crystal lattice, (thus easier to be identified in any fixed position), whilst in liquid state molecules, subject to weaker cohesion forces, are stimulated by a less thermal motion, that is, they are more irregular. In order to transit from solid to liquid state the system has to absorb heath (energy, enthalpy) at constant temperature, therefore its entropy variation shall be positive, i.e., entropy increases in correspondence of the passage to a phase characterized by a greater disorder.

We apply to entropy as the natural chaos, the microscopic disorder of a system, which allows enthalpy (Information, Culture, etc.,) to be acknowledged by, and transferred to ecosophic system (i.e., we can argue that entropy and enthalpy are in a reverse function compared to that given by Information Theory.)

This transfer can take place through a more or less state of entropy, and the entropy level will be the standard of transfer efficiency. In a very personal and subjective scale of entropy, we consider the U.S. as the highest entropic system, and Australia as the lowest.

**5.3 – Theory of Ideal Efficiency**

An Ecosophic System evolves in such a direction as to increase its degree of efficiency. Efficiency is defined as the quotient of the sum of the system’s benefits, B_{i, }divided by the sum of its cost effects C_{j.}

**ΣB _{i}**

**Efficiency = E = ──**

**ΣC _{j.}**

Benefit effects include all the valuable result of the system’s functioning. Cost effects include either individual or system cost.

Taking this trend to its limit, we can assume the notion of *Ideal Efficiency* is obtained when the B_{i }are maximum and the C_{j }are minimum. The theory thus states that as the system evolves, the sum of the B_{i} trend upward and the sum of C_{j }trend downward.

From Mechanics we can assume, as stated by Stan Kaplan “A technical system evolves in such a direction to increase its degree of ideality”.

Chaos Theory supports the aforementioned statement through the Theory of “Strange Attractors”.

From Economics we can assume the following theories:

Cost/Benefit Analysis

Profit Maximization

Scarcity (as a prerequisite to any economic behaviour)

The ratio Cost/Benefit indicates the efficiency of any business, i.e., any human action. In effect any benefit can be material or immaterial, real or presumed. Therefore, a benefit is a very personal appreciation, which can be related to tastes, ethics, religion, ideals, and/or any further material and/or immaterial aspect. Efficiency states that in every human action benefits must ever been greater than costs. In any contrary case we have to admit that our action is useless, and it cannot be communicated to, and/or exchanged with anybody else. Therefore, benefit is equal to utility.

Profit Maximization indicates the relationship we can trace between cost and benefit of any human exchange. That is, if we want to increase the efficiency of our action we need to increase benefits (if we can); otherwise we need to reduce costs.

In any business, producer can increase selling price, which is the benefit of his business (if he can), otherwise he has to reduce his production costs. In the other hand, consumers have to increase the benefits of the product (if they can), otherwise they need to reduce the cost (selling price) of the product. Therefore, in any exchange we can see a bargaining on the selling price, which is at the same time benefit for the producer and cost for the consumer. Only consumers really know the benefit they can get from any product, just because benefit is a very personal appreciation.

Scarcity is a prerequisite to any exchange, because in case of a free product there is no cost, which ting is contrary to any efficiency (Cost/Benefit ratio) analysis.

In effect, if we admit the possibility of satisfying a need for free, we must admit than somebody else has worked for free in order to produce the product that we have consumed. In this case we have abused of another person for satisfying our need, which aspect is not ethic at all.

**5.4 – Theory of Harmonization of Rhythms**

Dynamics can be visualized in term of geometric shapes called attractors. (If you start a dynamical system from some initial point and watch what it does in the long run, you often find that it ends up wandering around on some well-defined shape in phase shape. A system that settles down to a steady state has an attractor that is just a point. A system that settles down to repeating the same behaviour periodically has an attractor that is a closed loop. That is, closed loops correspond to oscillators. The butterfly effect implies that the detailed motion on a strange attractor cannot be determined in advance. But this doesn’t alter the fact that it is an attractor.

In his 1935 article “Synchronous Flashing of Fireflies” in the journal *Science* the America biologist Hugh Smith provides a compelling description of the phenomenon:

*Imagine a tree thirty-five to forty feet high, apparently with fireflies on every leaf, and all the fireflies flashing in perfect unison at the rate of about three times in two seconds, the tree being in complete darkness between flashes. Imagine a tenth of a mile of river front with an unbroken line of mangrove trees with fireflies on every leaf flashing in synchronism, the insect on the trees at the end of the line acting in perfect unison with those between. Then, if one’s imagination is sufficiently vivid, he may form some conception of this amazing spectacle.*

Why do the flashes synchronize? Asks Ian Stewart

In 1990, Renato Mirollo and Steven Strogatz showed that synchrony is the rule for mathematical models in which every firefly interacts with every other. Again, the idea is to model the insects as a population of oscillators coupled together -this time by visual signals. The chemical cycle used by each firefly to create a flash of light is represented as an oscillator. The population of fireflies is represented by a network of such oscillators with fully symmetric coupling -that is, each oscillator affects all of the others in exactly the same manner. The most unusual feature of this model, which was introduced by the American biologist Charles Peskin in 1975, is that the oscillators are pulse-coupled. That is, an oscillator affects its neighbours only at the instant when it creates a flash of light.** **

The mathematical difficulty is to disentangle all these interactions, so that their combined effect stands out clearly.

Mirollo and Strogatz proved that no matter what the initial conditions are, eventually all the oscillators become synchronized. The proof is based on the idea of* absorption*, which happens when two oscillators with different phases “lock together” and thereafter stay in phase with each other. Because the coupling is fully symmetric, once a group of oscillators has locked together, it cannot unlock. A geometric and analytic proof shows that a sequence of these absorptions must occur, which eventually lock *all* the oscillators together.

The big message in both locomotion and synchronization is that nature’s rhythms are often linked to symmetry, and that the patterns that occur can be classified mathematically by invoking the general principles of symmetry breaking. The principles of symmetry breaking do not answer every question about the natural world, but they do provide a unifying framework, and often suggest interesting new questions. In particular, they both pose and answer the question: Why these patterns but not others?

The lesser message is that mathematics can illuminate many aspects of nature that we do normally think of as being mathematical. This is a message that goes back to the Scottish zoologist D’Arcy Thompson, whose classic but maverick book *On Growth and Form* set out in 1917, an enormous variety of more or less plausible evidence for the role of mathematics in the generation of biological form and behaviour. In an age when most biologists seem to think that the only interesting thing about an animal is its DNA sequence, it is a message that needs to be repeated, loudly and often.

**6. – FINAL CONCLUSION**

As in Medicine it’s not responsibility of the analyst to issue the diagnosis, so in nomology is not responsibility of the analyst to come to the definition of any human law.

Our duty was to demonstrate that positive law cannot be something special than any other theory.

Theory and law (either common or civil) have the same logic. That is “Quod Erat Demonstrandum”.

### Enrico Furia.

*School of World Business Law*

Douglas Downing, Dictionary of Mathematics Terms, 2^{nd} ed. Barron’s Ed. Series, New York, 1995.

John Ayto, Dictionary of Word Origins, Arcade Publishing New York, 1990.

Douglas Downing, Dictionary of Mathematics Terms, Second Edition Barron’s, New York, 1990

Giacomo Devoto, Dizionario Etimologico, Le Monnier, Firenze 1968.

Ecosophy is intended as a global knowledge, which includes all aspect of life of living beings. For further description, see of the same author: “Introduction to the Ecosophic Set”; HYPERLINK “http://www.worldbusinesslaw.net/”www.worldbusinesslaw.net

Nicholas Georgescu-Roegen, (Konstanz 1906), Rumanian economist, who first applied Thermodynamics laws to Economics. See *Economics and Economic Process, *(1971)

Stan Kaplan, *An Introduction to TRIZ, *Ideation International Inc., 1996

Ian Stewart, Nature’s Numbers, Basic Books, New York, 1995