Matrixial Logic: Forms of Inequality

PREFACE

In I Want To Love But: Realising the Power of You, I called this sort of communication The Geeky Bits.

The ideas presented are intended to be accessible to the general reader, not only readers with a background in logic, philosophy or science.

Matrixial Logic (ML) is a symbolic logic.

•You know what a matrix is: a set of connections. Logic is showing how things connect;

•The symbols are arbitrarily chosen correlates to linked sets of ideas.

ML is a system of symbolic logic, a tool for organising thinking about thinking, which I first developed at university. That was a long time ago, when everything was in black and white. Well, Polaroid, certainly.

I found it a useful tool in getting through a law degree at Oxford. I’ve since found its utility as a tool in many walks of life: from business, to personal relations.

ML is the tool which made possible the discovery of ASR1 technology and Congruence Quotient2 technology: both of which form the CBT core of I Want To Love But.

ML is a philosophical tool: a method for organising how we think. It is in principle applicable to anything that we think about.

ML does not tell you what you should think about anything. It does not prescribe the contents of thoughts, nor what conclusions you might decide to reach in your thinking about anything.

ML instead provides a set of rules for rational thought. If you wish to think rationally, you can use the rules of ML.

I do not claim that use of the rules of ML is the only way in which you can think rationally. I cannot make such a claim, because I cannot conceive of how the truth-value of such a claim could be established.

In ML terminology, I cannot think of a frame of reference in which one can solve:

(A) ≠ (notA) = E

>

(Knowledge) ≠ (notKnowledge) = E?

So, at least I’m taking my own medicine.

The bitter pill, is that ML reveals certain structures, stances, or modes of reasoning to be fallacious. That is helpful, because it allows us to gain clarity.

This is not a survey of the massive field of symbolic logic. I do not say that any of those modes of symbolic discourse are right, or wrong.3 Nor does ML rely upon any of them.

The core method of ML can be analogised to syllogistic reasoning, but that is correlation, not cause.

Once you become familiar with using ML, you should find that you can use that tool in any of your thinking. In the course of this book, we’ll look at curated applications of ML: which reveals its wide range.

I entirely accept that, in publishing this monograph, I place the premises, propositions and conclusions (the Structure) subject to the conventions of philosophic discourse.

I try to keep a balance in this monograph between length and detail of explanation, and effective communication. The metric I use in weighing that balance, is whether I think that the reader has been given sufficient information to apply the ML tools.

Without committing the special pleading fallacy, I note the commonplace that philosophic discourse has a tendency to become burdened with demands for proof which are not observed in other disciplines.

I therefore offer, under the principle of humility, an election which lies entirely within the sovereign power of the reader:

•If ML tools should prove to be of practical utility in reasoning, then to appreciate such use for its own value;

•To reduce such appreciation by so much as the reader is unsatisfied, for whatever reason, as to some part of the ML rules below explained;

•To place such appreciation subject to any conditionality the reader considers appropriate;

•To refuse to consider applying ML tools until the reader can acquire by that reader’s standards such other information as is sufficient to satisfy such conditionality;

•To conclude, subject to further information, that as presently formulated, the ML tools have no value.

I appreciate the investment of any of your time in perusal of the remainder of this monograph.

I have found the ML tools explained in these following pages to be useful. If that be not so for you, then my hope is at least that, in determining the causes of such failure of effect, such exercise brings new information to you, which may itself be of some utility.

Paul Chaplin

2020

How to Read This Book

Turn the pages right to left, start at the top left corner and work your way across and down.

I’m having fun. And I want you, the reader to have some fun too. But, I shall make no bones about it: whether or not you are a professional logician, some of these Chapters are going to be hard going.

Indeed, labourers in that field will probably find it harder than others, because what they read is so challenging to ideas which are stamped into the DNA of classical and modern logic.

This is very much a book of parts:

Chapter 1 sets out some background. This is for the lay reader.

Part I

Thesis

Chapter 2 sets out the foundations of Matrixial Logic. There is no anticipation that much of what you read in

Chapter 2 will make much sense to you. Please treat Chapter 2 as if you were learning a new language, or the rules of a board game: with the merit that it has a very small alphabet and syntax.

Antithesis

Chapter 3 is one long Socratic dialogue. I use peak points from classical logic, and modern logic, to explain, by argument: (i) what problems there are4 with those ideas; (ii) more of how ML works; (iii) why ML works; (iv) what you gain from using ML.

Synthesis

Chapter 4 is the fulcrum. Some people may not need all of Chapters 2 and 3 to follow what is being shown in Chapter 4. Even so, the previous Chapters should come alive, once you can look back at them from the mountain peak of Chapter 4.

Part II

Chapter 5 is, in effect, a showcase for the practical applications of ML. In this case, in the field of linguistics. If ML can make material advances in our thinking about how language works, then that alone was worth the price of admission.

Chapter 6 is a bridge. The concepts here expand in Secret Self, but are of small application in the later Chapters.

Chapter 7 is another fulcrum thesis. It sets out how Matrixial Logic radically alters our views of time.

Chapter 8 digests the earlier elements of Part II, providing a working model of the Human Self. This is a summary Chapter, which signposts the elaborations in Secret Self.

Part III

Chapter 9 uses what we have learned about ML to proffer some propositions about key concepts in western thought.

Chapter 10 provides arial footage of the city of western philosophy, following detonation of the chain-reaction thermonuclear devices, in the previous Chapters.

The Appendix sets out in 20 short Propositions, what Matrixial Logic says of itself.

You can find a handy set of Tables, as a post script. These provide a listing of the Symbols used in the book, as well as how to type them. There’s also a Glossary of ML terms.

There’s no separate bibliography, as it would be as long as the book itself. Throughout the Chapters, you’ll find footnote references to all the key texts.

Clues

You’ll find references to Clues in the text.

I’m not trying to be smart or annoying.5 These are seeded deliberately, to induce you to go down lines of thought. So that those other thoughts, prompted by the clues, syncopate with what’s in the main text.

When you are reading, there’s lots of way stations. Places where you’re deliberately invited to put the book down, and have a think. The clues are one set of lay-bys on the journey.

Experiences

All the way through the book, you’ll find Experiences that I ask you to undertake. It is really important that you follow them.

For a start, they give you an entertaining diversion from the endless lines of equations and sometimes stuffy language of logic.

More importantly, through the Experiences, you will get an understanding of the equation ideas, not just in your intellect, but in your emotions.

You also get to remember those ideas, in a way that no amount of blackboard repetition would.

These are experiments, in the ordinary scientific sense:

•We have a theory;

•We invent tests of that theory: operations with pre-defined outcomes, derived from the theory;

•We test the outcomes against the theory.

By doing the experiments, you the reader gain important knowledge. The way you look at the world will change. Sometimes a little: sometimes a lot.

Lines of arguments in this book are challenging to many precepts: in logic, philosophy, psychology and science. Sceptical response is understandable.

The Experiences provoke significant difficulties for sceptisism. As we sometimes say, following an Experience: well, whatever is going on here, something is going on.

Matrixial Logic provides an explanation. Now, you can seek to refuse the explanation. But you can’t deny the Experience. Refuting the ML explanation poses some difficulty: because you only got to do the Experience by reason of the practical operation of ML.

If there were only Experiences limited in some dimension, then they might be capable of being disregarded as just an artefact of something odd. But you will undergo Experiences in many different areas of your interaction with yourself, and the world.

The Experiences are not tricks: you can see every element of them. There are no hidden wires. You just get a simple set of written instructions: you follow them, and you decide what the result is.

The Experiences do not operate under hypnosis. There is no induction such as would precipitate a hypnotic state. There’s just a few lines of written instructions. Indeed, you will find that you experience a widening of awareness.

These Experiences will show you that logic is not the opposite of emotion. 6 It is logic in action, in our life.

Logic is just as important as love, and is as universal as love. It just lasts longer, and we may one day learn to talk about it, without embarrassment.

__________

1 Anxiety-Soothing Rhythm

2 Five Slices

3 Although I do make clear that ML provides a perspective in which they are not agents for clarity

4 As entirely admitted by modern logicians

5 Although I probably succeed in one, but not the other. Sorry

6 Sorry Commander Spock

CHAPTER 1

FORMS OF EQUALITY

Matrixial Logic is new. You can google the phrase and no results will return.

ML is a different way of doing logic. Rather than abstracting at higher orders of levels from reality, the better to free logic from the confusatory shackles of the ordinary, ML is rooted in reality.

Logic, after all, is the pursuit of reality in understanding. Logic can and should avail itself in that pursuit, of all the technical tools furnished by advanced civilisation.

But logic must guard against the mechanical fallacy: that which mistakes machinery for understanding.

As we shall survey in this Chapter, logic has, since its period of Greek innovation, sought so often to perfect the mechanical fallacy. It was only in the 20th century that First Order Logic retreated expressly from that pursuit.

With no disrespect, FOL has retreated from the battlefield of reality, and now engages only in mythicism of forms of equality. The self-recusal is understandable.

From Leibnitz and Lambert to Russell and Godel, logic became trapped in a hall of mirrors. Haunted by problems incubated by self-reflecting. Until such a brilliant thinker as Russell could terminate his reflections with the conclusion7 that the only meaningful word in a sentence is the.

Logic is a scientific pursuit. Fuelled by spectātiō internal and external, coupled with insight and intuition, and informed by technological discovery. Then tested in theorems against reality: resulting in verification or falsification.

We appreciate that the tests of what is scientific, as adumbrated in logical positivism by the Vienna Circle,8 cannot provide a definitive account.

One of the functions of the Experiences, is to allow experiments to be conducted which move beyond the circular dictates of verification and falsification.

The reader acquires new knowledge: a union of the subjective and the objective.9 This goes beyond belief in the reality of a fact. It involves knowing the reality of a matter.

If it be asked how logical theorems can be tested against reality, the course of this book should provide adequate answer.

Achieving understanding of reality can be pursued from an armchair. But adoption of such understanding by those placed contingently at the axis of social change, can alter the historic fate of nations.

History of Symbolic Logic

Symbolic logic (SL) is commonly thought of as beginning with the Modern.

However, we must begin with Aristotle.10 The compilation of his 6 works concerning logic, The Organon, left a legacy which still engages us today.

The core is syllogistic reasoning, demarcated under four different types of Categorical Sentences: universal affirmative (A), particular affirmative (I), universal negative (E) and particular negative (O). We will have cause to return to these Categorical Sentences later.

Aristotle was happy to denote place-holders for idea by letters. This is not of course the same as designating operators for functions of ideas.

Such designation, in principles, raises the question of the relationship between the symbols, functions and reality:

Inferences

:

Equation

:

Symbols

:

Ideas

:

Reality

Modern symbolic logic expressly disavows conjunction of logical inferences, with underlying reality. That is considered to be a forlorn project of failed Higher Order Logic.

ML does not take this position. In ML, we are creating equation rules, which have hylomorphic relationship to reality.

Hylomorphism, (from Greek hylē, matter; morphē, form), in philosophy, metaphysical view according to which every natural body consists of two intrinsic principles, one potential, namely, primary matter, and one actual, namely, substantial form. It was the central doctrine of Aristotle’s philosophy of nature.

Matter and form are parts of substances, but they are not parts that you can divide with any technology. Instead matter is formed into a substance by the form it has. According to Aristotle, matter and form are not material parts of substances. The matter is formed into the substance it is by the form it is.

Surprisingly, as John MacFarlane pointed out, the father of both formal logic and hylomorphism was not the father of logical hylomorphism (MacFarlane, 2000: 255).11

One immediate and obvious difficulty which we meet is that it is not easy to explain why Aristotle uses letters of the alphabet, like ‘A’, ‘B’, ‘C’, instead of concrete terms if he did not distinguish between logical form and logical matter. Here is an exemplary formulation for the first syllogism in the first figure (Barbara) from the Prior Analytics: if A belongs to every B and B belongs to every C, it is necessary for A to belong to every C (Pr. An., 25b37-9).12

There will be much more to say about this in later Chapters.

It has been noted that the history of Euler Circles or Venn Diagrams goes back to the middle ages. Margaret Baron traces13 the ideographic concept back to Ramon Lull:14

This really does look like John Venn, 600 years early. Leibnitz15 is credited with the first invention of symbolic logic. However, his contribution is commonly understood not to have contributed to the development of SL for the next 200 years. As Margaret Baron wrote in 1969:16

Boole, writing in the mid-Victorian period, nevertheless referred to Leibnitz as if the two had shaken hands across the centuries.

Sadly forgotten is the contribution of JH Lambert.17 Such was his repute during his lifetime that his contemporary Kant18 called him der unvergleichlicher Mann. (the incomparable man). Which was the 18th century equivalent of Jimmy Page bigging up Jimi Hendrix.

His first logic was Neues Organon oder Gedanken über die Erforschung und Bezeichnung des Wahren und dessen Unterscheidung vom Irrthum und Schein.19 (New Organon, or Thinking Concerning the Exploration and Designation of Truth, and of the Distinction between Error and Appearance).20 Lambert updated his symbolic logic project over the years. Lambert was to have an influence on Venn.21

Boole is more famous for Boolean Logic, which in its later development became in effect the basis of the binary system. Translated into circuit mechanics, this became the basis of modern computing. I type these words with this technology thanks, in original part, to Boole.

The Author must admit to a childhood fascination for George Boole (1854-1864). Like the Author, he was a son of Lincolnshire. The Mechanics Institute, at which he taught, is actually adjacent to the Lincoln Library: the locus of the Author’s self-education in childhood.22

The Author recollects not being able to count the number of times walked past Boole’s house at 3 Pottergate, about 10 minutes’ walk from that childhood library.

(By Logicus - Own work, Public Domain)23

Boole wrote: 24

In every discourse, whether of the mind conversing with its own thoughts, or of the individual in his intercourse with others, there is an assumed or expressed limit within which the subjects of its operation are confined.

The most unfettered discourse is that in which the words we use are understood in the widest possible application, and for them the limits of discourse are co-extensive with those of the universe itself.

But more usually we confine ourselves to a less spacious field. Sometimes, in discoursing of men we imply (without expressing the limitation) that it is of men only under certain circumstances and conditions that we speak, as of civilized men, or of men in the vigour of life, or of men under some other condition or relation.

Now, whatever may be the extent of the field within which all the objects of our discourse are found, that field may properly be termed the universe of discourse. Furthermore, this universe of discourse is in the strictest sense the ultimate subject of the discourse.

No doubt it is sentimentality, but the Author still finds those words expressive of a beautiful idea: that the discourse of our mind is existentially in accord with the universe. Please forgive the digression. Back to the brief history of SL.

As Chris Dixon writes in How Aristotle Created the Computer:25

Boole is often described as a mathematician, but he saw himself as a philosopher, following in the footsteps of Aristotle. The Laws of Thought begins with a description of his goals, to investigate the fundamental laws of the operation of the human mind:

The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the symbolical language of a Calculus, and upon this foundation to establish the science of Logic... and, finally, to collect... some probable intimations concerning the nature and constitution of the human mind.

He then pays tribute to Aristotle, the inventor of logic, and the primary influence on his own work:

In its ancient and scholastic form, indeed, the subject of Logic stands almost exclusively associated with the great name of Aristotle. As it was presented to ancient Greece in the partly technical, partly metaphysical disquisitions of The Organon, such, with scarcely any essential change, it has continued to the present day.

Professor De Morgan26 said of Doctor Boole in his Budget of Paradoxes:27

I might legitimately have entered it among my paradoxes, or things counter to general opinion: but it is a paradox which, like that of Copernicus, excited admiration from its first appearance.

That the symbolic processes of algebra, invented as tools of numerical calculation, should be competent to express every act of thought, and to furnish the grammar and dictionary of an all-containing system of logic, would not have been believed until it was proved.

When Hobbes, in the time of the Commonwealth, published his ‘Computation or Logique,’ he had a remote glimpse of some of the points which are placed in the light of day by Mr. Boole.

The unity of the forms of thought in all the applications of reason, however remotely separated, will one day be matter of notoriety and common wonder; and Boole’s name will be remembered in connexion with one of the most important steps towards the attainment of this knowledge.

As Chris Dixon continues:

Boole’s goal was to do for Aristotelean logic what Descartes had done for Euclidean geometry: free it from the limits of human intuition by giving it a precise algebraic notation. To give a simple example, when Aristotle wrote: All men are mortal.

Boole replaced the words men and mortal with variables, and the logical words all and are with arithmetical operators:

x = x * y

Which could be interpreted as "Everything in the set x is also in the set y." The Laws of Thought created a new scholarly field -mathematical logic - which in the following years became one of the most active areas of research for mathematicians and philosophers. Bertrand Russell called the Laws of Thought the work in which pure mathematics was discovered.

The evolution of computer science from mathematical logic culminated in the 1930s, with two landmark papers: Claude Shannon’s A Symbolic Analysis of Switching and Relay Circuits, and Alan Turing’s "On Computable Numbers, With an Application to the Entscheidungsproblem." In the history of computer science, Shannon and Turing are towering figures, but the importance of the philosophers and logicians who preceded them is frequently overlooked.

A well-known history of computer science describes Shannon’s paper as possibly the most important, and also the most noted, master’s thesis of the century. Shannon wrote it as an electrical engineering student at MIT. His adviser, Vannevar Bush, built a prototype computer known as the Differential Analyzer that could rapidly calculate differential equations. The device was mostly mechanical, with subsystems controlled by electrical relays, which were organized in an ad hoc manner as there was not yet a systematic theory underlying circuit design. Shannon’s thesis topic came about when Bush recommended he try to discover such a theory.

…Shannon’s insight was that Boole’s system could be mapped directly onto electrical circuits. At the time, electrical circuits had no systematic theory governing their design. Shannon realized that the right theory would be exactly analogous to the calculus of propositions used in the symbolic study of logic.

We must mention in the same breath as Boole, William Stanley Jevons,28 with whom Boole had a correspondence later in Boole’s life.

Jevons described Boole’s SL work as29 using dark and symbolic processes. However, Jevons gave perhaps the most succinct summary of the operational principles of Bool’s SL:30

Dr. Boole’s remarkable investigations prove that, when once we view the proposition as an equation, all the deductions of the ancient doctrine of logic, and many more, may be arrived at by the processes of algebra. Logic is found to resemble a calculus in which there are only two numbers, o and I, and the analogy of the calculus of quality or fact and the calculus of quantity proves to be perfect.

Jevons himself explained his motivation in creating an SL,:31

In this small treatise I wish to submit to the judgment of those interested in the progress of logical science a notion which has often forced itself upon my mind during the last few years. All acts of reasoning seem to me to be different cases of one uniform process, which may perhaps be best described as the substitution of similars. This phrase clearly expresses that familiar mode in which we continually argue by analogy from like \ to like, and take one thing as a representative of another. The chief difficulty consists in showing that all the forms of the old logic, as well as the fundamental rules of mathematical reasoning, may be explained upon the same principle; and it is to this difficult task I have devoted the most attention.

The new and wonderful results of the late Dr. Boole’s mathematical system of Logic appear to develop themselves as most plain and evident consequences of the self-same process of substitution, when applied to the Primary Laws of Thought.

Should my notion be true, a vast mass of technicalities may be swept from our logical text-books, and yet the small remaining part of logical doctrine will prove far more useful than all the learning of the Schoolmen.

We would respectfully appropriate an element of Jevon’s catechism as representing our own agenda. However, we are not so bold as to claim that Matrixial Logic can serve as the imperator of All acts of reasoning. That was not our purpose in developing it.

We do not say that it cannot so serve. But we cannot say that we have fathomed the limits of Matrixial Logic sufficiently to warrant efficacy in such boundless application.

Jevons, being made of sterner Victorian stuff, felt it competent to claim:32

that these Laws are true both in the nature of thought and things. Given that science is in the mind and not in the things, the Laws of Thought seem to be purely subjective and only verified in the observation of the external world. However, Jevons argues that it is impossible to prove the fundamental laws of logic by reasoning, since they are already presupposed by the notion of a proof. Hence, the Laws of Thought must be presupposed by science as the prior conditions of all thought and all knowledge. Furthermore, our thoughts cannot be used as a criterion of truth, since we all know that mistakes are possible and omnipresent. Hence, we need to presuppose objective Laws of Thought in order to discriminate between correct and incorrect reasoning. It follows that Jevons regards the Laws of Thought as objective laws.

Jevons began his analysis by objecting to the non-commutative property of Aristotle’s Dictum de Omni et Nullo: the foundation of the syllogistic system of logic.

•The principle that whatever is universally affirmed of a kind is affirmable as well for any subkind of that kind;

•The related principle that whatever is denied of a kind is likewise denied of any subkind of that kind.

Venn (the overlapping circles guy: although that work was really made famous by his successor) wrote:33

To my thinking, he and Boole stand quite supreme in this subject in the way of originality; and, if the latter had knowingly built on the foundation laid by his predecessor instead of beginning anew for himself, it would be hard to say which of the two had actually done the most " (p. xxxi)

We must be indebted to Venn for his collation of historical attempts at symbolic logic.

We cannot improve on Venn’s own summary. He uses the paradigm of (E) The Universal Negative.34 As Venn says, this is an existential proposition. Indeed, as Matrixial Logic equations show, the (E) Categorical Sentence is distinct among Aristotle's Categorical Sentences, in calling upon the null hidden form of [E]35, rather than [E] as a form of substance.36

It is notable that, in all these logic schemes, the notation expresses a single underlying mechanical principle, which is also the existential principle: A=A. The law of identity:

Returning to Jevons, he continued:37

But it is hardly too much to say that Aristotle committed the greatest and most lamentable of all mistakes in the history of science when he took this kind of proposition as the true type of all propositions, and founded thereon his system. It was by a mere fallacy of accident that he was misled ; but the fallacy once committed by a master-mind became so rooted in the minds of all succeeding logicians, by the influence of authority, that twenty centuries have thereby been rendered a blank in the history of logic. [13]

Jevons was not complaining about the law of identity. He was, instead, complaining about its allegedly too limited scope.

Jevons was seeking a mathematic logic. As he continued:

Aristotle’s dictum in accordance therewith. It may then be formulated somewhat as follows:

Whatever is known of a term may be stated of its equal or equivalent. Or in other words, Whatever is true of a thing is true of its like. [14]

We can see here in Jevons, what we were later to see with Frege, Meinod, Russell and their successors. The attempt to defuse the inequality lying at the core of Aristotle’s Categorical Syllogisms.38

To substitute for the logical analyses of Substance, a formic logic which relies upon extrusions from pure equality.39

That program describes the history of symbolic logic in the 20th century, and the expressly limited landscape of First Order Logic.

The rebuttal offered by Matrixial Logic is that: this is an enterprise at which one is bound to fail, by succeeding.

ML is not, and does not seek to be, a closed system. Any closed system defeats itself with uncomputability, the moment it seeks to apply logic to the untrammelled landscape of reality. Indeed, no closed system can even justify itself.

ML is not, then, a system in which you can simply plug values into equations and have a answer pop out, under some iteration of a law of equality.

Instead, ML deals in forms of inequality. There are definite rules as to how to invoke and manipulate equations. But ML equations illumunate: they do not effect equalising definition.

What then, is the popint of a symbolic logic which does not by its own operation define outcomes?

Let’s find out by meeting Matrixial Logic.

__________

7 In 1959

8 Carnap, Kuhn, Popper, primarily in this respect

9 The only way to couch these matters at this stage in the book

10 (384-322 BC)

11 The Roots Of Logical Hylomorphism. Elenaragalina–Chernaya (2015)

12 Dragalina–Chernaya op. cit

13 Baron, Margaret E. (May 1969). A Note on The Historical Development of Logic Diagrams. The Mathematical Gazette. 53 (384): 113–125. doi:10.2307/3614533. JSTOR 3614533

14 (c. 1235-1315)

15 (1646-1716)

16 See later

17 (1728-1777)

18 (1724-1804)

19 (1764)

20 Author Trans.

21 See later

22 See Strange Days Chapter in I Want To Love But. The Author (2019)

23 https://commons.wikimedia.org/w/index.php?curid=6888771

24 The Laws of Thought (1854). From Cork, Ireland, rather than Lincoln

25 https://www.theatlantic.com/technology/archive/2017/03/aristotle-computer/518697/

26 (1806-1871)

27 Published (1915)

28 (1835–1882)

29 Pure Logic and Other Minor Works (1890) §174

30 The substitution of similars, the true principle of reasoning, derived from a modification of Aristotle’s dictum (1869)

31 op. cit

32 https://stanford.library.sydney.edu.au/entries/william-jevons/

33 Symbolic Logic (1881) p31

34 See Chapter 3

35 In effect as a Node: see Chapter 4

36 See Chapter 2

37 The Substitution Of Similars, The True Principle Of Reasoning, Derived From A Modification Of Aristotle’s Dictum. (1869)

38 See Chapter 3

39 See Chapter 4, and the discussion of Arithmetic

CHAPTER 2

MATRIXIAL FRAMEWORK

Introductory

Matrixial Logic is a symbolic logic.

In Matrixial Logic: we are examining types of relationship:

(1) In E Logic: Things which are Opposites , and both exist;

(2) In B Logic: Things which cause or relate to other things, and which exist, but as a Process;

(3) In C Logic: things which do not fall within either category of relationship.

Matrixial Logic seeks to symbolise and explain objects of logical enquiry under the 3Hrubric:

•Holistic

•Heuristic

•Hylomorphic

It should be added that the hylo is not limited to things, but includes processes, and paramorphic relations.41

In this introductory Chapter, we will need to define some terms, in ways which are not entirely accurate. ML requires a ladder of understanding. Once you get to the top, the bottom rungs look quite different than when you began the ascent.

But, if you were called upon to grasp the lower rungs in accordance with perspectives provided by the topped plateau, it may be almost impossible to begin the ascent at all.

Vocabulary Note:

The word things is used in preference to phenomena.

The reason is that the latter term has42 300 years of philosophical baggage attached to it. We do not say that such baggage has no utility within the discourses in which it has arisen.

Rather, that thing gains in neutrality what it sacrifices in specificity.

If, in reading the former word, you prefer to substitute the latter, then we hope such substitution affects not the meaning and effect of the former in its relevant context. We must hope because we do not envisage being able to conduct such a comprehensive survey of usage of the latter, so as to be able to advance a claim to that effect.

Notation Note

You will see, in the pages that follow, endless iterations of symbols in equations, and extractions from equations. We try to maintain consistency of iteration, except where we are deliberately inconsistent so as to emphasise something.

Because Matrixial Logic involves layers, matrices, of concepts and operations of logic, one concept may have different reference symbols, depending where in the matrix the concept is being operated with.

The Author readily confesses an occasional slip, inconsistency, between usage. Especially in narrative. If the reader notices these, then at least what is being communicated in substance, is having effect.

We will also use a mode of narrative writing in which we repeat the concept word, together with the symbol. Our intent is not that this should become wearisome, but so that such reiteration will allow the reader to become instantly familiar with the concept-symbol match: and also on occasion deliberately with intent to provoke thought.

Assertions

In this Chapter, we will make several unestablished context-setting assertions. In later Chapters we will unpack those assertions.

The reader is invited to reserve judgment, and to withhold agreement, conditional upon the effectiveness of such unpacking.

Exemplar Referenced to Daily Practice

An example of symbolic logic that you use every day:43

Arithmetic

1 + 2 = 3

You are so used to doing this, in your head, that you probably never think about how the bits of it are put together.

Proposition: Numbers 1–10 Relate to Each Other.

Let 2 be (A) in E (the 3 form of number):

(1) ≠ (2) = 3

What the ≠ not equals sign does, is show that one side of the sign is not the same as the other.

We don’t notice in our ordinary thinking, but that’s the sign we’re kind of using when we do arithmetic.

The different signs: + - x /

These are