4. Use of Dots

Fletcher Hanks’ Stardust

© Respective copyright holders.

The Use of Dots –

1. To bracket off propositions.

2. To indicate the logical product of two propositions.

The general principle is that a large number of dots indicate an outside bracket and a smaller number of dots indicate an inside bracket.


Group I      consists of dots adjoining a sign of implication (É), of equivalence (≡), of disjunction (v), or of equality by definition (= Df).


Group II     consists of dots following brackets indicative of an apparent variable such as (x), or (x,y), or (Эx), or (Эx,y), or [(ίx) (φx)] or analogous expressions.


Group III      consists of dots which stand between propositions in order to indicate a logical product. The scope of any collection of dots extends backwards or forwards beyond any smaller number of dots or equal number of dots from a group of less force until we reach the end of the proposition or a greater number of dots or an equal number of dots belonging to a group of equal or superior force.

 Dots indicating a logical product have a scope which works backwards and forwards. Other dots work only away from adjacent signs of disjunction, implication or equivalence, or forward from others in Group II.

“├ : p v q . É . q v q”

shows what is asserted is the whole of what follows the assertion sign.

“├ :. p É q . É : q É r . É . p É r”

means “if p implies q, then if q implies r, p implies r.”



“p É q . É . q É r : É . p É r”

will mean “if ‘p implies q’ implies ‘q implies r’ then

p implies r.”  (This is in general untrue.)

“p É q . q É r . É . p É r”

will mean “p implies q; and if q implies r then

p implies r.”

In this formula the first dot indicates a logical product, hence the second dot extends back to the beginning of the proposition.

“p É q : q É r . É . p É r”

will mean “p implies q; and if q implies r, then p implies r.” (this in general is not true). Here the two dots indicate a logical product, since the two dots do not occur anywhere else, the scope of these two dots extend backwards to the beginning of the proposition, and forwards to the end.

“p v q . É :. p . v . q É r : É p v r”

will mean “if either p or q is true, then if either p or ‘q implies r’ is true, it follows that either p or r is true.” If this is to be asserted, we must put four dots before the assertion sign thus:

“┌ :: p . v . q . É :. p . v . q É r : É . p v r.”

To assert an equivalent proposition to the phrase: “if either p or q is true, and either p or ‘p implies r’ is true, and either p or ‘q implies r’ is true,” we write:

“├ :. p v q : p . v . q É r : É . p v r.”

here the first pair of dots indicates a logical product, while the second pair does not. Thus the second pair of dots passes over the first pair, and back until we reach the three dots after the assertion sign.

In reading a proposition, the dots should be noticed first, as they show its structure.

In a proposition containing, several signs if implication, or equivalence, the one with the greatest number of dots before or after is the principle one: everything that goes before this one is stated by the proposition to imply or be equivalent to everything that comes after it.

 

 

Script, Pencils and Inks: Russ Manning  © Respective copyright/trademark holders.

The dots system and other symbols are carefully thought out to compress the language of propositional equations to as short a form as possible. In fact it seems the entire work is a response to the basic problem of writing about or speaking about very complex subjects, that of the severe limitation of the amount of time and attention the reader or listener is willing or able to devote to taking it in. As the nomenclature is absorbed and practiced very complex expressions may be absorbed and evaluated instantly, or practically so, without the need for the time consuming demands of conventional written language or speech.

Also by this system error manifests itself clearly, so one is not taxed by having to decide whether or not to believe the material presented.

                     Artist Unknown: Little Dot© Respective copyright/trademark holders.

 

 

03. Equivalence and Other Terms



Script:  Roy Thomas 

Pencils: Barry Windsor-Smith 

Inks: Sal Buscema © Respective

copyright/trademark holders.

Equivalence (mutual implication)

P implies q and q implies p

p ≡ q = (p É q) . (p É q)

“formal implication” thus “formal equivalence.”

It must not be supposed that two propositions which are equivalent are in any sense identical, or even remotely concerned with the same topic.

Newton was a man ≡ the sun is hot:  is true

Newton was not a man ≡ the sun is not hot: is false

Truth-Values

True if a proposition is true

False if a proposition is false

If proposition p occurs in any proposition f(p), the truth value of f(p) will depend not on a particular proposition p, but only its truth value.

if p ≡ q then f(p) ≡ f(q)

f(p) may be called a “truth function” when argument p is a proposition and the truth value of f(p) depends only on the truth value of p.

“A believes p” is a function of p which will vary its truth value for different arguments having the same truth value.

One may believe one true proposition without believing another and may believe one false proposition without believing another. This related to the characteristic of mathematics namely, mathematics being always concerned with extensions rather than intentions.  

Assertion-Sign “├”  what follows is asserted and distinguishes a complete proposition which is asserted from any subordinate propositions which are not asserted.

├ p É q   unless (p É q) is true the assertion is in error.

Inference – “├ p and ├ (p É q)” infers “├ q” –inference cannot be reduced to symbols.

To draw attention to an inference – “├ p É ├ q” may be read “p, therefore q.” this does not explicitly state what is part of its meaning, that p implies q.

This is a mere abbreviation of “├ p and (p É q) and ├ q.”

An inference is the dropping of a true premise;

it is the dissolution of an implication.

Jim Meddick’s Robotman  © Respective copyright/trademark holders.

In our dominant culture in North America, and in fact in most of the world, people follow and prefer to be ruled by laws, statutes, rules, guidelines, precepts, scriptures, principles, contracts, compacts, guarantees, covenants, commandments and all other such static recordings of behavioral intentions or requirements to govern the actions and agreements of human beings among ourselves in all matters and for every reason. It may be held that all this regulation is necessary because persons are good or bad and must be restrained from exercising their base human instincts that politicians and religions typically heap upon human character. My contention, supported by Whitehead and Russell in Principia Mathematica, is that all of this regulation has evolved from the inadequacy of language to convey abstract concepts, and the resulting misunderstandings that occur constantly and universally among all persons in dominant culture. Both from the inadequacy of language itself and more so people’s varying capacity to use the language they have, or to transliterate between one language and another.

 

Above are the beginning concepts that are baby steps towards melting away the constraints of our minds perception. The forms of functions and propositions will be repeated and expanded ahead, and the superb text will provide all the instruction a scholar requires. Concentrate on learning the material, but do not be too concerned if you can’t seem to memorize it, or bring it clearly to mind even some long way in. This is the nature of metaphysics, page after page of information with no tangible subject or story to bring it to mind. Eventually things should click if you study with diligence. Also vocabulary is crucial, and I urge you to turn to a good dictionary if any words used here are not completely clear. It can truly make all the difference.   

Script: Dave Wood; Jack Kirby  Pencils: Jack Kirby  Inks: Wally Wood 

Challengers of the Unknown © Respective copyright/trademark holders. 

02. Systematic Ambiguity

Script: Stan Lee  Pencils and Inks: Steve Ditko 

 Spider-man © Respective copyright/trademark holders.

(Systematic ambiguity)

The Contradictory Function: with argument p (any proposition) is the       proposition which is contradictory of p, asserting p is not p.

~p = not p or the negation of the proposition p.

The Logical Sum: is a propositional function with two arguments p and q and the proposition asserting p and q disjunctively, asserting that at least one of the two p and q is true.

p v q is the logical sum of p and q.

p v q means at least p or q is true, not exclusive of the case that both are true.

The Logical Pretext: asserts both p and q conjunctively, therefore both p and q are true, denoted p `q or p:q or p:.q or p::q all mean both p and q are true.

Also true for ~p or ~q must be false:

p . q = ~(~p v ~q)

The Implicative Function: is a proposition with two arguments p and q, and that either ‘not p’ or q is true or

~p v q. Thus if p is true, ~p is false, and accordingly the only alternative left by the proposition ~p v q is that q is true. In other words if p and ~p v q are both true, then q is true. In this sense the proposition ~p v q will be quoted as p implies q.

The symbol employed for p implies q or “~p v q” is p É q

p É q may also be read “if p, then q.”

The association of implication with the use of an apparent variable produces an extension called “formal implication.” (explained later) It is an idea derived from “implication” as defined above.

When it is necessary explicitly to discriminate “implication” from “formal implication,” it is called “material implication.”

These four functions of propositions are the fundamental constant (i.e. definite) propositional functions with propositions as arguments, and all other constant propositions as arguments in the present work, are formed out of them by successive steps. No variable propositional functions of this kind occur in this work.



Dan DeCarlo The greatest of all time comic book artist!  Archie Andrews and Betty Cooper  

© Respective copyright/trademark holder

In approaching this subject I urge you to see it as a work of metaphysics rather than mathematics. Memorization is not necessary or particularly useful in understanding it. Following posts will contain thousands of words expressing very concise and useful information, but all without a single subject to serve as glue to hold it together. This “glue” is to be your mind and your thought process. All of the material to come must be processed into our consciousness, and the device to make this happen is the work, Principia Mathematica, itself.

The authors carefully point out that the reason for this work is to train the mind to function without rules. They describe rules for various purposes of learning the material, but are clear that these can and must be left behind as the dharma takes hold and is put to practice. As the heading Systematic Ambiguity implies the purpose of dealing with any complex problem begins with establishing propositions of various functions and their relations. It is the essential function of this work to set out a clear vocabulary and hierarchy we may use to think of and communicate complexity in a manageable and intelligible manner.

*Note* Please be patient, the Indian connection will become apparent eventually if it is to clear already.

Script Pencils and Inks: The Venerable George Herriman  © Respective copyright/trademark holders

01. Notes on the System's Construction

 

 Notes in Learning the “Language”

 of the Construction of

Alfred North Whitehead’s and Bertrand Russell’s

Principia Mathematica to *56

Detailed for his use by Comic Book Shaman

Fletcher Hanks’ Stardust

© Respective copyright holders.

Denoting Variables and Constants:

Small letters of ordinary alphabet are all used as variables

(except “p” and “s” after *40 when p and s are assigned as constants.)

Capital letters used as constants:

       B, C, D, E, F, I and J.

 Small Greek letters used as constants: є epsilon, ί iota, and later  η eta, θ theta, ω omega.

 Certain Greek capitals will be introduced as constants, but not as variables.

p, q, r : propositional letters that will stand for variable propositions to *40: onwards “p” must not be used as a variable.

f, g, φ phi, ψ psi, χ chi, θ theta, and (until *33) F : Functional letters that will not be used for variable functions.

 The small Greek letters not named above will be used for variables whose values are classes, and will be referred to as ‘Greek letters.’

 The small common letters other than p, q, r, s, f and g will be used for variables whose values are not known to be functions, classes or relationships. These will be referred to as ‘small Latin letters.’

 Later:    variable classes: small Greek letters

              variable relationships: Capitals

              variables not given as necessarily classes or relations: small Latin letters

Dan DeCarlo The greatest of all time comic book artist!

Archie Andrews and Betty Cooper  © Respective copyright/trademark holders

These early posts are more abbreviated in relation to Whitehead and Russell’s original text than later posts will be. I hear Bertrand Russell’s voice so clearly in this introduction that I would judge that he actually wrote this, but expressing, I suppose both men’s thought discussed at length between them. I stress this is only my feeling, and the point is of no importance, but the text delights me to a great degree, as has every other Russell work I have read, and in the same fashion.

Beginning here to define terms that will express the work, these are largely familiar from the study of mathematics. However the authors are persistent in saying within the work, and I am reinforcing here with my own voice, that the title Principia Mathematica is a sort of joke. The work is one of metaphysics, in which the introduction constructs a wondrous web of means of thought and expression of fine distinction. Then the Body of the work expresses this new metaphysical construct in discussing mathematics.

In every discussion or analysis of Whitehead and Russell’s Principia Mathematica soon they want to change, modernize and simplify the work to conform to other agendas they have, usually in making the present work into one of mathematics. This is a mistake, for the goal of this work is to develop a condition of the mind in the reader that did not exist before beginning it. If one feels that it is necessary to change or simplify this work, you are playing in the wrong sandbox.

Stan Lee (co-plot, dialogue), Plot, Pencils and Inks: Bill Everett  Doctor Strange © Respective copyright/trademark holders.