5. On Definitions

5.

 Definitions

definiendum: that which is defined (noted to the left).

 definiens: that which it is defined as meaning (to the right with =)

“p É q . = . ~p v q  Df.”

Three primitive ideas have been introduced above which are not “defined” but only descriptively described. Their primitiveness is only relative to our expansion of logical connection and is not absolute; though of course such an exposition gains in importance according to the simplicity of its primitive ideas.

 These ideas are symbolized by “~p” and “p v q” and by “├” prefixed to a proposition.

Three definitions have been introduced:

p . q . = . ~(~p v ~ q)  Df,

p . É q .= . ~ p v q  Df,

p ≡ q . = . p É q . q ) p  Df.

Primitive propositions: some propositions must be taken without
proof, since all inference proceeds from propositions previously
asserted. These, like the primitive ideas are to some extent a matter
of arbitrary choice. Though, as in the previous case, a logical
system grows in importance according as the primitive propositions
are few and simple.
 
It will be found that owing to the weakness of the imagination in
dealing with simple abstract ideas no very great stress can be laid
upon their obviousness.
 
They are obvious to the instructed mind, but then so are many
propositions which cannot be quite true, as being disproved by their
contradictory consequences.
 
The following are the primitive propositions employed in the
calculus of propositions. The letters “Pp: stand for “primitive
proposition.”

(1)     Anything implied by a true premise is true  Pp.

This is the rule which justifies inference.

(2)     ├ : p v p . É . p  Pp.

i.e. if p or p is true, then p is true.

   (3)  ├ : q  . É . q v p  Pp.

i.e. if q is true then p or q is true.

   (4)   ├ : p v q . É . q v p  Pp.

i.e. if p or q is true, then q or p is true.

   (5)   ├ : p v É q v r) . É . q v (p v r)  Pp

i.e. if either p is true or “q or r” is true then either q is true or “p or r” is true.

   (6)   ├ :. q É r . É : p v q . É . p É r  Pp’

i.e. if q implies r, then “p or q” implies “p or r.”

   (7)   “The axiom of identification of real variables.”

When we have separately asserted two different functions of x where x is undetermined it is often important to know whether we can identify the x in one assertion with the x in the other. i.e. if     φ(x,y,z…) is a constituent in one assertion, and φ(x,u,v…) is a constituent in another.

*3·03, *1·7, and *1·72 (which is the statement of this axiom).

Some simple propositions:

The law of the excluded middle: ├ . p v ~p.  (*2·11)

The law of contradiction: ├ . ~(p . ~p).  (*3·24)

The law of double negation: ├ . p ≡ ~(~p).   (*4·13)


The principle of transposition: this principle has various forms, namely:

       (*4·1)   ├ : p É q .≡ . ~q É ~p,

       (*4·11) ├ : p ≡ q . ≡ . ~p ≡ ~q,

       (*4·14) ├ :. p . q . É . r : ≡ : p . ~r . ) . ~q. as well as other variations of these.

The handsome gent in the photo at the top of this post is a Tlingit shaman circa 1900. His culture was already in flux from half a century of contact with European culture. Centuries old realities were fast changing by concepts and technologies that brought about changes that were unexpected, and affected each person in individual ways. Cultures neither rise nor decline by broad strokes, it is the way that persons perceive, understand and communicate large and small changes among themselves that makes present and future dimensional realities

The base idea of this work is not specifically to be always correct in all assertions, inferences or propositions; more important is to be sure that that these are not in error. Perhaps these seem like the same thing, but they are heads and tails on the same coin. One may choose to see a coin as a prospective gumball,  or take a more existential view that there are two  dimensional-aspects whose differences are recognizable and definable as significant fine distinctions that are useful or not as each person ultimately decides  for herself.

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.