12. Propositions connecting real and apparent variables

Harold R. (Hal) Foster’s Prince Valiant

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Propositions connecting real and apparent variables

 The most important propositions connecting real and apparent variables are the following:

(1) " When a propositional function can be asserted, so can the proposition that all values of the function are true." Stated more briefly, if less exactly, "what holds of any, however chosen, holds of all." This translates itself into the rule that when a real variable occurs in an assertion, we may turn it into an apparent variable by putting the letter representing it in brackets immediately after the assertion-sign.

(2) " What holds of all, holds of any," i.e.

├ : (x). φx . É . φy.

This states " if φx is always true, then φy is true."

(3) " If φy is true, then φx is sometimes true," i.e.

├ : φy . É . ($x). φx.

An asserted proposition of the form "($x) . φx" expresses an "existence theorem," namely " there exists an x for which φx is true." The above proposition gives what is in practice the only way of proving existence-theorems: we always have to find some particular y for which φy holds, and thence to infer "($x). φx." If we were to assume what is called the multiplicative axiom, or the equivalent axiom enunciated by Zermelo, that would, in an important class of cases, give an existence-theorem where no particular instance of its truth can be found.

In virtue of " ├ : (x). φx . É. φy" and "├: φy . É . ( x). φx," we have

"├ : (x) . φx . É . ($x). φx," i.e. "what is always true is sometimes true. "This would not be the case if nothing existed; thus our assumptions contain the assumption that there is something. This is involved in the principle that what holds of all, holds of any; for this would not be true if there were no " any."

(4)"If φx is always true, and yx is always true, then 'φx . yx' is always true,"' i.e.

├ :. (x). φx : (x). yx: É (x) . φx . yx.

(This requires that φ and y should be functions which take arguments of the same type. We shall explain this requirement at a later stage.) The converse also holds; i.e. we have

├ :. (x). φx . yx . É : (x). φx : (x). yx.

It is to some extent optional which of the propositions connecting real and apparent variables are taken as primitive propositions. The primitive propositions assumed, on this subject, in the body of the work (*9), are the following:

(1) ├ : φx . É . ($z) . φz

(2) ├ : φx v φy . É . ($z) . φz,

i.e. if either φx is true, or φy is true, then ($z) . φz is true. (On the necessity for this primitive proposition, see remarks on *9'11 in the body of the work.)

(3) If we can assert φy, where y is a real variable, then we can assert    (x) . φx; i.e. what holds of any, however chosen, holds of all.

Script: Bob Haney  Pencils and Inks: Bernard Baily. ™ © Respective copyright/trademark holders.

In tribal Indian cultures, before European contact, great general continuity of experience and understanding existed of each individual’s place in the world they inhabited.  Certainly this was true within the tribe, and largely between tribes within a given region. The region containing the range of a tribe held great importance for its relation to the identity of every individual. It was understood by all, that the resources of any region were finite, and to be husbanded so that all could share as needed, not just for the family or tribe, but for the resources themselves. For all plants, animals, reptiles, fish and even minerals were living beings with spirits to themselves that were equally sacred as the lives of human beings.

There is evidence that these shared values and perceptions were virtually universal across the continent, allowing of course, for regional and tribal variations. This simplicity and homogenous set of experiences, beliefs and understandings allowed the Indian people to live an existence that includes what I have termed 4D thought, that is, a broad shared set of knowledge that allowed everyone to know the world they lived in and to clearly see how they could fit into it. Language was useful of course, but it was not called upon to convey commercial or technical complexities, rhetoric beyond story, or separation of classes.

The evolution of metaphysical principals, such as those examined in this blog, are valiant attempts to bring order the dissipated perceptions of European language based thought predicated by language in practically every aspect of life. As we contiue, this will become quite apparent I hope.

Script: ?  Pencils: Dick Beck  Inks: Don Perlin ?  © Respective copyright holders.

hanBLOGlaka

Logical Philosophy In American Indian

thought and Perception

Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders.

9. Apparently Variable

Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders.

Apparent variables

The symbols “(x) . φx” denotes a definitive proposition, and there is no distinction between “(x) . φx ” and “(y) . φy” when they occur in the same context. Thus “x” in “(x) . φx” is not an ambiguous constituent in any expression in which “(x) . φx ” occurs; and such an expression does not cease to convey a determinant meaning by reason of the ambiguity if the x in “φx ”.

The range of x in “(x) . φx ” or “($x) . φx” extends over the complete field of the values of x for which “φx” has meaning, and accordingly the meaning of  “(x) . φx ” or “($x) . φx” involves the supposition that such a field is determinant. The x which occurs in “(x) . φx ” or “($x) . φx” is called an “apparent variable” (after Peano). It follows from the meaning of “($x) . φx” that the x in that expression is also an apparent variable. A proposition in which x occurs as an apparent variable is not a function of x. Thus e.g. “(x) . x=x” will mean “everything is equal to itself.” This is an absolute content, not a function of a variable x. This is why the x is called an apparent variable in such cases.

As established in the second edition, there is no need of the distinction between real and apparent variables, nor of the primitive idea “assertion of a propositional function.” On all occasions ahead where, in Principia Mathematica, our authors have an asserted proposition of the form “├ . fx” or “├ . fp” this is to be taken as meaning “├ (x) . φx”; but in “├ ($x) . φx “ it is necessary to indicate explicitly the fact that “some” x (not “all” x’s) is involved.

 

Script: Gardner F. Fox  Pencil and Inks: Sheldon Moldoff (signed

)Cliff Cornwall™ © Respective copyright/trademark holders.

The text I am working from thus far is the abridged Principia Mathematica to *56, and this edition is drawn from the first edition. There are a number of revisions in the second edition, which were suggested by various readers in the years between editions. As the authors predicted this, and any metaphysical system is and must be in in flux to be useful to its purpose. Even though it became apparent that the distinction between real and apparent variables is not necessary within the system of propositions described ahead, it is nonetheless useful to understand the distinction. Even though the difference is semantic, it is still real, and a detail that may be useful in some not yet anticipated capacity.  

Script: R. D. Blackmore (original author); Ruth A. Roche (adaptation)

Pencils and Inks: Matt Baker © Respective copyright/trademark holders.

 

han

BLOG

laka

Logical Philosophy In American Indian thought and Perception

Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders.



 

 

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

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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.