20. 'Evident Relativity' from Analogues Propositions in Classes

Harold R. (Hal) Foster’s Prince Valiant

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The following are some propositions concerning classes which are analogues of propositions previously given concerning propositions:

┝ . α ∩ β = -( -α ∪ -β),

i.e. the common part of α and β is the negation of " not-α or not -β ";

┝ . x Î (α ∪ - a),

i.e. "x is a member of a or not -a";

┝ . x ~ Î (a ∩ -a),

i.e. "x is not a member of both a and not -a";

┝ . a = -(-a),

┝ : a Ì β . º . – β Ì -a,

┝ : a = β . º . –a = -β,

┝ : a = a ∩ a,

┝ : a = a ∪ a.

The two last are the two forms of the law of tautology.

The law of absorption holds in the form

┝ : a Ì β . º .  a = a ∩ β.

Thus for example "all Cretans are liars" is equivalent to "Cretans are identical with lying Cretans."

Just as we have   ┝ : p É q . q É r . p É r,

so we have          ┝ : a Ì β . β Ì ϒ . a Ì ϒ.

This expresses the ordinary syllogism in Barbara* (with the premises interchanged); for "a Ì β" means the same as "all a's are β's," so that the above proposition states: "If all a's are β's, and all β's are _ϒ's, then all a's are _ϒ's." (It should be observed that syllogisms are traditionally expressed with "therefore," as if they asserted both premises and conclusion. This is, of course, merely a slipshod way of speaking, since what is really asserted is only the connection of premises with conclusion.)

The syllogism in Barbara when the minor premise has an individual subject is

┝ : x Î. β Ì _ϒ  . É .x Î ϒ,

e.g."if Socrates is a man, and all men are mortals, then Socrates is a mortal." This, as was pointed out by Peano, is not a particular case of "a Ì β. Ì ϒ . É .   a Ì ϒ," since       "x Î β" is not a particular case of "a Ì β." This point is important, since traditional logic is here mistaken. The nature and magnitude of its mistake will become clearer at a later stage.

For relations, we have precisely analogous definitions and propositions.

We put

‘R ⩀  S = x^,y^(xRy . xSy)  Df,

which leads to ┝ : x^,y ^(R ⩀ S) y . º . xRy. xSy.

 

Similarly R  ⊍  S = : x^,y^(xRy. v . xSy)  Df,

                ∸ R = x^,y^{ ~(xRy)}  Df,

              R ⪽ S . = : xRy . É _x,y . xSy  Df.

Generally, when we require analogous but different symbols for relations and for classes, we shall choose for relations the symbol obtained by adding a dot, in some convenient position, to the corresponding symbol for classes. (The dot must not be put on the line, since that would cause confusion with the use of dots as brackets.) But such symbols require and receive a special definition in each case. A class is said to exist when it has at least one member: "a exists" is denoted by "∃ ! a." Thus we put

∃ ! a . = . (∃x) . x Î a  Df.

* All men are mortal. (MaP)

  All Greeks are men. (SaM) ∴

  All Greeks are mortal. (SaP)

(footnote graphic courtesy of Wikipedia)

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As we collect these analogues propositions concerning classes, propositions and relations we begin to acquire an understanding of their natures. The rule against ‘mixing apples and oranges’ goes away as we learn just how these natures interact within our system of equations, in which we may clearly see their transitive interactions by direct means or indirectly by their negation. In working within our system it is important to remember that our purpose is not to seek truth or falsehood in our equations and propositions combining propositions, classes and relations et. al.; in fact the goal is always to seek errors in our calculations.

We shall examine the nature of truth and falsehood and its spectrum of effect in later posts. Also we will find by this system a third so called moral/ethical existential condition: nonsense. Within our system, nonsense represents the contents of our systems equations that cannot be proven true or false, but eventually, always lead to error. This in effect, introduces the element of time as a real constant in the equation, which leads us laughing into Evident Relativity. This is a term, Evident Relativity, I had never seen or considered before beginning to write this paragraph. But I found that I could not put ‘theory of relativity’ because as we proceed I believe it will be evident Einstein’s general and special relativity, and nearly ninety years of earnest mathematical equation-al support in the cosmological fields of physics, astrophysics, and quantum physics, we come ‘round at last to some basic human equations that Indians have always lived by, and that have permeated even our dominant culture in North America; but without credence, because of its essential separation of science and art.

Read up here if you’d like, always remember Wikipedia is not an end, but a sign on a trail (start here) to the process of attaining knowledge.

The evidence that science yet lacks to ‘prove’ the theory of general and special relativity in terms of science may be derived or become evident from the arts and human experience here on planet Earth by means of our system described by Whitehead and Russell in their work Mathematica Principia. What our dominant culture and our scientists require and lack; is external, observable and reproducible data that will withstand all objective and subjective scrutiny. I can’t promise that I will ‘prove’ anything mathematicians before my time have not in nearly a century; but by shaman’s perception and interpretation I shall attempt to see that their work led us to this cunning trick, of demonstrating that relativity is evident.