Sunday, January 27, 2013

Stop The World (silent)


Harold R. (Hal) Foster’s Prince Valiant

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  “A ‘shaman’ knows that he cannot change, and yet he makes it his business to try to change, nevertheless. The ‘shaman’ is never
disappointed  when he fails to change. That is the only advantage a ‘shaman’ has over the average man.”
   - Don Juan Matus


Artist Unknown (Zia Pueblo) Painted Stone Fetish © Respective copyright holders.


Argosy June 2 1923 © Respective copyright holders.


  “The hardest thing in the world is to assume the mood of a warrior. It is of no use to be sad and complain and feel justified in doing so, believing that someone else is always doing something to us. No one is doing anything to anybody, much less to a warrior.”
 -Don Juan Matus
1949 Indians vs Yankees Program © Respective copyright holders.

Porky Pig Four Color Comics #112 July 1946 © Respective copyright holders.


Harold R. (Hal) Foster’s Prince Valiant

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Tlingit Shaman circa 1900 © Respective copyright holders.
“Only as a warrior can one withstand the path of knowledge. A warrior cannot complain or regret anything. his life is an endless challenge, and challenges cannot possibly be good or bad. Challenges are simply challenges.“
 -Don Juan Matus

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Kitche Postcard Bob Petley circa 1950s  © Respective copyright holders.


   “An average man is too concerned with liking people or with being liked himself. A warrior likes, that is all. He likes whatever and whomever he wants, for the hell of it.“
 -Don Juan Matus

Awa Tsireh (San Ildefonso) Stylized Snake Dancer; watercolor on paperboard 1930
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Harold R. (Hal) Foster’s Prince Valiant

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Saturday, January 26, 2013

47. Supplimental Post Expanding #s 1-2


Harold R. (Hal) Foster’s Prince Valiant
© Respective copyright/trademark holders.

This blog sprung up as I was deep in study of Principia Mathematica. Early on, I confess I was a bit at sea with the system, and I stubbornly read each paragraph, each page many times over, and then copied each word in longhand, carefully checking each passage for errors. I created a study guide to refer to, to help me learn the many symbols and notations.

It was for me, a journey from uncertainty to comprehension to clarity; and as I started this blog I forgot the struggle was what, in time, made things clear for me. I jumped out with my study guide, but I now realize that it is inadequate. I hope this and following supplimental posts will prove useful.

Script: Hannigan  Artwork: Perlin & Marcos   HellcatÔ & © Respective copyright/trademark holders.

CHAPTER I.

PRELIMINARY EXPLANATIONS OF IDEAS AND NOTATIONS. THE notation adopted in the present work is based upon that of Peano, and the following explanations are to some extent modeled on those which he prefixes to his Formnulario Mathematico. His use of dots as brackets is adopted, and so are many of his symbols.

Variables. The idea of a variable, as it occurs in the present work, is more general than that which is explicitly used in ordinary mathematics. In ordinary mathematics, a variable generally stands for an undetermined number or quantity. In mathematical logic, any symbol whose meaning is not determinate is called a variable, and the various determinations of which its meaning is susceptible are called the values of the variable. The values may be any set of entities, propositions, functions, classes or relations, according to circumstances. If a statement is made about "Mr. A and Mr. B," "Mr. A" and "Mr. B" are variables whose values are confined to men. A variable may either have a conventionally-assigned range of values, or may (in the absence of any indication of the range of values) have as the range of its values all determinations which render the statement in which it occurs significant. Thus when a text-book of logic asserts that "A is A," without any indication as to what A may be, what is meant is that any statement of the form "A is A" is true. We may call a variable restricted when its values are confined to some only of those of which it is capable; otherwise, we shall call it unrestricted. Thus when an unrestricted variable occurs, it represents any object such that the statement concerned can be made significantly (i.e. either truly or falsely) concerning that object. For the purposes of logic, the unrestricted variable is more convenient than the restricted variable, and we shall always employ it. We shall find that the unrestricted variable is still subject to limitations imposed by the manner of its occurrence, i.e. things which can be said significantly concerning a proposition cannot be said significantly concerning a class or a relation, and so on. But the limitations to which the unrestricted variable is subject do not need to be explicitly indicated, since they are the limits of significance of the statement in which the variable occurs, and are therefore intrinsically determined by this statement. This will be more fully explained later*1.

Chapter II of the Introduction.

To sum up, the three salient facts connected with the use of the variable are: (1) that a variable is ambiguous in its denotation and accordingly undefined: (2) that a variable preserves a recognizable identity in various occurrences throughout the same context, so that many variables can occur together in the same context each with its separate identity: and (3) that either the range of possible determinations of two variables may be the same, so that a possible determination of one variable is also a possible determination of the other, or the ranges of two variables may be different, so that, if a possible determination of one variable is given to the other, the resulting complete phrase is meaningless instead of becoming a complete unambiguous proposition (true or false) as would be the case if all variables in it had been given any suitable determinations.

The uses of various letters. Variables will be denoted by single letters, and so will certain constants; but a letter which has once been assigned to a constant by a definition must not afterwards be used to denote a variable. The small letters of the ordinary alphabet will all be used for variables, except p and s after *40, in which constant meanings are assigned to these two letters. The following capital letters will receive constant meanings: B, C, D, E, F, I and J. Among small Greek letters, we shall give constant meanings to ε, ι, sand (at a later stage) to η, θ and ω. Certain Greek capitals will from time to time be introduced for constants, but Greek capitals will not be used for variables. Of the remaining letters, p, q, r will be called propositional letters, and will stand for variable propositions (except that, from *40 onwards, p must not be used for a variable); ƒ, g, φ, ψ, χ, θ and (until *33) F will be called functional letters, and will be used for variable functions.

The small Greek letters not already mentioned will be used for variables whose values are classes, and will be referred to simply as Greek letters. Ordinary capital letters not already mentioned will be used for variables whose values are relations, and will be referred to simply as capital letters. Ordinary small letters other than p, q, r, s, ƒ, g will be used for variables whose values are not known to be functions, classes, or relations; these letters will be referred to simply as small Latin letters.

After the early part of the work, variable propositions and variable functions will hardly ever occur. We shall then have three main kinds of variables: variable classes, denoted by small Greek letters; variable relations, denoted by capitals; and variables not given as necessarily classes or relations, which will be denoted by small Latin letters.

In addition to this usage of small Greek letters for variable classes, capital letters for variable relations, small Latin letters for variables of type wholly undetermined by the context (these arise from the possibility of "systematic ambiguity," explained later in the explanations of the theory of types), the reader need only remember that all letters represent variables, unless they have been defined as constants in some previous place in the book. In general the structure of the context determines the scope of the variables contained in it; but the special indication of the nature of the variables employed, as here proposed, saves considerable labor of thought.

The fundamental functions of propositions. An aggregation of propositions, considered as wholes not necessarily unambiguously determined, into a single proposition more complex than its constituents, is a function with propositions as arguments. The general idea of such an aggregation of propositions, or of variables representing propositions, will not be employed in this work. But there are four special cases which are of fundamental importance, since all the aggregations of subordinate propositions into one complex proposition which occur in the sequel are formed out of them step by step.

They are (1) the Contradictory Function, (2) the Logical Sum, or Disjunctive Function, (3) the Logical Product, or Conjunctive Function, (4) the Implicative Function. These functions in the sense in which they are required in this work are not all independent; and if two of them are taken as primitive undefined ideas, the other two can be defined in terms of them. It is to some extent-though not entirely-arbitrary as to which functions are taken as primitive. Simplicity of primitive ideas and symmetry of treatment seem to be gained by taking the first two functions as primitive ideas.

The Contradictory Function with argument p, where p is any proposition, is the proposition which is the contradictory of p, that is, the proposition asserting that p is not true. This is denoted by ~p. Thus ~p is the contradictory function with p as argument and means the negation of the proposition p. It will also be referred to as the proposition not-p. Thus ~p means not-p, which means the negation of p.

The Logical Sum is a propositional function with two arguments p and q, and is the proposition asserting p or q disjunctively, that is, asserting that at least one of the two p and q is true. This is denoted by p v q. Thus p v q is the logical sum with p and q as arguments. It is also called the logical sum of p and q. Accordingly p v q means that at least p or q is true, not excluding the case in which both are true.

The Logical Product is a propositional function with two arguments p and q, and is the proposition asserting p and q conjunctively, that is, asserting that both p and q are true. This is denoted by p . q, or - in order to make the dots act as brackets in a way to be explained immediately - by p : q, or by p :. q, or by     p :: q. Thus p . q is the logical product with p and q as arguments. It is also called the logical product of p and q. Accordingly p . q means that both p and q are true. It is easily seen that this function can be defined in terms of the two preceding functions. For when p and q are both true it must be false that either ~p or ~q is true. Hence in this book p . q is merely a shortened form of symbolism for

~(~p v ~q).

If any further idea attaches to the proposition "both p and q are true," it is not required here.

The Implicative Function is a propositional function with two arguments p and q, and is the proposition that either not-p or q is true, that is, it is the proposition ~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 stating that p implies q. The idea contained in this propositional function is so important that it requires a symbolism which with direct simplicity represents the proposition as connecting p and q without the intervention of ~p. But "implies" as used here expresses nothing else than the connection between p and q also expressed by the disjunction "not-p or q." The symbol employed for "p implies q," i.e. for "p v q," is "p q." This symbol 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." This is explained later: it is an idea derivative from "implication" as here defined. When it is necessary explicitly to discriminate "implication"from "formal implication," it is called "material implication." Thus "material implication" is simply "implication" as here defined. The process of inference, which in common usage is often confused with implication, is explained immediately.

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

Equivalence. The simplest example of the formation of a more complex function of propositions by the use of these four fundamental forms is furnished by "equivalence." Two propositions p and q are said to be "equivalent" when p implies q and q implies p. This relation between p and q is denoted by "pq." Thus "pq" stands for "(p q). (q p)." It is easily seen that' two propositions are equivalent when, and only when, they are both true or are both false. Equivalence rises in the scale of importance when we come to "formal implication" and thus to "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. Thus "Newton was a man" and "the sun is hot" are equivalent as being both true, and "Newton was not a man" and "the sun is cold" are equivalent as being both false. But here we have anticipated deductions which follow later from our formal reasoning. Equivalence in its origin is merely mutual implication as stated above.

Truth-values. The "truth-value" of a proposition is truth if it is true, and falsehood if it is false*2. It will be observed that the truth-values of p v q, p . q, p q, ~p, pq depend only upon those of p and q, namely the truth-value of "p v q" is truth if the truth-value of either p or q is truth, and is falsehood otherwise; that of "p . q" is truth if that of both p and q is truth, and is falsehood otherwise; that of "p q" is truth if either that of p is falsehood or that of q is truth; that of "~p" is the opposite of that of p; and that of "pq" is truth if p and q have the same truth-value, and is falsehood otherwise. Now the only ways in which propositions will occur in the present work are ways derived from the above by combinations and repetitions. Hence it is easy to see (though it cannot be formally proved except in each particular case) that if a proposition p occurs in any proposition ƒ(p) which we shall ever have occasion to deal with, the truth-value of ƒ(p) will depend, not upon the particular proposition p, but only upon its truth-value; i.e. if pq, we shall have ƒ(p) ≡ ƒ(q). Thus whenever two propositions are known to be equivalent, either may be substituted for the other in any formula with which we shall have occasion to deal.

We may call a function ƒ(p) a "truth-function "when its argument p is a proposition, and the truth-value of ƒ(p) depends only upon the truth-value of p. Such functions are by no means the only common functions of propositions. For example, "A believes p" is a function of p which will vary its truth-value for different arguments having the same truth-value: A may believe one true proposition without believing another, and may believe one false proposition without believing another. Such functions are not excluded from our consideration, and are included in the scope of any general propositions we may make about functions; but the particular functions of propositions which we shall have occasion to construct or to consider explicitly are all truth-functions. This fact is closely connected with a characteristic of mathematics, namely, that mathematics is always concerned with extensions rather than intensions. The connection, if not now obvious, will become more so when we have considered the theory of classes and relations.

* Post 26.

*2 This phrase is due to Frege.
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Harold R. (Hal) Foster’s Prince Valiant
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