Juvenis

The definition of slavery in my title I learned from Abraham Lincoln (1809-65), 16th President of The United States, signer of the 1833 "Emancipation Proclamation" to free African-American slaves, Lincoln said it in the Seventh and Last Debate with Stephen A. Douglas, Alton, Illinois, October 15, 1858, the famous debates that put Lincoln on the road to Presidency: "That is the issue that will continue in this country when these poor tongues of Judge Douglas and myself shall be silent. It is the eternal struggle between these two principles -- right and wrong -- throughout the world. They are the two principles that have stood face to face from the beginning of time; and will ever continue to struggle. The one is the common right of humanity and the other the divine right of kings. It is the same principle in whatever shape it develops itself. It is the same spirit that says, 'You work and toil and earn bread, and I'll eat it.' No matter in what shape it comes, whether from the mouth of a king who seeks to bestride the people of his own nation and live by the fruit of their labor, or from one race of men as an apology for enslaving another race [or other people, it is the same tyrannical principle." One form of present day Hidden Slavery: most of America's independent inventors lose their patent rights in the courts. American Contract Law permits corporations to fight independent inventors (and their inheritors) for up to 99 years -- as in one case I found online -- until the independent side can no longer afford legal action. (Irony: The Fourteenth Amendment to The Constitution to provide civil rights to former slaves was interpreted by The Supreme Court to make corporations legal! Are corporations our inventor plantations? And do the banks know this, in denying loans to striving inventors?) Today's Americans and Media mavens ignore the fact that America's most famous inventor, Thomas Edison (1847-1931), spent more money defending his patents in the courts than he made from licenses. Today's Americans and Media mavens ignore past inventors (especially women and African-Americans) who endowed our lives: they work and toil while we eat their bread. Historians denigrate invention by citing as the only "Industrial Revolution" what should be called The Thermodynamic Industtrial Revolution</i>, beginning in 1776 with an effective steam engine, ignoring The Mechanical Industrial Revolution of 12th-13th centuries when monks built thousands of wind mills and water mills over Europe and Britain (cited in "The Medieval Machine: The Industrial Revolution of the Middle Ages", J. Gimpel. Also ignoring The Electrical Industrial Revolution, beginning in 1880 when Thomas Edison wired the streets of NYC. Ignoring The Electronic Industrial Revolution, beginning with 1920's radio stations. And The Photonic-Nanotech Industrial Revolution, begun recently. Today's Americans and Media mavens ignore Sam Slater (1768-1835), Father of the American Industrial Revolution, whose inventiveness founded the first major American industry: textiles. (Wikipedia has this.) A 1980's newspaper reported that the only kitchen convenience not created by independent inventors in basement or garage was the garbage disposal. A 1980's TV network program, "If Japan Can Do It, Why Can't We?", reported many cases wherein American inventors were ignored in America but paid and honored in other countries. (Jack Kilby, American inventor of the microchip, was honored in Japan for this decades before being honored here.) Have you heard of Margaret Knight (1838-1914)? You've used her invention: the box-type paper shopping bag. Knight built the machine making these, and her original machine is in The Smithsonian Museum. Google("women inventors+jonhays") to read of her and others. Today, the only extant history of early American inventors, "Yankee Science in the Making" (1958), was written by a Dutch-American mathematician, Dirk Sruik (1894-2000). Many inventors listed by Struik invented guns for the American Revolution and for developing our new country, but America's two largest gun-firms are now owned by British interests, a fact ignored by The National Rifle Association as well as by Americans and Media mavens. Also ignored are debts to scientists. The TV screen and computer screen are consequences of the 1905 "Photo-Electric Effect Law" of Albert Einstein (1875-1955). Thanks, Al!--Years ago I formulated a "find cuurent independent inventors test". Each year I'd calculate the number of years since end of WWII. Then go back the same number of years before WWII. I'd consult the current "World Almanac" (popular with Americans) to see the number of independent inventors listed. Usually, I'd find A DECLINE OF ABOUT 400 PER CENT! If that were a comparable decline in newspapers or TV station or motion picture theaters, you'd hear about it! Does any one care about this hidden slavery and its effect upon our lives?

In PHYSICS, speed is a scalar, or 0-vector, measure of motion. Standing in the middle of a street, you see a car racing at (the speed of) 50 mph. But it makes a difference as to whether it is coming toward you or going away: DIRECTION was missing in that "50 mph" statment. Adjoining DIRECTION to SPEED creates a (1-)vector measure of motion</I>. I argue we've a similar problem in relating LOGIC to everyday speech. "It was raining at the Washington Monument, 4/12/08" is A POSITIVE SIMPLE DECLARATIVE SENTENCE CAPABLE OF VERIFICATION -- qualifying it as a STATEMENT (a.k.a. proposition) in STATEMENT LOGIC (a.k.a. 0-ORDER PREDICATE LOGIC). But "Was it raining at the Washington Monument, 4/12/08?" does not qualify, being not in DECLARATIVE MOOD. INTERROGATIVES, SUBJUNCTIVES, IMPERATIVES, PETITIVES all fail to qualify -- IN SCALAR LOGIC. But we can fit them into VECTOR LOGIC, by PROVIDING FOR THE MOOD OF STATEMENT as well as STATEMENT. STATEMENT LOGIC IS TRUTH-FUNCTIONAL, meaning that THE TRUTH-VALUE OF A COMPOUND STATEMENT DEPENDS ONLY ON THE TRUTH-VALUE OF ITS COMPONENTS, not ORDER or CONTEXT or "whatever". My VECTOR LOGIC CONSERVES TRUTH-FUNCTIONALITY, as follows. In STATEMENT LOGIC, we may denote a STATEMENT by a single letter, say, S, a form which mathematicians call a "1-tuple". Now, in our initial form of VECTOR LOGIC, we introduce a 2-tuple, [M, S], wherein SECOND COMPONENT, S, IS A STANDARD ASSERTION, and the FIRST COMPONENT IS AN ASSERTION THAT SPEAKER SPEAKS IN MOOD "M". Let's adopt Bertrand Russell's "turnstile" symbol, |-, for DECLARATIVE MOOD, and standard question mark, ?, for THE INTERROGATIVE MOOD. Thus, we transform the discussion above into: "It is raining at the Washington Monument, 6/12/00.": [|-, S]</FONT>. "Is it raining at the Washington Monument, 6/12/00?": [?, S]</FONT>. We can write the IMPERATIVE "Go to the store!" as [!, S]</FONT>, where, now S denotes "You are going to the store." We can write the SUBJUNCTIVE "Would that you were going to the store." as [%, S], using "%" to denote THE SUBJUNCTIVE MOOD. We can write the PETITIVE "Please go to the store." as [*, S]</FONT>, with "*" denoting THE PETITIVE MOOD. (Other "mood operators" can be introduced.) Please notice that EACH VECTOR COMPONENT IS TRUTH-FUNCTIONAL: [The speaker speaks in mood M, The speaker makes statement S]. Using VECTOR LOGIC, we BYPASS THE LANGUAGE PROBLEMS OF SCALAR LOGIC, and ENCOMPASS MUCH OF DAILY SPEECH! American mathematician-logician-philosopher, Charles Saunders Peirce, founded SEMIOTICS: STUDY OR THEORY OF SIGNS: SYNTACTICS (SYNTAX) IS THE THEORY OF SIGNS WITHOUT REFERENTS. SEMANTICS: THEORY OF SIGNS ADJOINING REFERENTS TO SYNTACTIC SIGNS. PRAGMATICS: THEORY OF SIGNS THAT CONSIDERS THE SIGN-USER. Most people indulge in two confusions here. First, that Peirce promulgated "pragmatic philosophy": MEANING IS IN USE. He did not! And was so upset at this nonsense that he said he might change his term to "pragmaticism", since "no one would steal such an ugly term". Secondly, people say that two people "have a semantic difference", instead of saying that it's "a pragmatic difference". Thus, our VECTOR LOGIC IS A PRAGMATIC LOGIC. Vector-Logic resolves a long-standing problem in formulation of programming languages in BNF ("Backus-Naur-Form", first developed to define terms in ALGOL programming language. (Elsewhere, I show how to use BNF to develop a Methodocopoeia for teachers.) All of a programing language can be formulated in the DECLARATIVE mood, fitting BNF, except for assignments, which are IMPERATIVE. (In the C Programing Language I used for 12 years at the Naval Research Laboratory, "x = 2" is an assignment, hence, IMPERATIVE; "x = = 2" is an equivalence, HENCE, declarative.) But vectored BNF eliminates the problem in scalar BNF, by simply changing the "mood" component. Thus, a "hole" is removed. vector-Logic also resolves an even more famous problem. In the 1960's, many argued that COMPUTER TRANSLATION OF LANGUAGES would greatly simplify the publication of technical or literary-dramatic literature. But only a very limited success was ever achieved for years. (Note: a universal language for publication once existed: Latin. Even the great Karl Friedrich Gauss -- considered "greatest of all mathematicians" and a skilled linguist -- wrote in beautiful Latin, and his works could be read all over the world, including in China. But 19th century NATIONALISM "killed" Latin as a universal language. Result: It costs our Government and Universities and Publishers millions of dollars a year to fight the translation problem, and they still can't keep up!) Critics of "machine translation" bandied an Enlish problem: "Time flies." This has the usual meaning that "Time gets away from you if you're no careful." But a computer translated it into a matter of "measuring the flight of certain insects". However, this ambiguity only arises in scalar logic, as used in English Grammar. In Vector-logic, the two obviously differ: [|-, "Time flies."] for the first version; [!, "Time flies."], for the second version.--The Austrian logician, Gotthold Frege, in effect, treated statement logic as zero-order predicate logic and created first-order predicate logic, second-order predicate logic, tc. He did so by creating two QUANTIFIERS: The UNIVERSAL QUANTIFIER ("for all instances or members of the given set"), denoted by capital "A" turned upside down. The EXISTENTIAL QUANTIFIER ("ONE INSTANCE EXISTS"), denoted by capital "E" turned left. The VECTOR-LOGIC thus described can be carried over into "the predicate calculus".--Frege also noted that a NAME has two linguistic ROLES in language: USE, as in "Boston is a busy town", for the name "Boston". And MENTION, as in "'Boston' is a 6-letter word. (Note that the latter distinction can be made, in written form, by putting the term in quotation marks, showing that we are talking about the NAME rather than USING it. VECTOR-LOGIC also invokes these distinctions. The "extra" component we introduced plays a USE ROLE: THE SPEAKER IS USING A MOOD IN SPEAKING. However, in the spirit of Frege, we can also invoke the MENTION ROLE: THE MODES OF SPEECH. That is, "The sign-user mentions statenent S is mode D", using this letter for "mode", so as not to confuse it with "mood" -- MODES OF DISCUSSION, viz.: ALETHIC MODES or MODES OF TRUTH are, principally, necessary, possible, impossible, contingent. (For purposes of discussion below, let's adopt, A, as the alethic mode operator.) TENSE MODES enable TEMPORAL aspects of sentences. (Aristotle considered this problem: a declaration which is TRUE at a given time, and FALSE at others. I avoided this by putting time into my sentence about "raining".) Let's denote the temporal mode operator as "T". DEONTIC MODES deal with obligation, can, may, etc. (This is the MODE used in ETHICAL and MORAL discussions, as well as LOGICAL and THEOLOGICAL ones.) Let's denote it by "D". EPISTEMIC MODES deal with terms such as knowing, believing. Let's denote it by "E". The PREFERENCE MODE is needed in economic and political discussion. Let's use "P". The FREE MODE allow fictional imagination to "roam free". Let's use "F". Etc.--VECTOR-LOGIC via MODES enables us to RESOLVE A CLASSICAL PARADOX: "The Morning Star Paradox", often stated in SYLLOGISTIC FORM as in "All men are mortal. Socrates is a man. Therefore, Socrates is mortal. The "Morning Star Paradox" runs: "The Shepherd knows that Venus is The Morning Star. The Morning Star is The Evening Star. Therefore, "The Shepherd knows that Venus is The Evening Star." Then this would follow, in scalar logic, by TRANSITIVITY: "V = M. M = S. S = V." But the Shepherd may not actually know this. In terms of Vector-Logic, it would take the mood-form: [|-, "V = M"]. [|-. "M = S"]. [|-, "S = V"]. And transitivity would support it in the 2nd or syntactic component. But in the MODE form: [A, "V = M"]. [A, "M = S"]. [A. "S = V"]. In both cases, the 1st component is CONSTANT ("a parameter"), which shifts the logic-question entirely to the 2nd component-argument, which is VALID in Scalar-Logic. However, the "Morning Star Paradox" must be considered only in the mode-form of Vector-Logic and we must invoke the epistemic mode, for which we adopted (above) the operator, E. Using our two mode-operators, we have: [E<, "V = M"]. [A, "M = S"]. [E, "M = S"]. The mode shift is epistemic ("knowing") to alethic to epistemic -- or E to A to E. Lack of tranitivity shifts the burden to the 1st component, with NOTHING LOGICAL TO SUPPORT A SHIFT OF MODE. Therefore, THE ARGUMENT IS INVALID, hence, NO PARADOX! Other ambiguities and paradoxes could be resolved similarly. (Somewhere, in the "time capsule" of my past cluttered files, there was a C Programs I wrote years ago, automatically translating from well-formed English sentences into Vector-Logic, and vice versa. If JAVASCRIPT had existed then, it would have been easier and more might have been accomplished. I CHALLENGE you to TAKE UP THIS WORK! And invite comment/argument.

Two neglected math concepts: EXTENSION, INTENSION. Easily explicated in set language. Say, in natural number system, N, numbers less than seven. An extension is {0,1,2,3,4,5,6}. An intension is {x: x in N, x < 7}, "set of any x such that x is in N and less than 7". Seems ok. But Problem: extension is univalent: one referent; intension may be multivalent: descriptive statement may fit many referents. Axioms are intensional. The familiar Peano Axioms invoke the standard integer system. But online is a nonstandard version fitting Peano Axioms such that its every integer is greater than any standard integer. (A "stolen identity" case?) Banach-Tarsky Paradox says Euclidean Geometry Axioms allow moon to be cut into five parts, fitted together, put in your pocket. Standard Arithmetic is founded axiomatically: intensionally. But my extensional alternative exists online: generatics. William Rowan Hamilton rejected standard definition of a complex number as sum of real number plus real in product with square root of negative one. Hamilton invented the vector and its label -- ignored? -- to define complex number as bivector or reals with appropriate rules. Generatics uses this procedure to start with naturals, N, recursively (extensionally) defined; define integers, J, as bivectors of naturals with appropriate rules; define rationals, Q, as bivectors of integers with appropriate rules; define reals, R, limits of infinite vectors of rationals (decimal numbers) with appropriate rules; complex numbers in Hamilton's way. (As a teacher, I've given my best students clues whereby each independently discovered this.) But the bivectors are hidden by notation. Equivalence reduces naturals bivector to three types, explicated as positive/negative integers or zero, hidden by signs. Equivalence reduces integer bivectors to three types, hidden by solidus as fractions. Equivalence reduces infinite vectors of reals to three types hidden as of decimal numbers. But something happens with complex numbers. Using concept of "modul" (not module!) in "Introduction to Number Theory" by Oystein Ore, p. 159. A modul is a structure closed under subtraction. Ore notes naturals numbers aren't closed under subtraction, but integers, rationals, reals, and complex numbers are closed under subtraction, so each number forms a modul. Imaginaries form their own modul. Ore doesn't note that complex numbers form a BIMODUL: one for real part, one for imaginary part. Significance? VECTORS CAN NO LONGER BE HIDDEN BY NOTATION. Leads to MULTIVECTORS (a.k.a. arithmetic of Clifford Numbers), with Gibbs-Heaviside vectors as special case. Three products (inner, outer, multiproduct) distiguish these. Online in "Arithmetic Redux", I derive these in high school level. But all is extensional: univalent: single referent! I invite comment/argument.