SETTING UP VECTOR LOGIC

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 argue that we have a similar problem in relating LOGIC to everyday speech.

"It was raining at the Washington Monument, 6/12/00." 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, 6/12/00?" does not qualify, since it is not in the 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 THAT 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", a term also used in Relational Database Theory, as in ORACLE. Now, in our initial form of VECTOR LOGIC, we introduce a 2-tuple, [M, S], wherein the SECOND COMPONENT, S, IS A STANDARD ASSERTION, and the FIRST COMPONENT IS AN ASSERTION THAT THE SPEAKER SPEAKS IN MOOD "M".

Let's adopt Bertrand Russell's "turnstile" symbol, |-, for THE DECLARATIVE MOOD, and the standard question mark, ?, for THE INTERROGATIVE MOOD.

Thus, we transform the discussion above into:

We can write the IMPERATIVE "Go to the store!" as [!, S], 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], 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!

The American mathematician-logician, Charles Saunders Peirce, founded SEMIOTICS: THE STUDY OR THEORY OF SIGNS:

  1. SYNTACTICS (SYNTAX) IS THE THEORY OF SIGNS WITHOUT CONSIDERING THEIR REFERENTS.
  2. SEMANTICS IS THE THEORY OF SIGNS THAT ADJOINS REFERENTS TO SYNTACTIC SIGNS.
  3. PRAGMATICS IS THE THEORY OF SIGNS THAT ALSO 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.

My vector logic resolves a long-standing problem in formulation of programming languages in BNF. (BNF stands for "Backus-Naur-Form", first developed to define terms in the programming language of Algol. 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, which fits BNF, except for assignments, where 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.

My 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.

(I pause to tell you that a universal language for publication once existed: Latin. Even the great Karl Friedrich Gauss (x-y) -- considered "the 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 the NATIONALISM that increased during the 19th century "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" liked to bandy a problem sentence in English: "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 are obviously different:

  1. [|-, "Time flies."], for the first version.
  2. [!, "Time flies."], for the second version.

(Did you notice that the first is in declarative mood; the second, in imperative mood?)

And the distinction can be brought out in another way. Suppose we adopt in wordage, a convention used by mathematicians, writing, for example, "x1, x2", with SUSCRIPTS to make DISTINCTIONS. Lets put "n" in subscript for NOUN; "v", for VERB. I'll show you that "subscripted coordinates" resolves the language problem.

Using subscriptee coordinates, the two above cases become, respectively:

  1. "Timen fliesv."
  2. "Times fliesn."

See what happened NOUN and VERB CHANGED PLACES. So there! you carpers.

Now I show you a different stage in vector logic.