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ANOTHER STAGE IN VECTOR LOGIC

The Austrian logician, Gotthold Frege (x-y), in effect, treated statement logic as zero-order predicate logic and created , second-order perdicate logic, etc. He did so by creating two QUANTIFIERS:

  1. The UNIVERSAL QUANTIFIER ("for all instances or members of the given set"), denoted by capital "A" turned upside down.
  2. The EXISTENTIAL QUANTIFIER ("ONE INSTANCE EXISTS"), denoted by capital "E" turned left.

Frege also noted that a NAME has two ROLES in language:

  1. USE, as in "Boston is a busy town", for the name "Boston".
  2. MENTION, as in "'Boston' is a 6-letter word.
(Note that the latter disinction 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.

VECTORLOGIC 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.:

  1. The 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.)
  2. The 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".)
  3. The DEONTIC MODE deals with obligation, can, may, etc. (This is the MODE used in ETHICAL and MORAL discussions, as well as LOGAL and THEOLOGICAL ones.)
  4. The EPISTEMIC MODE deals with terms such as knowing, believing. (For purposes of that discussion we mentioned above, let's adopt, E, as the epistemic mode operator.)
  5. The PREFERENCE MODE is needed in economic and ppolitical discussion.
  6. The FREE MODE allow fictional imagination to "roam free".
  7. Etsettery.
VECTOR LOGIC enables us to RESOLVE A CLASSICAL PARADOX: "The Morning Star Paradox".

It seems to have the form of the famous SYLLOGISM, which runs:

  1. All men are mortal.
  2. Socrates is a man.
  3. Therefore, Socrates is mortal.

The "Morning Star Paradox" runs, thus:

  1. "The Shepherd knows that Venus is The Morning Star."
  2. "The Morning Star is The Evening Star."
  3. Therefore, "The Shepherd knows that Venus is The Evening Star."

But the Shepherd may not actually know this at all. Let's see what vector logic can tell us.

Suppose, on the contrary, the above argument were in the form:

  1. "Venus is the Morning Star."
  2. "The Morning Star is The Evening Star."
  3. "Therefore, Venus is The Evening Star."

Then this would follow, in scalar logic, by TRANSITIVITY:

  1. V = M.
  2. M = S.
  3. S = V.

In terms of vector logic, it would take the mood-form:

  1. [|-, "V = M"].
  2. [|-. "M = S"].
  3. [|-, "S = V"].

And transitivity would support it in the 2nd or syntactic component.

Approaching this again, please remember that, above, we adopted A as the alethic mode operator. Using this, the previous argument would take the mode-form:

  1. [A, "V = M"].
  2. [A, "M = S"].
  3. [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:

  1. [E, "V = M"].
  2. [A, "M = S"].
  3. [E, "M = S"].

The mode shift is epistemic ("knowing") Þ alethic Þ epistemic -- or E Þ A Þ E.

This shifts the burden to the 1st componemt, 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" assumed by my cluttered files, there should be C Programs I wrote fourteen years ago, automatically translating from well-formed English sentences into vector logic, and vice versa. If JAVASCRIPT had existed then, it would have been even easier and more might have been accomplished.

I CHALLENGE you to TAKE UP THIS WORK!

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