t-TABLES FOR PROVING t-wff's IN t-LOGIC

These can be proven by the traditionally axiomatic, first used in Euclidean geometyr: VERIFY EACH STEP AS PERMITTED BY AN AXIOM OR BY A THEOREM DERIVED FROM ONE OR MORE AXIOMS.

I hated to the study of this mathematics, which is known as "synthetic geometry", and have never used it or taught it, preferring "analytic geometry", applying algebra and arithmetic via coorddinates. So I was happy to learn that a nonaxiomatic method of proofing had developed in the 19th century, due to the work of British mathematician, George Boole (x-y), and American mathematician, Charles Saunders Peirce ("Purse", x-y).

It's known as "The Method of Truth-Tables", and I'll illustrate a few applications.

Let "P" denote a proposition or statement (declarative sentence verifiable as truth or false), and let "T, F" denote its two possible truth-values, either "T" (true) or "F" (false). Then we can express this by a 1-column, 2-row Table:

		   P
		   T
		   F

If we have two such statements, denoted "P, Q", they have, between them, four possible truth-values, expressed by a 2-column, 4-row Table:

		   P	           Q
	           T               T
                   F               T
	           T               F
	           F               F

Please note that the first column alternates every row; the second column alternates every two rows.

If we have three such statements, "P, Q, R", hey have, between them, eight possible truth-value, expressed by a 3-column, 8-row Table:

		   P          Q          R
		   T          T          T
                   F          T          T
                   T          F          T
                   F          F          T
                   T          T          F
                   F          F          F
                   T          T          F
                   T          T          F

Please note that the first column alternates every row; the second collumn, every two rows; the third column, every four rows.

In general, n statments, have a truth-table, consisting of n columns, 2n rows, wherein the columns alternate every single row, then every 2 rows, then every 4 rows, then every 8 row, etc. -- doubling äs you go". (CHALLENGE: Show how this follows from the "binomial coefficients", often given in a Pascal-table.)

I'll go back to he case of 2 statements to show you the SUBTABLES for NEGATION (denoted "~"), CONJUNCTION (denoted "&"), DISJUNCTION ("OR"), CONDITIONAL ("→"), and BICONDITIONAL ("«").

	P          Q          ~P          P&Q          P OR Q          P→Q          P«Q
        T          T          F            T              T             T             T
        F          T          T            F              T             T             F
        T          F          F            F              T             F             F
        F          F          T            F              F             T             T

Please note:

One more comment: A STATEMENT STRING IS VALID (TAUTOLOGICAL) IF, AND ONLY IF, ITS TOTAL FORM HAS ALL "T's".

I won't go any further, above, because (Goody-gumdrops!) there developed another representation of the Table, making use of numerals and allowing arithmetic. The value "T" is replaced by "1"; the value "F", by "0". These are sometimes called "Indicator Tables".

I'll illustrate indicator tables for t-logic, but note 20th century use of these numeric-valued tables for SETS -- "1" meaning an element was in a given set; "0", not in the set. This found widespread usesage in the application of SETS in PROBABILITY THEORY.

And I wish, furthermore, to note that I UNIQUELY applied indicator tables to LATTICE THEORY. I can say this for a very simple reason.

It has been STANDARD to DENOTE THE "TOP" OF A LATTICE BY "1" (speaking of "the 1 of a lattice"), and its "BOTTOM" BY "0" "("the 0 of a lattice"). Naturally, when I saw I could extend these numeric tables to LATTICES, I realized the confusion that would result in have "1" label THE HEADING of ONE OF THE SUBTABLES while also appearing as a row-value somewhere in all the SUBTABLES. Hence, I RELABELED THE "TOP" AS "MAX"; THE "BOTTOM" AS "MIN" -- keeping "1,0" row-values. WELL!!! In the 35 years since I began, and widely searching the literature -- books, journals -- I've never seen my distinctions applied elsewhere. (If you've seen otherwise, wise me up.) Hence, I say I can conclude that I UNIQUELY APPLY indicator tables to LATTICE THEORY.

One more comment, before going to indicator tables. I call these "t-tables", because they apply to "t-sets", which do not allow multiple tokenage for DEGREE or ORDER, but consider only KIND or TYPE. In t-set theory, EVERY SUBSET OF A GIVEN SET HAS A COMPLEMENT. It means the same thing in t-lattices, which are better known as "complemented distributive lattices" or (by distortion of his work) "Boolean lattices". In t-logic, this means that the NONCONSTRUCTIVE PROOF OF TERTIUM NON DATUR (CONTRADICTION OF THE NEGATION OF A STATEMENT PROVES THE STATEMENT) AND THE NONCONSTRUCTIVE AXIOM OF ORDER ("ALL EQUIVALENCE SETS CAN BE ORDERED SUCH THAT A REPRESENTATIVE OF EACH CAN BE CHOSEN" -- also known as "Bertrand Russell's socks).

But this causes an immediate problem in using set-language in arithmetic. 6 has proper factor 2 and 3, which we can write in set-notation as {2, 3} But 12 = 4 x 3 = 2 x 2 x 3, with double tokens of factor 2. But, in set-language: {2, 2, 3} = {2,3} -- since STANDARD SET THEORY (t-set theory) DOESN'T RECOGNIZE MULTIPLE TOKENS OF MEMBERS. This collapses an infinity of distinctins into MUD. And it meant that THE NEW MATH (with this set theory) DID NOT SUPPORT THE OLD MATH (with its arithmetic FACTOR THEORY).

So, in 1957, I rebelled against this -- thereby becoming PERSONA NON GRATA with the National Science Foundation, which supported THE NEW MATH. I discovered NO MATHEMATICAL reason for this, only PHILOSOPHICAL BIAS. So I arcticulated o-logomath, which already existed in GENERAL FACTOR THEORY and in Dedekind's derivation of distributive lattice theory from complemented distributive lattice theory. (In a distributive lattice, not all elments have to have a complement.) These two models guided me in developing o-logomath.

Given this, I can make my point by returning to the subject oF indicator tables. I UNIQUELY EXTENDED THEM -- as t-tables -- to o-tables. What's the difference? You'll see, below, that in a t-indicator-table, each SUBTABLE PRESIDES OVER A SINGLE COLUMN OF VALUES. In am o-indicator-table, a SUBSTABLE MAY PRESIDE OVER TWO OR MORE COLUMNS TO EXPRESS ITS GIVEN DEGREE OR ORDER. Thus, since 4 = 2 x 2, its SUBTABLE PRESIDES OVER TWO COLUMNS, EACH COPYING THE SINGLE COLUMN for 2. Dig? This is discussed elsewhere.

Now to rewrite the truth-tables above (for P, Q) as indicator-tables.

	P          Q          ~P          P&Q          P OR Q          P→Q          P«Q
        1          1          0            1              1             1             1
        0          1          1            0              1             1             0
        1          0          0            0              1             0             0
        0          0          1            0              0             1             1

OK, what does MODUS PONENS (MP) LOOK LIKE IN THESE TERMS?

Using again "P,Q" for our statements, we have:

		((P→Q) & Q) → Q

We'll build MP up in the SUBTABLES: P; Q; (P→Q); ((P→Q) & Q); finally, ((P→Q) & Q) → Q.

	P          Q        (P→Q)    ((P→Q) & P)      ((P→Q) & P) → Q        
        1          1          1            1                  1         
        0          1          1            0                  1
        1          0          0            1                  1
        0          0          1            0                  1

The LAST COLUMN HAS ALL ONES, hence, MP IS INDICATOR-TABLE-PROVEN AS A VALIDITY (TAUTOLOGY)!

Are you ready for a t-TABLE-test?.