The standard indicator table ("truth table") involves only distinctions of type (kind) not of order (degree), that is, t-Tables. On the other hand, while these tables are applied in statement logic, set theory, probability theory, I am perhaps unique in applying them to factors of natural numbers or integers.For, when doing so, distinctions of order (degree) immediately arise since you must deal not only with t-numbers ("square-free" numbers) -- that is, numbers such as 30 = 2*3*5, containing each prime factor only once -- but you must also deal with o-numbers ("nonsquare-free" numbers) such as 60 = 2*2*3*5, in which a prime factor appears multiply.
("The New Math" promoted a set theory that ignores multiple tokens of type which was incompatible with "The Old Math" of arithmetical factoring. This failure of "The New Math" to support "The Old Math" was a technical reason for its failure.)
So, I realized that t-tables can easily be extended to o-tables. For simplicity, I will consider tables for the t-number of 6 = 2*3 and the o-number of 12 = 2*2*3.
t-TABLE OF 6 = 2 * 3 1 2 3 6 0 0 0 0 0 0 1 1 0 1 0 1 Now we extend from the t-number 6 to the o-number of 12: 1 1 1 1
o-TABLE OF 12 = 2 * 2 * 3 1 2 3 4 6 12 0 - 0 - 0 - 0 0 0 - 0 0 0 - 0 - 1 - 0 0 1 - 1 0 0 - 1 - 0 - 1 1 1 - 1 1 1 - 1 - 1 - 1 1 1 - 1 1 Concerning this table, note:
What does the blank in this o-table signfify? I'll answer that by referring to a hisorical problem. Some one asked (as many kids have asked, since then), "What is the difference between zero and the empty set?" Answer: "It's the difference between having a wallet with no money in it, and not having a wallet." So, I'll give you an answer in terms of how we would build this in circuitry. The difference between zero and blank is the difference between a circuit with the switch turned off, and not having a circuit.
- 4 = 2 * 2 has two complete columns for the double tokens of prime 2, and each column is the same as the column for 2;
- 12 = 6 * 2 has two columns, the first being duplicate of the column for 6, the second being duplicate of the column for 2;
- only the numbers 4 and 12 have two complete columns, the other numbers having a second column of blanks (explained below).
Given this prototype, we can write a a t-table or o-table for any number whatever, as well as for o-sets, o-lattices, o-logic. Thus, my o-math encompasses both "The Old Math" and "The New Math".
After I worked this out in 1958, I realized that an entry in an o-table, when read right-to-left could be interpreted as a probability in binary numeration. Thus, I had independently conceived a digital form of what was to be developed by Lofti Zadeh in the 60's as an analogic form of "fuzzy set theory", leading to "fuzzy logic".