Symbolic Logic and Pythagorean Tables: Boolean Operations and Mathematical Foundations

metaphysical sophistries; perhaps the current Simplicio should reflect on the language used by any electronic computer... Zero posits One, $0 + 1 = 1$ and if One posits Zero it finds only itself: $1 + 0 = 1$, One that finds itself returns to Zero: $1 + 1 = 0$. The reality to which we have just reconnected is expressed through the following Pythagorean table:

$\begin{array}{c|cc} + & 0 & 1 \\ \hline 0 & 0 & 1 \\ 1 & 1 & 0 \\ \end{array}$ Tab.9

It is evident that table (Tab.9) is a substantialization of table (Tab.8), with the equivalences $0 \leftrightarrow \alpha$, $1 \leftrightarrow \beta$ and where the composition law is the supreme generative law of addition. Another symbolism, which connects to what we have just shown, is that given by the great law of even and odd, which bipartitions the set of all integers. In fact, through the correspondence $\text{even} \leftrightarrow \alpha$, $\text{odd} \leftrightarrow \beta$, the Pythagorean table (Tab.8) expresses the law of bipartition of numbers into two classes: that given by all numbers divisible by 2, the even numbers, and that given by all numbers non-divisible by 2, the odd numbers. The composition is still the law of addition. The Pythagorean table (Tab.8) expresses, in other words, the well-known truth that the sum of an even number with another even number is an even number, the sum of an even number and an odd number is odd and that the sum of two odd numbers is an even number; that is:

$\begin{array}{c|cc} + & \text{even} & \text{odd} \\ \hline \text{even} & \text{even} & \text{odd} \\ \text{odd} & \text{odd} & \text{even} \\ \end{array}$ Tab.10

We believe it is important here to recall that the return of One to Zero also founds the multiplicative law of the opposite. In Reality, absolute Zero, that is absolute Good, is supra-essential fullness, which can let Being-as-essential-fullness be, that is One, which finds Good again in the generation of itself: $1 + 1 = 0$. In the reality of the generativity of Being alone, Zero manifests itself as pushing one into the opposite of one, that is into $-1$, and the law is that of multiplication. In this case table (Tab.8) with the evident equivalences becomes:

$\begin{array}{c|cc} \times & 1 & -1 \\ \hline 1 & 1 & -1 \\ -1 & -1 & 1 \\ \end{array}$ Tab.11

The symbolic speculation that addresses both the law of the opposite and of the inverse will be the subject of a forthcoming work. There is finally a last, fifth, essential modality in which two actions, $\alpha$, $\beta$, can be concatenated so that the result is still one of them; as in the case of the law of overcoming the identical, also in this last case it holds that the result of concatenation $\alpha \circ \beta$ is equal to that of concatenation $\beta \circ \alpha$. Let us see how this last table imposes itself on our meditation by simple metaphysical descent and how therefore taking it into consideration descends from the method of meditation itself. We believe that when it is understood that many practical truths have been discovered, we are dealing with meditative situations analogous to the one we are living together, in this instant. Such table is the following:

$\begin{array}{c|cc} \circ & \alpha & \beta \\ \hline \alpha & \alpha & \alpha \\ \beta & \alpha & \beta \\ \end{array}$ Tab.12

Table (Tab.12) is a support for meditating on the link between what is true and what is false. If we operate the substitutions $\alpha \leftrightarrow \text{true}$, $\beta \leftrightarrow \text{false}$ and the composition is read as the conjunction of two assertions, the Pythagorean table (Tab.12) expresses the law of the veracity of the conjunction of two affirmations each of which can be true or false. We have, in fact:

$\begin{array}{c|cc} \land & \text{false} & \text{true} \\ \hline \text{true} & \text{false} & \text{true} \\ \text{false} & \text{false} & \text{false} \\ \end{array}$ Tab.13

that is, if $p$ and $q$ are two assertions, then the veracity of the single affirmation obtained by conjoining $p$ and $q$ depends on the veracity of both assertion $p$ and assertion $q$ in the way indicated in table (Tab.13). We note that if instead we consider the logical alternative, that is given two propositions $p$ and $q$ we form that single proposition which affirms "$p$ or $q$", it is well known that the affirmation "$p$ or $q$" is false in the only case in which both $p$ and $q$ are false assertions. The Pythagorean table of the truth value of the logical alternative depending on the truth values of the individual propositions $p$ and $q$ is the following:

$\begin{array}{c|cc} \lor & \text{false} & \text{true} \\ \hline \text{false} & \text{false} & \text{true} \\ \text{true} & \text{true} & \text{true} \\ \end{array}$ Tab.14

Apparently table (Tab.14) is different from table (Tab.12). In reality it is obtained from the latter only by exchanging the position in it of $\alpha$ with $\beta$; that is

$\begin{array}{c|cc} \circ & \beta & \alpha \\ \hline \beta & \beta & \beta \\ \alpha & \beta & \alpha \\ \end{array}$ Tab.15

The same table (Tab.12), but with the substitutions already seen $\alpha \leftrightarrow \text{even}$ $\beta \leftrightarrow \text{odd}$ and with the interpretation of the composition law between two actions through the well-known operation of product between two numbers, allows us to visualize the law of behavior of even and odd in multiplication:

$\begin{array}{c|cc} \times & \text{even} & \text{odd} \\ \hline \text{even} & \text{even} & \text{even} \\ \text{odd} & \text{even} & \text{odd} \\ \end{array}$ Tab.16

In other words, table (Tab.16) translates the well-known law that the product of...