Introducing Extended Additive Logic: 1-bit Case

08 May, 2026

For pedagogical reasons, I shall begin with the simplest case of Extended Additive Logic (EAL). We will start with the 1-bit case, which will not reveal any departures from the classical paradigm, but will at least help us get used to the notation and terminology.

1-bit EAL

Truth Values

The truth values of EAL are binary vectors. In the 1-bit case they are 1-dimensional binary vectors $0$ and $1 .$ The truth values are ordered lexicographically, with $1$ being the maximal element, and $0$ being the minimal element.

\begin{matrix} 0 < 1 \end{matrix}

We will call $1$ the unit of this system. The unit is always the maximal element, and it is always an idempotent. On the other hand, $0$ is an absorbing nilpotent. These terms will seem somewhat useless in the 1-dimensional case but we will see how significant they are even in 2-bit EAL. I will discuss nilpotents and idempotents in more detail later.

Addition

The first logical connective we will consider is addition. Two binary vectors can be added by adding up their components modulo two, and in our 1-dimensional case it is very simple:

\begin{matrix} \oplus & 0 & 1 \\ 0 & 0 & 1 \\ 1 & 1 & 0 \end{matrix}

This operation is identical to the exclusive OR ( $X O R$ ) operation, regardless of dimension.

Multiplication

Recall that $1$ is an idempotent. This implies $1 \otimes 1 = 1 .$ Since $0$ is absorbing, $0 \otimes 1 = 0$ and $1 \otimes 0 = 0$ hold. Of course, $0 \otimes 0$ is also equal to $0 .$ This way, we have inferred the entire multiplication table:

\begin{matrix} \otimes & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 1 \end{matrix}

Somewhat disappointingly, it turns out that $\otimes$ is the ordinary boolean $A N D .$ However, this is only true of the 1-bit model. Things get more interesting when the truth values have two or more dimensions.

Max and Min

Since the truth values are ordered lexicographically, we can take the maximum (denoted $\lor$ ) and minimum (denoted $\land$ ) of any two truth values. This is fairly self-explanatory.

\begin{matrix} \lor & 0 & 1 \\ 0 & 0 & 1 \\ 1 & 1 & 1 \end{matrix} \begin{matrix} \land & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 1 \end{matrix}

Implication

The strong implication is notated as $a ↠ b$ and is defined as the greatest truth value $v$ such that $a \otimes v \leq b .$

\begin{matrix} ↠ & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{matrix}

The weak implication is notated as $a \to b$ and is defined similarly: it is the greatest truth value $u$ such that $u \otimes a \leq b .$

\begin{matrix} \to & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{matrix}

In the general case, these can be very different operations, but in our 1-dimensional example they turn out to be identical, because multiplication here obeys the commutative law.

Properties

Observe that multiplication ( $\otimes$ ) distributes over

addition: $a \otimes (b \oplus c) = (a \otimes b) \oplus (a \otimes c),$
maximum: $a \otimes (b \lor c) = (a \otimes b) \lor (a \otimes c)$ and
minimum: $a \otimes (b \land c) = (a \otimes b) \land (a \otimes c) .$

In the 1-dimensional case, these seem somewhat trivial. But we will see later how they become powerful when working in 2 or more dimensions. In particular, distributivity over minimum and maximum is always retained, while distributivity over addition can break!

Conclusion and Parting Remarks

It should be clear that this case of EAL is identical to ordinary boolean logic. Indeed, boolean logic is, as we will see, simply a 1-dimensional shadow of EAL.

I highly recommend the following resources to gain some intuition for the general family of logical structures to which EAL belongs:

Big thanks to Sophie for editing and formatting advice.