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Before we continue with this intinerary, however, I would like to highlight another sort of differential aspect that concerns the ''boundary operator'' or the ''marked connective'' that serves as one of the two basic connectives in the cactus language for ZOL.

Before we continue with this intinerary, however, I would like to highlight another sort of differential aspect that concerns the ''boundary operator'' or the ''marked connective'' that serves as one of the two basic connectives in the cactus language for ZOL.

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For example, consider the proposition <math>f\!</math> of concrete type <math>f : P \times Q \times R \to \mathbb{B}</math> and abstract type <math>f : \mathbb{B}^3 \to \mathbb{B}</math> that is written <math>\texttt{(} p, q, r \texttt{)}</math> in cactus syntax.  Taken as an assertion in what Peirce called the ''existential interpretation'', the proposition <math>\texttt{(} p, q, r \texttt{)}</math> says that just one of <math>p, q, r\!</math> is false. It is instructive to consider this assertion in relation to the logical conjunction <math>pqr\!</math> of the same propositions.  A venn diagram of <math>\texttt{(} p, q, r \texttt{)}</math> looks like this:

Consider the proposition f of concrete type f : !P! x !Q! x !R! -> B

and abstract type f : B^3 -> B that is written as "(p, q, r)" in the

cactus syntax.  Taken as an assertion in what C.S. Peirce called the

"existential interpretation", the so-called boundary form "(p, q, r)"

asserts that one and only one of the propositions p, q, r is false.

It is instructive to consider this assertion in relation to the

conjunction "p q r" of the same propositions.  A venn diagram

for the boundary form (p, q, r) is shown in Figure 11.

{| align="center" cellpadding="10"

| [[Image:Minimal Negation Operator (p,q,r).jpg|500px]]

|                |                        |                |

|               |                        |                |

Figure 11.  Boundary Form (p, q, r)

In relation to the center cell indicated by the conjunction pqr

In relation to the center cell indicated by the conjunction pqr

the region indicated by (p, q, r) is comprised of the "adjacent"

the region indicated by (p, q, r) is comprised of the "adjacent"