Cactus Language for Propositional Logic (cont.)
Table 1 shows the cactus graphs, the corresponding cactus expressions, their logical meanings under the so‑called existential interpretation, and their translations into conventional notations for a sample of basic propositional forms.
Table 1. Syntax and Semantics of a Calculus for Propositional Logic
The simplest expression for logical truth is the empty word, typically denoted by 
![\begin{matrix} a + b \quad = \quad \texttt{(} a \texttt{,} b \texttt{)} \\[8pt] a + b + c \quad = \quad \texttt{(} a \texttt{,(} b \texttt{,} c \texttt{))} \quad = \quad \texttt{((} a \texttt{,} b \texttt{),} c \texttt{)} \end{matrix}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Bmatrix%7D++a+%2B+b+%5Cquad+%3D+%5Cquad+%5Ctexttt%7B%28%7D+a+%5Ctexttt%7B%2C%7D+b+%5Ctexttt%7B%29%7D++%5C%5C%5B8pt%5D++a+%2B+b+%2B+c++%5Cquad+%3D+%5Cquad+%5Ctexttt%7B%28%7D+a+%5Ctexttt%7B%2C%28%7D+b+%5Ctexttt%7B%2C%7D+c+%5Ctexttt%7B%29%29%7D++%5Cquad+%3D+%5Cquad+%5Ctexttt%7B%28%28%7D+a+%5Ctexttt%7B%2C%7D+b+%5Ctexttt%7B%29%2C%7D+c+%5Ctexttt%7B%29%7D++%5Cend%7Bmatrix%7D&bg=ffffff&fg=000000&s=1&c=20201002)
It is important to note the last expressions are not equivalent to the 3‑place form
Resources
- Logic Syllabus
- Minimal Negation Operator
- Survey of Differential Logic
- Survey of Animated Logical Graphs
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-ary scope. The syntactic formulas of that calculus map into a family of graph-theoretic structures called “painted and rooted cacti” which lend visual representation to the functional structures of propositions and smooth the path to efficient computation.
to mean exactly one of the propositions 
is false, in short, their minimal negation is true. An expression of that form is associated with a cactus structure called a lobe and is “painted” with the colors 
to mean all the propositions 
may be used for the logical operators.
is called an arc from point 
to point 
and a self‑loop 
is called a sling at 
and 
can be represented as digraphs on the six points of their common world set 
The arcs are given as follows.
to 
to 
to 
to 
and 
evaluate the signs in 
according to their own frames of reference.
and 
can be represented as digraphs on the four points of their common syntactic domain 
Since 
arcs in both directions between the points of 
and arcs in both directions between the points of 
and arcs in both directions between the points of 
are shown in the next two Figures.
is an equivalence relation on a set 
then every element ![[x]_E](https://s0.wp.com/latex.php?latex=%5Bx%5D_E&bg=ffffff&fg=000000&s=0&c=20201002)
or the simpler form ![[x]](https://s0.wp.com/latex.php?latex=%5Bx%5D&bg=ffffff&fg=000000&s=0&c=20201002)
when the subscript 
are equivalent under 
is a sign relation whose connotative component 
is an equivalence relation on 
let ![[s]_L](https://s0.wp.com/latex.php?latex=%5Bs%5D_L&bg=ffffff&fg=000000&s=0&c=20201002)
be the equivalence class of 
under 
In short, ![[s]_L = [s]_{L_{SI}}.](https://s0.wp.com/latex.php?latex=%5Bs%5D_L+%3D+%5Bs%5D_%7BL_%7BSI%7D%7D.&bg=ffffff&fg=000000&s=0&c=20201002)
A statement that the signs 
in a sign relation 
it is permissible to let ![[o]_L](https://s0.wp.com/latex.php?latex=%5Bo%5D_L&bg=ffffff&fg=000000&s=0&c=20201002)
be defined as ![[s]_L.](https://s0.wp.com/latex.php?latex=%5Bs%5D_L.&bg=ffffff&fg=000000&s=0&c=20201002)
This lets the notation for semiotic equivalence classes harmonize more smoothly with the frequent use of similar devices for the denotations of signs and expressions.
and 
will serve to illustrate their use and utility.
and 
have many interesting properties over and above those possessed by sign relations in general. Some of those properties have to do with the relation between signs and their interpretant signs, as reflected in the projections of 
‑plane, notated as 
and 
respectively. The dyadic relations on 
induced by those projections are also referred to as the connotative components of the corresponding sign relations, notated as 
respectively. Tables 6a and 6b show the corresponding connotative components.
This type of equivalence relation is called a semiotic equivalence relation (SER) because it equates signs having the same meaning to some interpreter.
partitions the collection of signs into semiotic equivalence classes. This constitutes a strong form of representation in that the structure of the interpreters’ common object domain 
is reflected or reconstructed, part for part, in the structure of each one’s semiotic partition of the syntactic domain 
Information about the ennotative aspect of meaning is obtained from 
and the interpretant domain 
The ennotative component of a sign relation 
or 
is defined as follows.
is echoed unchanged in 
and all the data of 
is echoed unchanged in 
respectively. The rows of each Table list the ordered pairs 
in the corresponding projections, 
Information about the connotative aspect of meaning is obtained from 
or 
is defined as follows.
in the corresponding projections, 
Information about the denotative aspect of meaning is obtained from 
or 
is defined as follows.
in the corresponding projections, 
various rows of the Tables specify, for example, that 
to denote 
to denote 
whether it happens to be a sign relation or not, there are six dyadic relations obtained by projecting 
‑space 
The six dyadic projections of a triadic relation 
‑plane 
is written briefly as 
or written more fully as 
and is defined as the set of all ordered pairs 
in the set 
are known as the object domain, the sign domain, and the interpretant domain, respectively, of the sign relation 
In those cases it becomes convenient to lump signs and interpretants together in a single class called a sign system or syntactic domain. In the forthcoming examples 
are identical as sets, so the same elements manifest themselves in two different roles of the sign relations in question.
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