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$\ldots$

\section{Cactus calculus}

\textbf{Temporary Note.} The remainder of this discussion uses the syntax for propositional calculus that is described in the entry on minimal negation operators.  Logical negation is written by enclosing an expression in parentheses, for example, $(x)$ is $\lnot x.$  Logical conjunction is written by concatenating expressions in the manner of algebraic products, for example, $x\ y\ z$ is $x \land y \land z.$ See Table A1 in \PMlinkname{Appendix 1}{DifferentialPropositionalCalculusAppendices} for equivalent expressions in this syntax and several others for the 16 propositional forms on two variables.

\begin{itemize}

A bracketed list of propositional expressions in the form $(e_1, e_2, \ldots, e_{k-1}, e_k)$ indicates that exactly one of the propositions $e_1, e_2, \ldots, e_{k-1}, e_k$ is false.

A concatenation of propositional expressions in the form $e_1\ e_2\ \ldots\ e_{k-1}\ e_k$ indicates that all of the propositions $e_1, e_2, \ldots, e_{k-1}, e_k$ are true, in other words, that their logical conjunction is true.

 

\multicolumn{3}{c}{\textbf{Table 4.  Syntax and Semantics of a Propositional Calculus}} \\[8pt]

x\ \operatorname{is~equal~to}\ y    \\

x, y, z                    \\

$\begin{matrix}

$((x),(y),(z))$         &

\operatorname{is~true}.      \\

$\begin{matrix}

\operatorname{species}\ x, y, z. \\

$\begin{matrix}

 

All other propositional connectives can be obtained through combinations of these two forms. Strictly speaking, the concatenation form is dispensable in light of the bracket form, but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.  The briefest expression for logical truth is the empty word, abstractly denoted $\varepsilon$ or $\lambda$ in formal languages, where it forms the identity element for concatenation.  It can be given visible expression in this context by means of the logically equivalent expression $``((~))",$ or, especially if operating in an algebraic context, by a simple $``1".$  Also when working in an algebraic mode, the plus sign $``+"$ may be used for exclusive disjunction.  For example, we have the following paraphrases of algebraic expressions by bracket expressions:

 

 

$$x + y + z \quad = \quad ((x, y), z) \quad = \quad (x, (y, z))$$

 

It is important to note that the last expressions are not equivalent to the triple bracket $(x, y, z).$

 

For more information about this syntax for propositional calculus, see the entries on \PMlinkname{minimal negation operators}{MinimalNegationOperator}, \PMlinkname{zeroth order logic}{ZerothOrderLogic}, and Table A1 in \PMlinkname{Appendix 1}{DifferentialPropositionalCalculusAppendices}.

\section{Formal development}

\section{Formal development}

A set of logical features, $\mathcal{A} = \{ a_1, \ldots, a_n \},$ affords a basis for generating an $n$-dimensional universe of discourse, written $A^\circ = [ \mathcal{A} ] = [ a_1, \ldots, a_n ].$  It is useful to consider a universe of discourse as a \PMlinkname{categorical}{Category} object that incorporates both the set of points $A = \langle a_1, \ldots, a_n \rangle$ and the set of propositions $A^\uparrow = \{ f : A \to \mathbb{B} \}$ that are implicit with the ordinary picture of a venn diagram on $n$ features.  Accordingly, the universe of discourse $A^\circ$ may be regarded as an ordered pair $(A, A^\uparrow)$ having the type $(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),$ and this last type designation may be abbreviated as $\mathbb{B}^n\ +\!\to \mathbb{B},$ or even more succinctly as $[ \mathbb{B}^n ].$  For convenience, the data type of a finite set on $n$ elements may be indicated by either one of the equivalent notations, $[n]$ or $\mathbf{n}.$

A set of logical features, $\mathcal{A} = \{ a_1, \ldots, a_n \},$ affords a basis for generating an $n$-dimensional universe of discourse, written $A^\circ = [ \mathcal{A} ] = [ a_1, \ldots, a_n ].$  It is useful to consider a universe of discourse as a \PMlinkname{categorical}{Category} object that incorporates both the set of points $A = \langle a_1, \ldots, a_n \rangle$ and the set of propositions $A^\uparrow = \{ f : A \to \mathbb{B} \}$ that are implicit with the ordinary picture of a venn diagram on $n$ features.  Accordingly, the universe of discourse $A^\circ$ may be regarded as an ordered pair $(A, A^\uparrow)$ having the type $(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),$ and this last type designation may be abbreviated as $\mathbb{B}^n\ +\!\to \mathbb{B},$ or even more succinctly as $[ \mathbb{B}^n ].$  For convenience, the data type of a finite set on $n$ elements may be indicated by either one of the equivalent notations, $[n]$ or $\mathbf{n}.$

Table 4 summarizes the notations that are needed to describe ordinary propositional calculi in a systematic fashion.

Table 5 summarizes the notations that are needed to describe ordinary propositional calculi in a systematic fashion.

\begin{center}\begin{tabular}{|l|l|l|l|}

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\multicolumn{4}{c}{\textbf{Table 4.  Propositional Calculus : Basic Notation}} \\[8pt]

\multicolumn{4}{c}{\textbf{Table 5.  Propositional Calculus : Basic Notation}} \\[8pt]

\hline

\hline

A proposition in a differential extension of a universe of discourse is called a \textit{differential proposition} and forms the analogue of a system of differential equations in \PMlinkname{ordinary calculus}{Calculus}.  With these constructions, the first order extended universe $\operatorname{E}A^\circ$ and the first order differential proposition $f : \operatorname{E}A \to \mathbb{B},$ we have arrived, in concept at least, at the foothills of differential logic.

A proposition in a differential extension of a universe of discourse is called a \textit{differential proposition} and forms the analogue of a system of differential equations in \PMlinkname{ordinary calculus}{Calculus}.  With these constructions, the first order extended universe $\operatorname{E}A^\circ$ and the first order differential proposition $f : \operatorname{E}A \to \mathbb{B},$ we have arrived, in concept at least, at the foothills of differential logic.

Table 5 summarizes the notations that are needed to describe the first order differential extensions of propositional calculi in a systematic manner.

Table 6 summarizes the notations that are needed to describe the first order differential extensions of propositional calculi in a systematic manner.

\begin{center}\begin{tabular}{|l|l|l|l|}

\begin{center}\begin{tabular}{|l|l|l|l|}

\multicolumn{4}{c}{\textbf{Table 5.  Differential Extension : Basic Notation}} \\[8pt]

\multicolumn{4}{c}{\textbf{Table 6.  Differential Extension : Basic Notation}} \\[8pt]

\hline

\hline