Differential Logic • 2

Cactus Language for Propositional Logic

The development of differential logic is facilitated by having a moderately efficient calculus in place at the level of boolean-valued functions and elementary logical propositions.  One very efficient calculus on both conceptual and computational grounds is based on just two types of logical connectives, both of variable k-ary scope.  The syntactic formulas of this 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.

The first kind of connective takes the form of a parenthesized sequence of propositional expressions, written \texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)} and meaning exactly one of the propositions e_1, e_2, \ldots, e_{k-1}, e_k is false, in short, their minimal negation is true.  An expression of this form maps into a cactus structure called a lobe, in this case, “painted” with the colors e_1, e_2, \ldots, e_{k-1}, e_k as shown below.

Lobe Connective

The second kind of connective is a concatenated sequence of propositional expressions, written e_1\ e_2\ \ldots\ e_{k-1}\ e_k and meaning all the propositions e_1, e_2, \ldots, e_{k-1}, e_k are true, in short, their logical conjunction is true.  An expression of this form maps into a cactus structure called a node, in this case, “painted” with the colors e_1, e_2, \ldots, e_{k-1}, e_k as shown below.

Node Connective

All other propositional connectives can be obtained through combinations of these two forms.  As it happens, the parenthesized form is sufficient to define the concatenated form, making the latter formally dispensable, but it’s convenient to maintain it as a concise way of expressing more complicated combinations of parenthesized forms.  While working with expressions solely in propositional calculus, it’s easiest to use plain parentheses for logical connectives.  In contexts where ordinary parentheses are needed for other purposes an alternate typeface \texttt{(} \ldots \texttt{)} may be used for the logical operators.

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This entry was posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Change, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamics, Equational Inference, Frankl Conjecture, Functional Logic, Gradient Descent, Graph Theory, Hologrammautomaton, Indicator Functions, Inquiry Driven Systems, Leibniz, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Surveys, Time, Topology, Visualization, Zeroth Order Logic and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

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