Differential Propositional Calculus • 2

Cactus Calculus

Table 6 outlines a syntax for propositional calculus based on two types of logical connectives, both of variable k-ary scope.

  • A bracketed list of propositional expressions in the form \texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)} indicates 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 all the propositions e_1, e_2, \ldots, e_{k-1}, e_k are true, in other words, their logical conjunction is true.

Table 6.  Syntax and Semantics of a Calculus for Propositional Logic

Syntax and Semantics of a Calculus for Propositional Logic

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 for more complicated bracket expressions.  While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives.  In contexts where parentheses are needed for other purposes “teletype” parentheses \texttt{(} \ldots \texttt{)} or barred parentheses (\!| \ldots |\!) may be used for logical operators.

The briefest expression for logical truth is the empty word, abstractly denoted \boldsymbol\varepsilon or \boldsymbol\lambda in formal languages, where it forms the identity element for concatenation.  It may be given visible expression in this context by means of the logically equivalent form \texttt{((} ~ \texttt{))}, 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:

\begin{matrix}  x + y ~=~ \texttt{(} x \texttt{,} y \texttt{)}  \\[6pt]  x + y + z ~=~ \texttt{((} x \texttt{,} y \texttt{),} z \texttt{)} ~=~ \texttt{(} x \texttt{,(} y \texttt{,} z \texttt{))}  \end{matrix}

It is important to note the last expressions are not equivalent to the triple bracket \texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}.

More information about this syntax for propositional calculus can be found at the following locations.

Resources

cc: CyberneticsOntolog ForumPeirce ListStructural ModelingSystems Science

This entry was posted in Amphecks, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Change, Computational Complexity, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamics, Dynamical Systems, Equational Inference, Functional Logic, Gradient Descent, Graph Theory, Group Theory, Hologrammautomaton, Indicator Functions, Logic, Logical Graphs, Mathematical Models, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Propositional Equation Reasoning Systems, Time, Visualization and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

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