Cf: Differential Logic ??? 2
At: http://inquiryintoinquiry.com/2020/03/23/differential-logic-%e2%80%a2-2/
Cactus Language for Propositional Logic
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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 (e_1, e_2,
..., e_k) and meaning exactly one of the propositions e_1, e_2, ..., 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, ..., e_k as shown below.
Lobe Connective
https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-ej-lobe-connective.jpg
The second kind of connective is a concatenated sequence of propositional expressions, written e_1 e_2 ... e_k and
meaning all of the propositions e_1, e_2, ..., 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, ..., e_k as
shown below.
Node Connective
https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-ej-node-connective.jpg
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 (...) may be
used for the logical operators.
Regards,
Jon
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