Cf: Differential Propositional Calculus : 2
At:
http://inquiryintoinquiry.com/2020/02/22/differential-propositional-calculus-%e2%80%a2-2/
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 (e_1, e_2, ..., e_k)
indicates exactly one of the propositions e_1, e_2, ..., e_k is false.
* A concatenation of propositional expressions in the form e_1 e_2 ... e_k
indicates all the propositions e_1, e_2, ..., e_k are true, in other words,
their logical conjunction ( https://oeis.org/wiki/Logical_conjunction ) is
true.
Table 6. Syntax and Semantics of a Calculus for Propositional Logic
https://inquiryintoinquiry.files.wordpress.com/2020/02/syntax-and-semantics-of-a-calculus-for-propositional-logic.png
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 ( ... ) or barred parentheses (| ... |) may be used for logical operators.
The briefest expression for logical truth is the empty word, abstractly denoted epsilon or 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 (( )), 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 (
https://oeis.org/wiki/Exclusive_disjunction ). For example, we have the following paraphrases of algebraic expressions:
x + y = (x, y)
x + y + z = ((x, y), z) = (x, (y, z))
It is important to note the last expressions are not equivalent to the triple
bracket (x, y, z).
More information about this syntax for propositional calculus can be found at
the following locations.
Resources
=========
* Minimal Negation Operators
( https://oeis.org/wiki/Minimal_negation_operator )
* Zeroth Order Logic
( https://oeis.org/wiki/Zeroth_order_logic )
* Table A1. Propositional Forms on Two Variables
(
https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Appendices#Table_A1._Propositional_Forms_on_Two_Variables
)
Regards,
Jon
inquiry into inquiry: https://inquiryintoinquiry.com/
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