ONT Re: Propositional Equation Reasoning Systems
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PERS. Note 6
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Let us now extend the CSP-GSB calculus in the following way:
The first extension is the "reflective extension of logical graphs" (RefLog).
It is generated by generalizing the negation operator "(_)" in a certain way,
naming "(_)" the "controlled", "moderated", or "reflective" negation operator
of order 1, and adding another such operator for each integer greater than 1.
In sum, these operators are symbolized by bracketed argument lists as follows:
"(_)", "(_,_)", "(_,_,_)", ..., where the number of slots is the order of the
reflective negation operator in question.
The formal rule of evaluation for a k-operator is:
o-----------------------------------------------------------o
| Evaluation Rule |
o-----------------------------------------------------------o
| |
| x_1 x_2 ... x_k |
| o----o-...-o----o |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| @ = @ |
| |
o-----------------------------------------------------------o
| |
| ( x_1, x_2, ..., x_k ) = <space> |
| |
o-----------------------------------------------------------o
| |
| iff |
| o |
| Just one of the x_1, x_2, ..., x_k = | = () |
| @ |
| |
o-----------------------------------------------------------o
The interpretation of these operators, read as assertions
about the values of their listed arguments, is as follows:
o-----------------------------------------------------------o
| Interpretation Rule |
o-----------------------------------------------------------o
| |
| x_1 x_2 ... x_k |
| o----o-...-o----o |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| @ |
| |
| A "lobe operator" of the form "( x_1, ..., x_k )" |
| enjoys two commonly employed interpretations for |
| propositional logic, that is, ways of reading it |
| as an assertion about, or a constraint upon, the |
| logical values of the ordered arguments, that is, |
| the mentioned variables x_j, for j = 1 through k. |
| |
| Existential Interpretation: |
| |
| "Just one of the k arguments is not true." |
| |
| Entitative Interpretation: |
| |
| "Not just one of the k arguments is true." |
| |
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Jon Awbrey
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