Talk:Futures Of Logical Graphs

Futures Of Logical Graphs • Document History

2005 • Inquiry List • Futures Of Logical Graphs • Exposition

2005 • Inquiry List • Futures Of Logical Graphs • Discussion

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The suggested perspective on algebraic expressions allows us to understand the algebraic law as saying that every one of the arithmetic expressions fitting the indicated algebraic pattern evaluates to the indicated arithmetic result.

Interpretation Rules

Version 1

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 "k-lobe operator" of the form "(x_1, ..., x_k)"         |
| enjoys two commonly employed interpretations for          |
| propositional logic, in other words, two ways of          |
| taking it as an assertion about, or a constraint          |
| upon, the logical values of the listed arguments,         |
| the mentioned variables x_j, for j = 1 through k.         |
|                                                           |
| Entitative  Interpretation:                               |
|                                                           |
|    "Not just one of the k arguments is true."             |
|                                                           |
| Existential Interpretation:                               |
|                                                           |
|    "Just one of the k arguments is not true."             |
|                                                           |
o-----------------------------------------------------------o

Version 2

The two ways of mapping cactus structures to logical meanings are summarized in the following Table, which compares the existential and entitative interpretations of the basic cactus structures.

${\text{Logical Interpretations of Cactus Structures}}$
${\text{Expression}}$	${\begin{matrix}{\text{Existential}}\\{\text{Interpretation}}\end{matrix}}$	${\begin{matrix}{\text{Entitative}}\\{\text{Interpretation}}\end{matrix}}$
$~$	$\mathrm {true}$	$\mathrm {false}$
${\texttt {(}}~{\texttt {)}}$	$\mathrm {false}$	$\mathrm {true}$
$x_{1}\ldots x_{k}$	$x_{1}\land \ldots \land x_{k}$	$x_{1}\lor \ldots \lor x_{k}$
${\texttt {(}}x_{1}{\texttt {,}}\ldots {\texttt {,}}x_{k}{\texttt {)}}$	${\begin{matrix}{\text{just one of}}\\[6px]x_{1},\ldots ,x_{k}\\[6px]{\text{is false}}\end{matrix}}$	${\begin{matrix}{\text{not just one of}}\\[6px]x_{1},\ldots ,x_{k}\\[6px]{\text{is true}}\end{matrix}}$

Version 3

The following Table summarizes the entitative and existential interpretations of the basic cactus structures,.

${\text{Logical Interpretations of Cactus Structures}}$
${\text{Cactus Graph}}$	${\text{Cactus Expression}}$	${\text{Entitative}}$ ${\text{Interpretation}}$	${\text{Existential}}$ ${\text{Interpretation}}$
	$~$	$\mathrm {false}$	$\mathrm {true}$
	${\texttt {(}}~{\texttt {)}}$	$\mathrm {true}$	$\mathrm {false}$
	$x_{1}\,\ldots \,x_{k}$	$x_{1}\lor \ldots \lor x_{k}$	$x_{1}\land \ldots \land x_{k}$
	${\texttt {(}}x_{1}{\texttt {,}}\ldots {\texttt {,}}x_{k}{\texttt {)}}$	${\begin{matrix}{\text{not just one of}}\\[6px]x_{1},\ldots ,x_{k}\\[6px]{\text{is true}}\end{matrix}}$	${\begin{matrix}{\text{just one of}}\\[6px]x_{1},\ldots ,x_{k}\\[6px]{\text{is false}}\end{matrix}}$

The abstract character of the cactus language relative to its logical interpretations makes it possible to give abstract rules of equivalence for transforming one cactus into another that partition the space of cacti into formal equivalence classes. These transformation rules and the resulting equivalence classes are “purely formal” in the sense of being indifferent to the logical interpretation, entitative or existential, one happens to choose.

Talk:Futures Of Logical Graphs

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Futures Of Logical Graphs • Document History

2005 • Inquiry List • Futures Of Logical Graphs • Exposition

2005 • Inquiry List • Futures Of Logical Graphs • Discussion

Work Area

Fragment

Interpretation Rules

Version 1

Version 2

Version 3

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