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# User:Jon Awbrey/Theme One Program • Work Area

## Work Area

### Basic Connectives

• The node connective joins a number of component cacti ${\displaystyle C_{1},\ldots ,C_{k}}$ to a node:
• The lobe connective joins a number of component cacti ${\displaystyle C_{1},\ldots ,C_{k}}$ to a lobe:

### Basic Reductions

• A node reduction is permitted if and only if every component cactus joined to a node itself reduces to a node.
• A lobe reduction is permitted if and only if exactly one component cactus listed in a lobe reduces to an edge.

### Reduction Rules

The easy way to visualize the values of these graphical expressions is just to notice the following equivalents:

### Syntax and Semantics of a Calculus for Propositional Logic

Table 1 collects a sample of basic propositional forms as expressed in terms of cactus language connectives.

 ${\displaystyle {\text{Graph}}}$ ${\displaystyle {\text{Expression}}}$ ${\displaystyle {\text{Interpretation}}}$ ${\displaystyle {\text{Other Notations}}}$ ${\displaystyle ~}$ ${\displaystyle \mathrm {true} }$ ${\displaystyle 1}$ ${\displaystyle {\texttt {(}}~{\texttt {)}}}$ ${\displaystyle \mathrm {false} }$ ${\displaystyle 0}$ ${\displaystyle a}$ ${\displaystyle a}$ ${\displaystyle a}$ ${\displaystyle {\texttt {(}}a{\texttt {)}}}$ ${\displaystyle \mathrm {not} ~a}$ ${\displaystyle \lnot a\quad {\bar {a}}\quad {\tilde {a}}\quad a^{\prime }}$ ${\displaystyle a~b~c}$ ${\displaystyle a~\mathrm {and} ~b~\mathrm {and} ~c}$ ${\displaystyle a\land b\land c}$ ${\displaystyle {\texttt {((}}a{\texttt {)(}}b{\texttt {)(}}c{\texttt {))}}}$ ${\displaystyle a~\mathrm {or} ~b~\mathrm {or} ~c}$ ${\displaystyle a\lor b\lor c}$ ${\displaystyle {\texttt {(}}a{\texttt {(}}b{\texttt {))}}}$ ${\displaystyle {\begin{matrix}a~\mathrm {implies} ~b\\[6pt]\mathrm {if} ~a~\mathrm {then} ~b\end{matrix}}}$ ${\displaystyle a\Rightarrow b}$ ${\displaystyle {\texttt {(}}a{\texttt {,}}b{\texttt {)}}}$ ${\displaystyle {\begin{matrix}a~\mathrm {not~equal~to} ~b\\[6pt]a~\mathrm {exclusive~or} ~b\end{matrix}}}$ ${\displaystyle {\begin{matrix}a\neq b\\[6pt]a+b\end{matrix}}}$ ${\displaystyle {\texttt {((}}a{\texttt {,}}b{\texttt {))}}}$ ${\displaystyle {\begin{matrix}a~\mathrm {is~equal~to} ~b\\[6pt]a~\mathrm {if~and~only~if} ~b\end{matrix}}}$ ${\displaystyle {\begin{matrix}a=b\\[6pt]a\Leftrightarrow b\end{matrix}}}$ ${\displaystyle {\texttt {(}}a{\texttt {,}}b{\texttt {,}}c{\texttt {)}}}$ ${\displaystyle {\begin{matrix}\mathrm {just~one~of} \\a,b,c\\\mathrm {is~false} .\end{matrix}}}$ ${\displaystyle {\begin{matrix}&{\bar {a}}~b~c\\\lor &a~{\bar {b}}~c\\\lor &a~b~{\bar {c}}\end{matrix}}}$ ${\displaystyle {\texttt {((}}a{\texttt {),(}}b{\texttt {),(}}c{\texttt {))}}}$ ${\displaystyle {\begin{matrix}\mathrm {just~one~of} \\a,b,c\\\mathrm {is~true} .\\[6pt]\mathrm {partition~all} \\\mathrm {into} ~a,b,c.\end{matrix}}}$ ${\displaystyle {\begin{matrix}&a~{\bar {b}}~{\bar {c}}\\\lor &{\bar {a}}~b~{\bar {c}}\\\lor &{\bar {a}}~{\bar {b}}~c\end{matrix}}}$ ${\displaystyle {\texttt {(}}a{\texttt {,(}}b{\texttt {,}}c{\texttt {))}}}$ ${\displaystyle {\begin{matrix}\mathrm {oddly~many~of} \\a,b,c\\\mathrm {are~true} .\end{matrix}}}$ ${\displaystyle a+b+c}$ ${\displaystyle {\begin{matrix}&a~b~c\\\lor &a~{\bar {b}}~{\bar {c}}\\\lor &{\bar {a}}~b~{\bar {c}}\\\lor &{\bar {a}}~{\bar {b}}~c\end{matrix}}}$ ${\displaystyle {\texttt {(}}x{\texttt {,(}}a{\texttt {),(}}b{\texttt {),(}}c{\texttt {))}}}$ ${\displaystyle {\begin{matrix}\mathrm {partition} ~x\\\mathrm {into} ~a,b,c.\\[6pt]\mathrm {genus} ~x~\mathrm {comprises} \\\mathrm {species} ~a,b,c.\end{matrix}}}$ ${\displaystyle {\begin{matrix}&{\bar {x}}~{\bar {a}}~{\bar {b}}~{\bar {c}}\\\lor &x~a~{\bar {b}}~{\bar {c}}\\\lor &x~{\bar {a}}~b~{\bar {c}}\\\lor &x~{\bar {a}}~{\bar {b}}~c\end{matrix}}}$

### Existential Interpretation

Table A illustrates the existential interpretation of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.

 ${\displaystyle {\text{Cactus Graph}}}$ ${\displaystyle {\text{Cactus Expression}}}$ ${\displaystyle {\text{Interpretation}}}$ ${\displaystyle \mathrm {~} }$ ${\displaystyle \mathrm {true} }$ ${\displaystyle {\texttt {(}}~{\texttt {)}}}$ ${\displaystyle \mathrm {false} }$ ${\displaystyle a}$ ${\displaystyle a}$ ${\displaystyle {\texttt {(}}a{\texttt {)}}}$ ${\displaystyle {\begin{matrix}{\tilde {a}}\\[2pt]a^{\prime }\\[2pt]\lnot a\\[2pt]\mathrm {not} ~a\end{matrix}}}$ ${\displaystyle a~b~c}$ ${\displaystyle {\begin{matrix}a\land b\land c\\[6pt]a~\mathrm {and} ~b~\mathrm {and} ~c\end{matrix}}}$ ${\displaystyle {\texttt {((}}a{\texttt {)(}}b{\texttt {)(}}c{\texttt {))}}}$ ${\displaystyle {\begin{matrix}a\lor b\lor c\\[6pt]a~\mathrm {or} ~b~\mathrm {or} ~c\end{matrix}}}$ ${\displaystyle {\texttt {(}}a{\texttt {(}}b{\texttt {))}}}$ ${\displaystyle {\begin{matrix}a\Rightarrow b\\[2pt]a~\mathrm {implies} ~b\\[2pt]\mathrm {if} ~a~\mathrm {then} ~b\\[2pt]\mathrm {not} ~a~\mathrm {without} ~b\end{matrix}}}$ ${\displaystyle {\texttt {(}}a,b{\texttt {)}}}$ ${\displaystyle {\begin{matrix}a+b\\[2pt]a\neq b\\[2pt]a~\mathrm {exclusive~or} ~b\\[2pt]a~\mathrm {not~equal~to} ~b\end{matrix}}}$ ${\displaystyle {\texttt {((}}a,b{\texttt {))}}}$ ${\displaystyle {\begin{matrix}a=b\\[2pt]a\iff b\\[2pt]a~\mathrm {equals} ~b\\[2pt]a~\mathrm {if~and~only~if} ~b\end{matrix}}}$ ${\displaystyle {\texttt {(}}a,b,c{\texttt {)}}}$ ${\displaystyle {\begin{matrix}\mathrm {just~one~of} \\a,b,c\\\mathrm {is~false} \end{matrix}}}$ ${\displaystyle {\texttt {((}}a{\texttt {)}},{\texttt {(}}b{\texttt {)}},{\texttt {(}}c{\texttt {))}}}$ ${\displaystyle {\begin{matrix}\mathrm {just~one~of} \\a,b,c\\\mathrm {is~true} \end{matrix}}}$ ${\displaystyle {\texttt {(}}a,{\texttt {(}}b{\texttt {)}},{\texttt {(}}c{\texttt {))}}}$ ${\displaystyle {\begin{matrix}\mathrm {genus} ~a~\mathrm {of~species} ~b,c\\[6pt]\mathrm {partition} ~a~\mathrm {into} ~b,c\\[6pt]\mathrm {pie} ~a~\mathrm {of~slices} ~b,c\end{matrix}}}$

### Entitative Interpretation

Table B illustrates the entitative interpretation of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.

 ${\displaystyle {\text{Cactus Graph}}}$ ${\displaystyle {\text{Cactus Expression}}}$ ${\displaystyle {\text{Interpretation}}}$ ${\displaystyle \mathrm {~} }$ ${\displaystyle \mathrm {false} }$ ${\displaystyle {\texttt {(}}~{\texttt {)}}}$ ${\displaystyle \mathrm {true} }$ ${\displaystyle a}$ ${\displaystyle a}$ ${\displaystyle {\texttt {(}}a{\texttt {)}}}$ ${\displaystyle {\begin{matrix}{\tilde {a}}\\[2pt]a^{\prime }\\[2pt]\lnot a\\[2pt]\mathrm {not} ~a\end{matrix}}}$ ${\displaystyle a~b~c}$ ${\displaystyle {\begin{matrix}a\lor b\lor c\\[6pt]a~\mathrm {or} ~b~\mathrm {or} ~c\end{matrix}}}$ ${\displaystyle {\texttt {((}}a{\texttt {)(}}b{\texttt {)(}}c{\texttt {))}}}$ ${\displaystyle {\begin{matrix}a\land b\land c\\[6pt]a~\mathrm {and} ~b~\mathrm {and} ~c\end{matrix}}}$ ${\displaystyle {\texttt {(}}a{\texttt {)}}b}$ ${\displaystyle {\begin{matrix}a\Rightarrow b\\[2pt]a~\mathrm {implies} ~b\\[2pt]\mathrm {if} ~a~\mathrm {then} ~b\\[2pt]\mathrm {not} ~a,~\mathrm {or} ~b\end{matrix}}}$ ${\displaystyle {\texttt {(}}a,b{\texttt {)}}}$ ${\displaystyle {\begin{matrix}a=b\\[2pt]a\iff b\\[2pt]a~\mathrm {equals} ~b\\[2pt]a~\mathrm {if~and~only~if} ~b\end{matrix}}}$ ${\displaystyle {\texttt {((}}a,b{\texttt {))}}}$ ${\displaystyle {\begin{matrix}a+b\\[2pt]a\neq b\\[2pt]a~\mathrm {exclusive~or} ~b\\[2pt]a~\mathrm {not~equal~to} ~b\end{matrix}}}$ ${\displaystyle {\texttt {(}}a,b,c{\texttt {)}}}$ ${\displaystyle {\begin{matrix}\mathrm {not~just~one~of} \\a,b,c\\\mathrm {is~true} \end{matrix}}}$ ${\displaystyle {\texttt {((}}a,b,c{\texttt {))}}}$ ${\displaystyle {\begin{matrix}\mathrm {just~one~of} \\a,b,c\\\mathrm {is~true} \end{matrix}}}$ ${\displaystyle {\texttt {(((}}a{\texttt {)}},b,c{\texttt {))}}}$ ${\displaystyle {\begin{matrix}\mathrm {genus} ~a~\mathrm {of~species} ~b,c\\[6pt]\mathrm {partition} ~a~\mathrm {into} ~b,c\\[6pt]\mathrm {pie} ~a~\mathrm {of~slices} ~b,c\end{matrix}}}$

### Existential and Entitative Interpretations of Cactus Structures

 ${\displaystyle {\text{Cactus Graph}}}$ ${\displaystyle {\text{Cactus Expression}}}$ ${\displaystyle {\text{Existential}}}$${\displaystyle {\text{Interpretation}}}$ ${\displaystyle {\text{Entitative}}}$${\displaystyle {\text{Interpretation}}}$ ${\displaystyle \mathrm {true} }$ ${\displaystyle \mathrm {false} }$ ${\displaystyle {\texttt {(}}~{\texttt {)}}}$ ${\displaystyle \mathrm {false} }$ ${\displaystyle \mathrm {true} }$ ${\displaystyle C_{1}\,C_{2}\,\ldots \,C_{k-1}\,C_{k}}$ ${\displaystyle C_{1}\land C_{2}\land \ldots \land C_{k-1}\land C_{k}}$ ${\displaystyle C_{1}\lor C_{2}\lor \ldots \lor C_{k-1}\lor C_{k}}$ ${\displaystyle {\texttt {(}}C_{1}{\texttt {,}}C_{2}{\texttt {,}}\ldots {\texttt {,}}C_{k-1}{\texttt {,}}C_{k}{\texttt {)}}}$ ${\displaystyle {\begin{matrix}{\text{just one of}}\\[6px]C_{1},C_{2},\ldots ,C_{k-1},C_{k}\\[6px]{\text{is false}}\end{matrix}}}$ ${\displaystyle {\begin{matrix}{\text{not just one of}}\\[6px]C_{1},C_{2},\ldots ,C_{k-1},C_{k}\\[6px]{\text{is true}}\end{matrix}}}$