User:Jon Awbrey/Figures and Tables 32

Syntax and Semantics of a Calculus for Propositional Logic

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

${\text{Syntax and Semantics of a Calculus for Propositional Logic}}$

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Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable $k$-ary scope.

${\text{Table 1. Syntax and Semantics of a Calculus for Propositional Logic}}$

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

Existential Interpretation

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${\text{Existential Interpretation}}$

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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.

${\text{Table A. Existential Interpretation}}$

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

Entitative Interpretation

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${\text{Entitative Interpretation}}$

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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.

${\text{Table B. Entitative Interpretation}}$

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

Logical Interpretations of Cactus Structures

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Logical Interpretations of Cactus Structures • En Ex

${\text{Logical Interpretations of Cactus Structures}}$

Logical Interpretations of Cactus Structures • Ex En

${\text{Logical Interpretations of Cactus Structures}}$

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Logical Interpretations of Cactus Structures • Columns En Ex

${\text{Logical Interpretations of Cactus Structures}}$

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

Logical Interpretations of Cactus Structures • Columns Ex En

${\text{Logical Interpretations of Cactus Structures}}$

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

Basic Connectives

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Node Connective

Lobe Connective

Basic Reductions

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Node Reduction

Lobe Reduction

Reduction Rules

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