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Syntax and Semantics of a Calculus for Propositional Logic

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Differential Logic • Variables {x, a, b, c}

Table 1 shows the cactus graphs, the corresponding cactus expressions, their logical meanings under the so-called existential interpretation, and their translations into conventional notations for a sample of basic propositional forms.

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

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

Logical Graphs • Variables {x, a, b, c}

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

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

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Differential Propositional Calculus • Variables {w, x, y, z}

${\text{Table 6.}}~~{\text{Syntax and Semantics of a Calculus for Propositional Logic}}$

${\text{Expression}}$	${\text{Interpretation}}$	${\text{Other Notations}}$
	${\text{True}}$	$1$
${\texttt {(}}~{\texttt {)}}$	${\text{False}}$	$0$
$x$	$x$	$x$
${\texttt {(}}x{\texttt {)}}$	${\text{Not}}~x$	${\begin{matrix}x'\\{\tilde {x}}\\\lnot x\end{matrix}}$
$x~y~z$	$x~{\text{and}}~y~{\text{and}}~z$	$x\land y\land z$
${\texttt {((}}x{\texttt {)(}}y{\texttt {)(}}z{\texttt {))}}$	$x~{\text{or}}~y~{\text{or}}~z$	$x\lor y\lor z$
${\texttt {(}}x~{\texttt {(}}y{\texttt {))}}$	${\begin{matrix}x~{\text{implies}}~y\\\mathrm {If} ~x~{\text{then}}~y\end{matrix}}$	$x\Rightarrow y$
${\texttt {(}}x{\texttt {,}}y{\texttt {)}}$	${\begin{matrix}x~{\text{not equal to}}~y\\x~{\text{exclusive or}}~y\end{matrix}}$	${\begin{matrix}x\neq y\\x+y\end{matrix}}$
${\texttt {((}}x{\texttt {,}}y{\texttt {))}}$	${\begin{matrix}x~{\text{is equal to}}~y\\x~{\text{if and only if}}~y\end{matrix}}$	${\begin{matrix}x=y\\x\Leftrightarrow y\end{matrix}}$
${\texttt {(}}x{\texttt {,}}y{\texttt {,}}z{\texttt {)}}$	${\begin{matrix}{\text{Just one of}}\\x,y,z\\{\text{is false}}.\end{matrix}}$	${\begin{array}{l}x'y~z~~~\lor \\x~y'z~~~\lor \\x~y~z'\end{array}}$
${\texttt {((}}x{\texttt {),(}}y{\texttt {),(}}z{\texttt {))}}$	${\begin{matrix}{\text{Just one of}}\\x,y,z\\{\text{is true}}.\\[5pt]{\text{Partition all}}\\{\text{into}}~x,y,z.\end{matrix}}$	${\begin{array}{l}x~y'z'~~\lor \\x'y~z'~~\lor \\x'y'z\end{array}}$
${\begin{matrix}{\texttt {((}}x{\texttt {,}}y{\texttt {),}}z{\texttt {)}}\\&\\{\texttt {(}}x{\texttt {,(}}y{\texttt {,}}z{\texttt {))}}\end{matrix}}$	${\begin{matrix}{\text{Oddly many of}}\\x,y,z\\{\text{are true}}.\end{matrix}}$	${\begin{array}{l}x+y+z\\[5pt]x~y~z~~~\lor \\x~y'z'~~\lor \\x'y~z'~~\lor \\x'y'z\end{array}}$
${\texttt {(}}w{\texttt {,(}}x{\texttt {),(}}y{\texttt {),(}}z{\texttt {))}}$	${\begin{matrix}{\text{Partition}}~w\\{\text{into}}~x,y,z.\\[5pt]{\text{Genus}}~w~{\text{comprises}}\\{\text{species}}~x,y,z.\end{matrix}}$	${\begin{array}{l}w'x'y'z'~~\lor \\w~x~y'z'~~\lor \\w~x'y~z'~~\lor \\w~x'y'z\end{array}}$

Differential Logic and Dynamic Systems • Variables {w, x, y, z}

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

${\text{Expression}}$	${\text{Interpretation}}$	${\text{Other Notations}}$
	${\text{True}}$	$1$
${\texttt {(}}~{\texttt {)}}$	${\text{False}}$	$0$
$x$	$x$	$x$
${\texttt {(}}x{\texttt {)}}$	${\text{Not}}~x$	${\begin{matrix}x'\\{\tilde {x}}\\\lnot x\end{matrix}}$
$x~y~z$	$x~{\text{and}}~y~{\text{and}}~z$	$x\land y\land z$
${\texttt {((}}x{\texttt {)(}}y{\texttt {)(}}z{\texttt {))}}$	$x~{\text{or}}~y~{\text{or}}~z$	$x\lor y\lor z$
${\texttt {(}}x~{\texttt {(}}y{\texttt {))}}$	${\begin{matrix}x~{\text{implies}}~y\\\mathrm {If} ~x~{\text{then}}~y\end{matrix}}$	$x\Rightarrow y$
${\texttt {(}}x{\texttt {,}}y{\texttt {)}}$	${\begin{matrix}x~{\text{not equal to}}~y\\x~{\text{exclusive or}}~y\end{matrix}}$	${\begin{matrix}x\neq y\\x+y\end{matrix}}$
${\texttt {((}}x{\texttt {,}}y{\texttt {))}}$	${\begin{matrix}x~{\text{is equal to}}~y\\x~{\text{if and only if}}~y\end{matrix}}$	${\begin{matrix}x=y\\x\Leftrightarrow y\end{matrix}}$
${\texttt {(}}x{\texttt {,}}y{\texttt {,}}z{\texttt {)}}$	${\begin{matrix}{\text{Just one of}}\\x,y,z\\{\text{is false}}.\end{matrix}}$	${\begin{matrix}x'y~z~&\lor \\x~y'z~&\lor \\x~y~z'&\end{matrix}}$
${\texttt {((}}x{\texttt {),(}}y{\texttt {),(}}z{\texttt {))}}$	${\begin{matrix}{\text{Just one of}}\\x,y,z\\{\text{is true}}.\\&\\{\text{Partition all}}\\{\text{into}}~x,y,z.\end{matrix}}$	${\begin{matrix}x~y'z'&\lor \\x'y~z'&\lor \\x'y'z~&\end{matrix}}$
${\begin{matrix}{\texttt {((}}x{\texttt {,}}y{\texttt {),}}z{\texttt {)}}\\&\\{\texttt {(}}x{\texttt {,(}}y{\texttt {,}}z{\texttt {))}}\end{matrix}}$	${\begin{matrix}{\text{Oddly many of}}\\x,y,z\\{\text{are true}}.\end{matrix}}$	$x+y+z$ ${\begin{matrix}x~y~z~&\lor \\x~y'z'&\lor \\x'y~z'&\lor \\x'y'z~&\end{matrix}}$
${\texttt {(}}w{\texttt {,(}}x{\texttt {),(}}y{\texttt {),(}}z{\texttt {))}}$	${\begin{matrix}{\text{Partition}}~w\\{\text{into}}~x,y,z.\\&\\{\text{Genus}}~w~{\text{comprises}}\\{\text{species}}~x,y,z.\end{matrix}}$	${\begin{matrix}w'x'y'z'&\lor \\w~x~y'z'&\lor \\w~x'y~z'&\lor \\w~x'y'z~&\end{matrix}}$