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The easy way to visualize the values of these graphical expressions is just to notice the following equivalents:
Table 1 collects a sample of basic propositional forms as expressed in terms of cactus language connectives.
a i m p l i e s b i f a t h e n b {\displaystyle {\begin{matrix}a~\mathrm {implies} ~b\\[6pt]\mathrm {if} ~a~\mathrm {then} ~b\end{matrix}}}
a n o t e q u a l t o b a e x c l u s i v e o r b {\displaystyle {\begin{matrix}a~\mathrm {not~equal~to} ~b\\[6pt]a~\mathrm {exclusive~or} ~b\end{matrix}}}
a ≠ b a + b {\displaystyle {\begin{matrix}a\neq b\\[6pt]a+b\end{matrix}}}
a i s e q u a l t o b a i f a n d o n l y i f b {\displaystyle {\begin{matrix}a~\mathrm {is~equal~to} ~b\\[6pt]a~\mathrm {if~and~only~if} ~b\end{matrix}}}
a = b a ⇔ b {\displaystyle {\begin{matrix}a=b\\[6pt]a\Leftrightarrow b\end{matrix}}}
j u s t o n e o f a , b , c i s f a l s e . {\displaystyle {\begin{matrix}\mathrm {just~one~of} \\a,b,c\\\mathrm {is~false} .\end{matrix}}}
a ¯ b c ∨ a b ¯ c ∨ a b c ¯ {\displaystyle {\begin{matrix}&{\bar {a}}~b~c\\\lor &a~{\bar {b}}~c\\\lor &a~b~{\bar {c}}\end{matrix}}}
j u s t o n e o f a , b , c i s t r u e . p a r t i t i o n a l l i n t o a , b , c . {\displaystyle {\begin{matrix}\mathrm {just~one~of} \\a,b,c\\\mathrm {is~true} .\\[6pt]\mathrm {partition~all} \\\mathrm {into} ~a,b,c.\end{matrix}}}
a b ¯ c ¯ ∨ a ¯ b c ¯ ∨ a ¯ b ¯ c {\displaystyle {\begin{matrix}&a~{\bar {b}}~{\bar {c}}\\\lor &{\bar {a}}~b~{\bar {c}}\\\lor &{\bar {a}}~{\bar {b}}~c\end{matrix}}}
o d d l y m a n y o f a , b , c a r e t r u e . {\displaystyle {\begin{matrix}\mathrm {oddly~many~of} \\a,b,c\\\mathrm {are~true} .\end{matrix}}}
a + b + c {\displaystyle a+b+c}
a b c ∨ a b ¯ c ¯ ∨ a ¯ b c ¯ ∨ 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}}}
p a r t i t i o n x i n t o a , b , c . g e n u s x c o m p r i s e s s p e c i e s a , b , c . {\displaystyle {\begin{matrix}\mathrm {partition} ~x\\\mathrm {into} ~a,b,c.\\[6pt]\mathrm {genus} ~x~\mathrm {comprises} \\\mathrm {species} ~a,b,c.\end{matrix}}}
x ¯ a ¯ b ¯ c ¯ ∨ x a b ¯ c ¯ ∨ x a ¯ b c ¯ ∨ x a ¯ b ¯ c {\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}}}
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.
a ~ a ′ ¬ a n o t a {\displaystyle {\begin{matrix}{\tilde {a}}\\[2pt]a^{\prime }\\[2pt]\lnot a\\[2pt]\mathrm {not} ~a\end{matrix}}}
a ∧ b ∧ c a a n d b a n d c {\displaystyle {\begin{matrix}a\land b\land c\\[6pt]a~\mathrm {and} ~b~\mathrm {and} ~c\end{matrix}}}
a ∨ b ∨ c a o r b o r c {\displaystyle {\begin{matrix}a\lor b\lor c\\[6pt]a~\mathrm {or} ~b~\mathrm {or} ~c\end{matrix}}}
a ⇒ b a i m p l i e s b i f a t h e n b n o t a w i t h o u t b {\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}}}
a + b a ≠ b a e x c l u s i v e o r b a n o t e q u a l t o b {\displaystyle {\begin{matrix}a+b\\[2pt]a\neq b\\[2pt]a~\mathrm {exclusive~or} ~b\\[2pt]a~\mathrm {not~equal~to} ~b\end{matrix}}}
a = b a ⟺ b a e q u a l s b a i f a n d o n l y i f b {\displaystyle {\begin{matrix}a=b\\[2pt]a\iff b\\[2pt]a~\mathrm {equals} ~b\\[2pt]a~\mathrm {if~and~only~if} ~b\end{matrix}}}
j u s t o n e o f a , b , c i s f a l s e {\displaystyle {\begin{matrix}\mathrm {just~one~of} \\a,b,c\\\mathrm {is~false} \end{matrix}}}
j u s t o n e o f a , b , c i s t r u e {\displaystyle {\begin{matrix}\mathrm {just~one~of} \\a,b,c\\\mathrm {is~true} \end{matrix}}}
g e n u s a o f s p e c i e s b , c p a r t i t i o n a i n t o b , c p i e a o f s l i c e s b , c {\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}}}
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.
a ⇒ b a i m p l i e s b i f a t h e n b n o t a , o r 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}}}
n o t j u s t o n e o f a , b , c i s t r u e {\displaystyle {\begin{matrix}\mathrm {not~just~one~of} \\a,b,c\\\mathrm {is~true} \end{matrix}}}
just one of C 1 , C 2 , … , C k − 1 , C k is false {\displaystyle {\begin{matrix}{\text{just one of}}\\[6px]C_{1},C_{2},\ldots ,C_{k-1},C_{k}\\[6px]{\text{is false}}\end{matrix}}}
not just one of C 1 , C 2 , … , C k − 1 , C k is true {\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}}}