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− | The underlined parenthetical expressions on the right are the cactus forms for the boolean functions that correspond to inclusive disjunction and logical equivalence, respectively. By way of a reminder, consult Table 1 on the page at this location: | + | The underlined parenthetical expressions on the right are the cactus forms for the boolean functions that correspond to inclusive disjunction and logical equivalence, respectively. Table 1 summarizes the basic elements of the cactus notation for propositional logic. |
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− | :* [http://stderr.org/pipermail/inquiry/2003-May/000478.html DLOG D1] | + | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:96%" |
| + | |+ <math>\text{Table 1. Syntax and Semantics of a Calculus for Propositional Logic}\!</math> |
| + | |- style="background:whitesmoke" |
| + | | <math>\text{Expression}\!</math> |
| + | | <math>\text{Interpretation}\!</math> |
| + | | <math>\text{Other Notations}\!</math> |
| + | |- |
| + | | <math>~</math> |
| + | | <math>\operatorname{True}</math> |
| + | | <math>1\!</math> |
| + | |- |
| + | | <math>(~)</math> |
| + | | <math>\operatorname{False}</math> |
| + | | <math>0\!</math> |
| + | |- |
| + | | <math>x\!</math> |
| + | | <math>x\!</math> |
| + | | <math>x\!</math> |
| + | |- |
| + | | <math>(x)\!</math> |
| + | | <math>\operatorname{Not}\ x</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | x' \\ |
| + | \tilde{x} \\ |
| + | \lnot x \\ |
| + | \end{matrix}</math> |
| + | |- |
| + | | <math>x\ y\ z</math> |
| + | | <math>x\ \operatorname{and}\ y\ \operatorname{and}\ z</math> |
| + | | <math>x \land y \land z</math> |
| + | |- |
| + | | <math>((x)(y)(z))\!</math> |
| + | | <math>x\ \operatorname{or}\ y\ \operatorname{or}\ z</math> |
| + | | <math>x \lor y \lor z</math> |
| + | |- |
| + | | <math>(x\ (y))\!</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | x\ \operatorname{implies}\ y \\ |
| + | \operatorname{If}\ x\ \operatorname{then}\ y \\ |
| + | \end{matrix}</math> |
| + | | <math>x \Rightarrow y\!</math> |
| + | |- |
| + | | <math>(x, y)\!</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | x\ \operatorname{not~equal~to}\ y \\ |
| + | x\ \operatorname{exclusive~or}\ y \\ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | x \neq y \\ |
| + | x + y \\ |
| + | \end{matrix}</math> |
| + | |- |
| + | | <math>((x, y))\!</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | x\ \operatorname{is~equal~to}\ y \\ |
| + | x\ \operatorname{if~and~only~if}\ y \\ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | x = y \\ |
| + | x \Leftrightarrow y \\ |
| + | \end{matrix}</math> |
| + | |- |
| + | | <math>(x, y, z)\!</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{Just~one~of} \\ |
| + | x, y, z \\ |
| + | \operatorname{is~false}. \\ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | x'y~z~ & \lor \\ |
| + | x~y'z~ & \lor \\ |
| + | x~y~z' & \\ |
| + | \end{matrix}</math> |
| + | |- |
| + | | <math>((x),(y),(z))\!</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{Just~one~of} \\ |
| + | x, y, z \\ |
| + | \operatorname{is~true}. \\ |
| + | & \\ |
| + | \operatorname{Partition~all} \\ |
| + | \operatorname{into}\ x, y, z. \\ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | x~y'z' & \lor \\ |
| + | x'y~z' & \lor \\ |
| + | x'y'z~ & \\ |
| + | \end{matrix}</math> |
| + | |- |
| + | | |
| + | <math>\begin{matrix} |
| + | ((x, y), z) \\ |
| + | & \\ |
| + | (x, (y, z)) \\ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{Oddly~many~of} \\ |
| + | x, y, z \\ |
| + | \operatorname{are~true}. \\ |
| + | \end{matrix}</math> |
| + | | |
| + | <p><math>x + y + z\!</math></p> |
| + | <br> |
| + | <p><math>\begin{matrix} |
| + | x~y~z~ & \lor \\ |
| + | x~y'z' & \lor \\ |
| + | x'y~z' & \lor \\ |
| + | x'y'z~ & \\ |
| + | \end{matrix}</math></p> |
| + | |- |
| + | | <math>(w, (x),(y),(z))\!</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | \operatorname{Partition}\ w \\ |
| + | \operatorname{into}\ x, y, z. \\ |
| + | & \\ |
| + | \operatorname{Genus}\ w\ \operatorname{comprises} \\ |
| + | \operatorname{species}\ x, y, z. \\ |
| + | \end{matrix}</math> |
| + | | |
| + | <math>\begin{matrix} |
| + | w'x'y'z' & \lor \\ |
| + | w~x~y'z' & \lor \\ |
| + | w~x'y~z' & \lor \\ |
| + | w~x'y'z~ & \\ |
| + | \end{matrix}</math> |
| + | |} |
| + | |
| + | <br> |
| | | |
| <pre> | | <pre> |