User:Jon Awbrey/Figures and Tables 2

Boolean Functions on Two Variables

Old Table

\(\text{Table 7.} ~~ \text{Propositional Forms on Two Variables}\)

\(\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}\)	\(\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}\)	\(\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}\)	\(\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}\)	\(\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}\)	\(\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}\)
	\(x\colon\)	\(1~1~0~0\)
	\(y\colon\)	\(1~0~1~0\)
\(f_{0}\)	\(f_{0000}\)	\(0~0~0~0\)	\((~)\)	\(\text{false}\)	\(0\)
\(f_{1}\)	\(f_{0001}\)	\(0~0~0~1\)	\((x)(y)\)	\(\text{neither}~ x ~\text{nor}~ y\)	\(\lnot x \land \lnot y\)
\(f_{2}\)	\(f_{0010}\)	\(0~0~1~0\)	\((x)\ y\)	\(y ~\text{without}~ x\)	\(\lnot x \land y\)
\(f_{3}\)	\(f_{0011}\)	\(0~0~1~1\)	\((x)\)	\(\text{not}~ x\)	\(\lnot x\)
\(f_{4}\)	\(f_{0100}\)	\(0~1~0~0\)	\(x\ (y)\)	\(x ~\text{without}~ y\)	\(x \land \lnot y\)
\(f_{5}\)	\(f_{0101}\)	\(0~1~0~1\)	\((y)\)	\(\text{not}~ y\)	\(\lnot y\)
\(f_{6}\)	\(f_{0110}\)	\(0~1~1~0\)	\((x, y)\)	\(x ~\text{not equal to}~ y\)	\(x \ne y\)
\(f_{7}\)	\(f_{0111}\)	\(0~1~1~1\)	\((x\ y)\)	\(\text{not both}~ x ~\text{and}~ y\)	\(\lnot x \lor \lnot y\)
\(f_{8}\)	\(f_{1000}\)	\(1~0~0~0\)	\(x\ y\)	\(x ~\text{and}~ y\)	\(x \land y\)
\(f_{9}\)	\(f_{1001}\)	\(1~0~0~1\)	\(((x, y))\)	\(x ~\text{equal to}~ y\)	\(x = y\)
\(f_{10}\)	\(f_{1010}\)	\(1~0~1~0\)	\(y\)	\(y\)	\(y\)
\(f_{11}\)	\(f_{1011}\)	\(1~0~1~1\)	\((x\ (y))\)	\(\text{not}~ x ~\text{without}~ y\)	\(x \Rightarrow y\)
\(f_{12}\)	\(f_{1100}\)	\(1~1~0~0\)	\(x\)	\(x\)	\(x\)
\(f_{13}\)	\(f_{1101}\)	\(1~1~0~1\)	\(((x)\ y)\)	\(\text{not}~ y ~\text{without}~ x\)	\(x \Leftarrow y\)
\(f_{14}\)	\(f_{1110}\)	\(1~1~1~0\)	\(((x)(y))\)	\(x ~\text{or}~ y\)	\(x \lor y\)
\(f_{15}\)	\(f_{1111}\)	\(1~1~1~1\)	\(((~))\)	\(\text{true}\)	\(1\)

New Tables

Template

\(\text{Boolean Functions on Two Variables}\)

\(\text{Boolean Function}\)	\(\text{Linguistic Formula}\)	\(\text{Entitative Graph}\)	\(\text{Existential Graph}\)
\(f_0\)	\(\text{false}\)	32px	32px
\(f_1\)	\(\text{neither}~ x ~\text{nor}~ y\)	\(f_1\)	\(f_1\)
\(f_2\)	\(y ~\text{without}~ x\)	\(f_2\)	\(f_2\)
\(f_3\)	\(\text{not}~ x\)	32px	32px
\(f_4\)	\(x ~\text{without}~ y\)	\(f_4\)	\(f_4\)
\(f_5\)	\(\text{not}~ y\)	32px	32px
\(f_6\)	\(x ~\text{not equal to}~ y\)	\(f_6\)	64px
\(f_7\)	\(\text{not both}~ x ~\text{and}~ y\)	\(f_7\)	\(f_7\)
\(f_8\)	\(x ~\text{and}~ y\)	\(f_8\)	32px
\(f_9\)	\(x ~\text{equal to}~ y\)	64px	\(f_9\)
\(f_{10}\)	\(y\)	32px	32px
\(f_{11}\)	\(\text{not}~ x ~\text{without}~ y\)	\(f_{11}\)	\(f_{11}\)
\(f_{12}\)	\(x\)	32px	32px
\(f_{13}\)	\(\text{not}~ y ~\text{without}~ x\)	\(f_{13}\)	\(f_{13}\)
\(f_{14}\)	\(x ~\text{or}~ y\)	\(f_{14}\)	\(f_{14}\)
\(f_{15}\)	\(\text{true}\)	32px	32px

Logical Cacti • Theme One Exposition

Up till now we've been working to hammer out a two-edged sword of syntax, honing the syntax of painted and rooted cacti and expressions (PARCAE), and turning it to use in taming the syntax of two-level formal languages.

But the purpose of a logical syntax is to support a logical semantics, which means, for starters, to bear interpretation as sentential signs that can denote objective propositions about some universe of objects.

One of the difficulties that we face in this discussion is that the words interpretation, meaning, semantics, and so on will have so many different meanings from one moment to the next of their use. A dedicated neologician might be able to think up distinctive names for all of the aspects of meaning and all of the approaches to them that will concern us here, but I will just have to do the best that I can with the common lot of ambiguous terms, leaving it to context and the intelligent interpreter to sort it out as much as possible.

As it happens, the language of cacti is so abstract that it can bear at least two different interpretations as logical sentences denoting logical propositions. The two interpretations that I know about are descended from the ones that C.S. Peirce called the entitative and the existential interpretations of his systems of graphical logics. For our present aims, I shall briefly introduce the alternatives and then quickly move to the existential interpretation of logical cacti.

Existential Interpretation

Table 13 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.

\(\text{Table 13.} ~~ \text{Existential Interpretation}\)

\(\text{Cactus Graph}\)	\(\text{Cactus Expression}\)	\(\text{Interpretation}\)
24px	\(\mathrm{~}\)	\(\mathrm{true}\)
24px	\(\texttt{(} ~ \texttt{)}\)	\(\mathrm{false}\)
	\(a\)	\(a\)
	\(\texttt{(} a \texttt{)}\)	\(\begin{matrix} \tilde{a} \\[2pt] a^\prime \\[2pt] \lnot a \\[2pt] \mathrm{not}~ a \end{matrix}\)
	\(a~b~c\)	\(\begin{matrix} a \land b \land c \\[6pt] a ~\mathrm{and}~ b ~\mathrm{and}~ c \end{matrix}\)
	\(\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}\)	\(\begin{matrix} a \lor b \lor c \\[6pt] a ~\mathrm{or}~ b ~\mathrm{or}~ c \end{matrix}\)
	\(\texttt{(} a \texttt{(} b \texttt{))}\)	\(\begin{matrix} a \Rightarrow b \\[2pt] a ~\mathrm{implies}~ b \\[2pt] \mathrm{if}~ a ~\mathrm{then}~ b \\[2pt] \mathrm{not}~ a ~\mathrm{without}~ b \end{matrix}\)
	\(\texttt{(} a, b \texttt{)}\)	\(\begin{matrix} a + b \\[2pt] a \neq b \\[2pt] a ~\mathrm{exclusive~or}~ b \\[2pt] a ~\mathrm{not~equal~to}~ b \end{matrix}\)
	\(\texttt{((} a, b \texttt{))}\)	\(\begin{matrix} a = b \\[2pt] a \iff b \\[2pt] a ~\mathrm{equals}~ b \\[2pt] a ~\mathrm{if~and~only~if}~ b \end{matrix}\)
	\(\texttt{(} a, b, c \texttt{)}\)	\(\begin{matrix} \mathrm{just~one~of} \\ a, b, c \\ \mathrm{is~false} \end{matrix}\)
	\(\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}\)	\(\begin{matrix} \mathrm{just~one~of} \\ a, b, c \\ \mathrm{is~true} \end{matrix}\)
	\(\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))}\)	\(\begin{matrix} \mathrm{genus}~ a ~\mathrm{of~species}~ b, c \\[6pt] \mathrm{partition}~ a ~\mathrm{into}~ b, c \\[6pt] \mathrm{pie}~ a ~\mathrm{of~slices}~ b, c \end{matrix}\)

Entitative Interpretation

Table 14 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.

\(\text{Table 14.} ~~ \text{Entitative Interpretation}\)

\(\text{Cactus Graph}\)	\(\text{Cactus Expression}\)	\(\text{Interpretation}\)
24px	\(\mathrm{~}\)	\(\mathrm{false}\)
24px	\(\texttt{(} ~ \texttt{)}\)	\(\mathrm{true}\)
	\(a\)	\(a\)
	\(\texttt{(} a \texttt{)}\)	\(\begin{matrix} \tilde{a} \\[2pt] a^\prime \\[2pt] \lnot a \\[2pt] \mathrm{not}~ a \end{matrix}\)
	\(a~b~c\)	\(\begin{matrix} a \lor b \lor c \\[6pt] a ~\mathrm{or}~ b ~\mathrm{or}~ c \end{matrix}\)
	\(\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}\)	\(\begin{matrix} a \land b \land c \\[6pt] a ~\mathrm{and}~ b ~\mathrm{and}~ c \end{matrix}\)
	\(\texttt{(} a \texttt{)} b\)	\(\begin{matrix} a \Rightarrow b \\[2pt] a ~\mathrm{implies}~ b \\[2pt] \mathrm{if}~ a ~\mathrm{then}~ b \\[2pt] \mathrm{not}~ a, ~\mathrm{or}~ b \end{matrix}\)
	\(\texttt{(} a, b \texttt{)}\)	\(\begin{matrix} a = b \\[2pt] a \iff b \\[2pt] a ~\mathrm{equals}~ b \\[2pt] a ~\mathrm{if~and~only~if}~ b \end{matrix}\)
	\(\texttt{((} a, b \texttt{))}\)	\(\begin{matrix} a + b \\[2pt] a \neq b \\[2pt] a ~\mathrm{exclusive~or}~ b \\[2pt] a ~\mathrm{not~equal~to}~ b \end{matrix}\)
	\(\texttt{(} a, b, c \texttt{)}\)	\(\begin{matrix} \mathrm{not~just~one~of} \\ a, b, c \\ \mathrm{is~true} \end{matrix}\)
	\(\texttt{((} a, b, c \texttt{))}\)	\(\begin{matrix} \mathrm{just~one~of} \\ a, b, c \\ \mathrm{is~true} \end{matrix}\)
	\(\texttt{(((} a \texttt{)}, b, c \texttt{))}\)	\(\begin{matrix} \mathrm{genus}~ a ~\mathrm{of~species}~ b, c \\[6pt] \mathrm{partition}~ a ~\mathrm{into}~ b, c \\[6pt] \mathrm{pie}~ a ~\mathrm{of~slices}~ b, c \end{matrix}\)

Logical Graphs

Old Versions

Example 1

o-----------------------------------------------------------o | | | o o o o o o | | \| | | | | | | o o o o o o o o o | | \|/ \|/ |/ | | | @ = @ = @ = @ | | | o-----------------------------------------------------------o | | | (()())(())() = (())(())() = (())() = ( ) | | | o-----------------------------------------------------------o

Example 2

o-------------------o-------------------o-------------------o | Object | Sign | Interpretant | o-------------------o-------------------o-------------------o | | | | | Falsity | "(()())(())()" | "(())(())()" | | | | | | Falsity | "(())(())()" | "(())()" | | | | | | Falsity | "(())()" | "()" | | | | | o-------------------o-------------------o-------------------o

Example 3

o-------------------o-------------------o-------------------o | Object | Sign | Interpretant | o-------------------o-------------------o-------------------o | | | | | Falsity | "(()())(())()" | "(()())(())()" | | | | | | Falsity | "(()())(())()" | "(())(())()" | | | | | | Falsity | "(()())(())()" | "(())()" | | | | | | Falsity | "(()())(())()" | "()" | | | | | o-------------------o-------------------o-------------------o | | | | | Falsity | "(())(())()" | "(()())(())()" | | | | | | Falsity | "(())(())()" | "(())(())()" | | | | | | Falsity | "(())(())()" | "(())()" | | | | | | Falsity | "(())(())()" | "()" | | | | | o-------------------o-------------------o-------------------o | | | | | Falsity | "(())()" | "(()())(())()" | | | | | | Falsity | "(())()" | "(())(())()" | | | | | | Falsity | "(())()" | "(())()" | | | | | | Falsity | "(())()" | "()" | | | | | o-------------------o-------------------o-------------------o | | | | | Falsity | "()" | "(()())(())()" | | | | | | Falsity | "()" | "(())(())()" | | | | | | Falsity | "()" | "(())()" | | | | | | Falsity | "()" | "()" | | | | | o-------------------o-------------------o-------------------o

Example 4

o-------------------o-------------------o-------------------o | a | b | (a , b) | o-------------------o-------------------o-------------------o | | | | | blank | blank | cross | | | | | | blank | cross | blank | | | | | | cross | blank | blank | | | | | | cross | cross | cross | | | | | o-------------------o-------------------o-------------------o

Example 5

o-------o-------o-------o-----------o | a | b | c | (a, b, c) | o-------o-------o-------o-----------o | | | | | | blank | blank | blank | cross | | | | | | | blank | blank | cross | blank | | | | | | | blank | cross | blank | blank | | | | | | | blank | cross | cross | cross | | | | | | | cross | blank | blank | blank | | | | | | | cross | blank | cross | cross | | | | | | | cross | cross | blank | cross | | | | | | | cross | cross | cross | cross | | | | | | o-------o-------o-------o-----------o

Example 6

o-------o-------o-------o-----------o | a | b | c | (a, b, c) | o-------o-------o-------o-----------o | | | | | | o | o | o | | | | | | | | | o | o | | | o | | | | | | | o | | | o | o | | | | | | | o | | | | | | | | | | | | | | | o | o | o | | | | | | | | | o | | | | | | | | | | | | | | | o | | | | | | | | | | | | | | | | | | | | | | o-------o-------o-------o-----------o

New Versions

Example 1

…

Example 2a

\(\text{Object}\)	\(\text{Sign}\)	\(\text{Interpretant}\)
\(\mathrm{Falsity}\)	\({}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime}\)	\({}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}\)
\(\mathrm{Falsity}\)	\({}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime}\)	\({}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}\)
\(\mathrm{Falsity}\)	\({}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime}\)	\({}^{\backprime\backprime} \texttt{()} {}^{\prime\prime}\)

Example 2b

\(\text{Object}\)	\(\text{Sign}\)	\(\text{Interpretant}\)
\(\begin{array}{l} \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \end{array}\)	\(\begin{array}{l} {}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime} \end{array}\)	\(\begin{array}{l} {}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{()} {}^{\prime\prime} \end{array}\)

Example 3

\(\text{Object}\)	\(\text{Sign}\)	\(\text{Interpretant}\)
\(\begin{array}{l} \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \end{array}\)	\(\begin{array}{l} {}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime} \end{array}\)	\(\begin{array}{l} {}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{()} {}^{\prime\prime} \end{array}\)
\(\begin{array}{l} \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \end{array}\)	\(\begin{array}{l} {}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime} \end{array}\)	\(\begin{array}{l} {}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{()} {}^{\prime\prime} \end{array}\)
\(\begin{array}{l} \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \end{array}\)	\(\begin{array}{l} {}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime} \end{array}\)	\(\begin{array}{l} {}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{()} {}^{\prime\prime} \end{array}\)
\(\begin{array}{l} \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \\[6pt] \mathrm{Falsity} \end{array}\)	\(\begin{array}{l} {}^{\backprime\backprime} \texttt{()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{()} {}^{\prime\prime} \end{array}\)	\(\begin{array}{l} {}^{\backprime\backprime} \texttt{(()())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{(())()} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \texttt{()} {}^{\prime\prime} \end{array}\)

Example 4

\(a\)	\(b\)	\(\texttt{(} a \texttt{,} b \texttt{)}\)
\(\begin{matrix} \texttt{Blank} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Cross} \end{matrix}\)	\(\begin{matrix} \texttt{Blank} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Cross} \end{matrix}\)	\(\begin{matrix} \texttt{Cross} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Cross} \end{matrix}\)

Example 5

\(a\)	\(b\)	\(c\)	\(\texttt{(} a \texttt{,} b \texttt{,} \texttt{c} \texttt{)}\)
\(\begin{matrix} \texttt{Blank} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Cross} \end{matrix}\)	\(\begin{matrix} \texttt{Blank} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Cross} \end{matrix}\)	\(\begin{matrix} \texttt{Blank} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Cross} \end{matrix}\)	\(\begin{matrix} \texttt{Cross} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Blank} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Cross} \\[6pt] \texttt{Cross} \end{matrix}\)

Example 6

\(a\)	\(b\)	\(c\)	\(\texttt{(} a \texttt{,} b \texttt{,} \texttt{c} \texttt{)}\)
\(\begin{matrix} \texttt{o} \\[6pt] \texttt{o} \\[6pt] \texttt{o} \\[6pt] \texttt{o} \\[6pt] \texttt{|} \\[6pt] \texttt{|} \\[6pt] \texttt{|} \\[6pt] \texttt{|} \end{matrix}\)	\(\begin{matrix} \texttt{o} \\[6pt] \texttt{o} \\[6pt] \texttt{|} \\[6pt] \texttt{|} \\[6pt] \texttt{o} \\[6pt] \texttt{o} \\[6pt] \texttt{|} \\[6pt] \texttt{|} \end{matrix}\)	\(\begin{matrix} \texttt{o} \\[6pt] \texttt{|} \\[6pt] \texttt{o} \\[6pt] \texttt{|} \\[6pt] \texttt{o} \\[6pt] \texttt{|} \\[6pt] \texttt{o} \\[6pt] \texttt{|} \end{matrix}\)	\(\begin{matrix} \texttt{|} \\[6pt] \texttt{o} \\[6pt] \texttt{o} \\[6pt] \texttt{|} \\[6pt] \texttt{o} \\[6pt] \texttt{|} \\[6pt] \texttt{|} \\[6pt] \texttt{|} \end{matrix}\)