Animated Logical Graphs • 64

If exegesis raised a hermeneutic problem, that is, a problem of interpretation, it is because every reading of a text always takes place within a community, a tradition, or a living current of thought, all of which display presuppositions and exigencies — regardless of how closely a reading may be tied to the quid, to “that in view of which” the text was written.

Paul Ricoeur • The Conflict of Interpretations

Re: Laws of Form • John Mingers
Re: Richard J. Lipton • The Art Of Math

Dear John,

It occurred to me a picture might save a few thousand words. A good place to start is the following Table from an earlier post on my blog.

The smart way to deal with parens + character strings in computing is to parse them into graph-theoretic data structures and then work on those instead of the strings themselves. Usually one gets some sort of tree structures for the parse graphs. In my work on logical graphs I eventually came to use the more general species of structure graph theorists call cactus graphs or cacti.

Referring to the Table —

Column 1 shows the logical graphs I use for the sixteen boolean functions on two variables, with the string forms underneath. The cactus string obtained by traversing the cactus graph uses parens + commas + variables in forms like $\texttt{(} x \texttt{,} y \texttt{)}$ and $\texttt{((} x \texttt{,} y \texttt{))}.$
Column 2 shows the venn diagram associated with the entitative interpretation of the graph in Column 1. This is the interpretation C.S. Peirce used in his earlier work on entitative graphs and the one Spencer Brown used in his Laws of Form.
Column 3 shows the venn diagram associated with the existential interpretation of the graph in Column 1. This is the interpretation C.S. Peirce used in his later work on existential graphs.

Logical Graphs • Entitative and Existential Venn Diagrams
$\text{Logical Graph}$	$\text{Entitative Interpretation}$	$\text{Existential Interpretation}$

$\texttt{(} ~ \texttt{)}$	$\text{true}$ $f_{15}$	$\text{false}$ $f_{0}$

$\texttt{(} x \texttt{)(} y \texttt{)}$	$\lnot x \lor \lnot y$ $f_{7}$	$\lnot x \land \lnot y$ $f_{1}$

$\texttt{(} x \texttt{)} y$	$x \Rightarrow y$ $f_{11}$	$x \nLeftarrow y$ $f_{2}$

$\texttt{(} x \texttt{)}$	$\lnot x$ $f_{3}$	$\lnot x$ $f_{3}$

$x \texttt{(} y \texttt{)}$	$x \Leftarrow y$ $f_{13}$	$x \nRightarrow y$ $f_{4}$

$\texttt{(} y \texttt{)}$	$\lnot y$ $f_{5}$	$\lnot y$ $f_{5}$

$\texttt{(} x \texttt{,} y \texttt{)}$	$x = y$ $f_{9}$	$x \ne y$ $f_{6}$

$\texttt{(} x y \texttt{)}$	$\lnot (x \lor y)$ $f_{1}$	$\lnot (x \land y)$ $f_{7}$

$x y$	$x \lor y$ $f_{14}$	$x \land y$ $f_{8}$

$\texttt{((} x \texttt{,} y \texttt{))}$	$x \ne y$ $f_{6}$	$x = y$ $f_{9}$

$y$	$y$ $f_{10}$	$y$ $f_{10}$

$\texttt{(} x \texttt{(} y \texttt{))}$	$x \nLeftarrow y$ $f_{2}$	$x \Rightarrow y$ $f_{11}$

$x$	$x$ $f_{12}$	$x$ $f_{12}$

$\texttt{((} x \texttt{)} y \texttt{)}$	$x \nRightarrow y$ $f_{4}$	$x \Leftarrow y$ $f_{13}$

$\texttt{((} x \texttt{)(} y \texttt{))}$	$x \land y$ $f_{8}$	$x \lor y$ $f_{14}$

	$\text{false}$ $f_{0}$	$\text{true}$ $f_{15}$

Take a gander at all that and I’ll discuss more tomorrow …

Regards,

Jon

Resources

cc: Cybernetics (1) (2) • Laws of Form • FB | Logical Graphs • Ontolog Forum (1) (2)
• Peirce List (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) • Structural Modeling (1) (2)
• Systems Science (1) (2)

	Survey of Animated L… on Logic Syllabus • Discussion…
	Survey of Animated L… on Logic Syllabus
	Logic Syllabus • Dis… on Genus, Species, Pie Charts, Ra…
	Logic Syllabus • Dis… on Logic Syllabus
	Survey of Inquiry Dr… on Inquiry Into Inquiry • On Init…
	Inquiry Into Inquiry… on Survey of Abduction, Deduction…
	Inquiry Into Inquiry… on Survey of Pragmatic Semiotic I…
	Inquiry Into Inquiry… on Survey of Inquiry Driven Syste…
	Inquiry Into Inquiry… on Inquiry Into Inquiry • On Init…
	The Failure Of AI De… on Inquiry Into Inquiry • On Init…

M	T	W	T	F	S	S
1	2	3	4	5	6	7
8	9	10	11	12	13	14
15	16	17	18	19	20	21
22	23	24	25	26	27	28
29	30	31