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 FormJohn M.
Re: Richard J. LiptonThe 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}
Cactus Stem
 
f₁₅(x,y) f₀(x,y)
\texttt{(} ~ \texttt{)}
 
\text{true}
f_{15}
\text{false}
f_{0}
Cactus (x)(y)
 
f₇(x,y) f₁(x,y)
\texttt{(} x \texttt{)(} y \texttt{)}
 
\lnot x \lor \lnot y
f_{7}
\lnot x \land \lnot y
f_{1}
Cactus (x)y
 
f₁₁(x,y) f₂(x,y)
\texttt{(} x \texttt{)} y
 
x \Rightarrow y
f_{11}
x \nLeftarrow y
f_{2}
Cactus (x)
 
f₃(x,y) f₃(x,y)
\texttt{(} x \texttt{)}
 
\lnot x
f_{3}
\lnot x
f_{3}
Cactus x(y)
 
f₁₃(x,y) f₄(x,y)
x \texttt{(} y \texttt{)}
 
x \Leftarrow y
f_{13}
x \nRightarrow y
f_{4}
Cactus (y)
 
f₅(x,y) f₅(x,y)
\texttt{(} y \texttt{)}
 
\lnot y
f_{5}
\lnot y
f_{5}
Cactus (x,y)
 
f₉(x,y) f₆(x,y)
\texttt{(} x \texttt{,} y \texttt{)}
 
x = y
f_{9}
x \ne y
f_{6}
Cactus (xy)
 
f₁(x,y) f₇(x,y)
\texttt{(} x y \texttt{)}
 
\lnot (x \lor y)
f_{1}
\lnot (x \land y)
f_{7}
Cactus xy
 
f₁₄(x,y) f₈(x,y)
x y
 
x \lor y
f_{14}
x \land y
f_{8}
Cactus ((x,y))
 
f₆(x,y) f₉(x,y)
\texttt{((} x \texttt{,} y \texttt{))}
 
x \ne y
f_{6}
x = y
f_{9}
Cactus y
 
f₁₀(x,y) f₁₀(x,y)
y
 
y
f_{10}
y
f_{10}
Cactus (x(y))
 
f₂(x,y) f₁₁(x,y)
\texttt{(} x \texttt{(} y \texttt{))}
 
x \nLeftarrow y
f_{2}
x \Rightarrow y
f_{11}
Cactus x
 
f₁₂(x,y) f₁₂(x,y)
x
 
x
f_{12}
x
f_{12}
Cactus ((x)y)
 
f₄(x,y) f₁₃(x,y)
\texttt{((} x \texttt{)} y \texttt{)}
 
x \nRightarrow y
f_{4}
x \Leftarrow y
f_{13}
Cactus ((x)(y))
 
f₈(x,y) f₁₄(x,y)
\texttt{((} x \texttt{)(} y \texttt{))}
 
x \land y
f_{8}
x \lor y
f_{14}
Cactus Root
 
f₀(x,y) f₁₅(x,y)
 
 
\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 FormFB | 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)

This entry was posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Deduction, Diagrammatic Reasoning, Duality, Equational Inference, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Peirce, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Theorem Proving, Visualization and tagged , , , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

3 Responses to Animated Logical Graphs • 64

  1. Pingback: Animated Logical Graphs • 66 | Inquiry Into Inquiry

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