Animated Logical Graphs • 47

Re: Richard J. LiptonThe Art Of Math
Re: Animated Logical Graphs • (30) (45) (46)

A logical concept represented by a boolean variable has its extension, the cases it covers in a designated universe of discourse, and its comprehension (or intension), the properties it implies in a designated hierarchy of predicates.  The formulas and graphs tabulated in previous posts are well-adapted to articulate the syntactic and intensional aspects of propositional logic.  But their very tailoring to those tasks tends to slight the extensional and therefore empirical applications of logic.  Venn diagrams, despite their unwieldiness as the number of logical dimensions increases, are indispensable in providing the visual intuition with a solid grounding in the extensions of logical concepts.  All that makes it worthwhile to reset our table of boolean functions on two variables to include the corresponding venn diagrams.

Venn Diagrams and Logical Graphs on Two Variables
\text{Boolean Function} \text{Entitative Graph} \text{Existential Graph}
f₀(x,y) Cactus Root
Cactus Stem
f_{0} \text{false} \text{false}
f₁(x,y) Cactus (xy)
Cactus (x)(y)
f_{1} \lnot (x \lor y) \lnot x \land \lnot y
f₂(x,y) Cactus (x(y))
Cactus (x)y
f_{2} \lnot x \land y \lnot x \land y
f₃(x,y) Cactus (x)
Cactus (x)
f_{3} \lnot x \lnot x
f₄(x,y) Cactus ((x)y)
Cactus x(y)
f_{4} x \land \lnot y x \land \lnot y
f₅(x,y) Cactus (y)
Cactus (y)
f_{5} \lnot y \lnot y
f₆(x,y) Cactus ((x,y))
Cactus (x,y)
f_{6} x \ne y x \ne y
f₇(x,y) Cactus (x)(y)
Cactus (xy)
f_{7} \lnot x \lor \lnot y \lnot (x \land y)
f₈(x,y) Cactus ((x)(y))
Cactus xy
f_{8} x \land y x \land y
f₉(x,y) Cactus (x,y)
Cactus ((x,y))
f_{9} x = y x = y
f₁₀(x,y) Cactus y
Cactus y
f_{10} y y
f₁₁(x,y) Cactus (x)y
Cactus (x(y))
f_{11} x \Rightarrow y x \Rightarrow y
f₁₂(x,y) Cactus x
Cactus x
f_{12} x x
f₁₃(x,y) Cactus x(y)
Cactus ((x)y)
f_{13} x \Leftarrow y x \Leftarrow y
f₁₄(x,y) Cactus xy
Cactus ((x)(y))
f_{14} x \lor y x \lor y
f₁₅(x,y) Cactus Stem
Cactus Root
f_{15} \text{true} \text{true}


cc: Cybernetics Communications (1) (2)FB | Logical Graphs • Ontolog Forum (1) (2)
cc: Peirce List (1) (2) (3) (4) (5) (6) (7) (8) • Structural Modeling (1) (2) • Systems (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.

5 Responses to Animated Logical Graphs • 47

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