Animated Logical Graphs • 47

Re: Richard J. Lipton • The 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
$\text{Boolean Function}$	$\text{Entitative Graph}$	$\text{Existential Graph}$

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

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

$f_{2}$	$\lnot x \land y$	$\lnot x \land y$

$f_{3}$	$\lnot x$	$\lnot x$

$f_{4}$	$x \land \lnot y$	$x \land \lnot y$

$f_{5}$	$\lnot y$	$\lnot y$

$f_{6}$	$x \ne y$	$x \ne y$

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

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

$f_{9}$	$x = y$	$x = y$

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

$f_{11}$	$x \Rightarrow y$	$x \Rightarrow y$

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

$f_{13}$	$x \Leftarrow y$	$x \Leftarrow y$

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

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

Resources

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)

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