## Animated Logical Graphs • 73

Re: Richard J. LiptonThe Art Of Math
Re: Animated Logical Graphs • (30)(45)(57)(58)(59)(60)(61)(62)(63)(64)(65)(66)(69)(70)(71)(72)

Last time we took up the four singleton orbits in the action of $T$ on $X$ and saw each consists of a single logical graph which $T$ fixes, preserves, or transforms into itself.  On that account these four logical graphs are said to be self-dual or $T$-invariant.

In general terms, it is useful to think of the entitative and existential interpretations as two formal languages which happen to use the same set of signs, each in its own way, to denote the same set of formal objects.  Then $T$ defines the translation between languages and the self-dual logical graphs are the points where the languages coincide, where the same signs denote the same objects in both.  Such constellations of “fixed stars” are indispensable to navigation between languages, as every argot-naut discovers in time.

Returning to the case at hand, where $T$ acts on a selection of 16 logical graphs for the 16 boolean functions on two variables, the following Table shows the values of the denoted boolean function $f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}$ for each of the self-dual logical graphs.

The functions indexed here as $f_{12}$ and $f_{10}$ are known as the coordinate projections $(x, y) \mapsto x$ and $(x, y) \mapsto y$ on the first and second coordinates, respectively, and the functions indexed as $f_{3}$ and $f_{5}$ are the negations $(x, y) \mapsto \tilde{x}$ and $(x, y) \mapsto \tilde{y}$ of those projections, respectively.

### Resources

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cc: Structural Modeling (1) (2) • Systems Science (1) (2)
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