[Inquiry] Re: Peirce's Logic Of Information

[Jon Awbrey]

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PLOI.  Note 20

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There's a way to finesse the use of existential quantifiers when
it comes to especially simple classes of existential propositions,
in this way stretching the usefulness of zeroth order logic slightly
further than it might initially be thought.  This is done through the
use of a certain interpretive convention, namely, we adopt the blanket
assumption that things of a given description may be taken to exist in
default of their propositional specification being explicitly denied or
else consequentially inconsistent.  In effect, it is never necessary to
say when something of a given description exists, only when it does not.

This "formal existence" convention of interpretation permits us to resolve,
or maybe just temporize, a number of the problems that we ran into earlier.

For example, the problem about Red => Color but Red =/= Color
all but magically disappers when rightly viewed in this light.
First, just so that I can use 1-letter labels in diagrams and
formulas without causing a confusion between the term "Color"
and the element 'C', let me change the term "Color" to "Hue".

Let us now contemplate the proposition Red => Hued, henceforth R => H,
and what it says in the context of a suitable universe of discourse X.
The import becomes strikingly evident in the existential graph syntax,
where R => H takes the form (R (H)), making manifest that it excludes
the existence of anything in the universe X from the region indicated
by the propositional specification R (H), that is, "Red and not Hued".
The situation can be diagrammed in a rough lattice fashion as follows:

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` X ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` `/`\` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` / ` \ ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` `/` ` `\` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` / ` ` ` \ ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` `/` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` H o ` ` ` ` ` o (H) ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` `/`\` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` / ` \ ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` `/` ` `\` ` ` ` ` `\` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` / ` ` ` \ ` ` ` ` ` \ ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` `/` ` ` ` `\` ` ` ` ` `\` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` R o ` ` ` ` ` o (R) ` ` ` o (R) ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `

In short, the formal existence of things under the propositional
descriptions H (R), Hued Non-Reds, and (H)(R), Non-Hued Non-Reds,
provides the asymmetry needed for a proper order relation R => H.

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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