[Inquiry] Re: Functional Logic

[Jon Awbrey]

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FL.  Note 5

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2.1.5.  Extending the Existential Interpretation to Quantificational Logic

Previously I introduced a calculus for propositional logic, fixing its meaning
according to what C.S. Peirce called the "existential interpretation".  As far
as it concerns propositional calculus this interpretation settles the meanings
that are associated with merely the most basic symbols and logical connectives.
Now we must extend and refine the existential interpretation to comprehend the
analysis of "quantifications", that is, quantified propositions.  In doing so
we recognize two additional aspects of logic that need to be developed, over
and above the material of propositional logic.  At the formal extreme there
is the aspect of higher order functional types, into which we have already
ventured a little above.  At the level of the fundamental content of the
available propositions we have to introduce a different interpretation
for what we may call "elemental" or "singular" propositions.

Let us return to the 2-dimensional universe X% = [x, y].
In order to construct a bridge between propositions and
quantifications it serves to create a set of qualifiers
L_uv : (B^2 -> B) -> B that take on the following forms:

   L_00 f   =   L_<(x)(y)> f

            =   !a!_1 f

            =   !Y!_<(x)(y)> f

            =   !Y!<(x)(y) => f>

            =   <f likes (x)(y)>

   L_01 f   =   L_<(x) y > f

            =   !a!_2 f

            =   !Y!_<(x) y > f

            =   !Y!<(x) y  => f>

            =   <f likes (x) y >

   L_10 f   =   L_< x (y)> f

            =   !a!_4 f

            =   !Y!_< x (y)> f

            =   !Y!< x (y) => f>

            =   <f likes  x (y)>

   L_11 f   =   L_< x  y > f

            =   !a!_8 f

            =   !Y!_< x  y > f

            =   !Y!< x  y => f>

            =   <f likes  x  y >

Intuitively, the L_uv operators may be thought of as qualifying propositions
according to the elements of the universe of discourse that each proposition
positively values.  Taken together, these measures provide us with the means
to express many useful observations about the propositions in X% = [x, y],
and so they mediate a subtext [L_00, L_01, L_10, L_11] that takes place
within the higher order universe of discourse X%2 = [X%] = [[x, y]].

Figure 15 summarizes the action of the L_uv on the f_i in X%2.

o-------------------------------------------------o
|                                                 |
|                        o                        |
|                       / \                       |
|                      /   \                      |
|                     /x   y\                     |
|                    / o---o \                    |
|                   o   \ /   o                   |
|                  / \   o   / \                  |
|                 /   \  |  /   \                 |
|                /     \ @ /     \                |
|               / x   y \ / x   y \               |
|              o  o---o  o  o---o  o              |
|             / \  \    / \    /  / \             |
|            /   \  @  /   \  @  /   \            |
|           /     \   /     \   /     \           |
|          /   y   \ /       \ /   y   \          |
|         o    @    o    @    o    o    o         |
|        / \       / \       / \   |   / \        |
|       /   \     /   \     /   \  @  /   \       |
|      /     \   /x   y\   /     \   /     \      |
|     /  x y  \ / o   o \ /  x y  \ / x   y \     |
|    o    @    o   \ /   o    o    o  o   o  o    |
|    |\       / \   o   / \   |   / \  \ /  /|    |
|    | \     /   \  |  /   \  @  /   \  @  / |    |
|    |  \   /     \ @ /     \   /     \   /  |    |
|    |   \ /   x   \ / x   y \ /   x   \ /   |    |
|    |    o    @    o  o---o  o    o    o    |    |
|    |    |\       / \  \ /  / \   |   /|    |    |
|    |    | \     /   \  @  /   \  @  / |    |    |
|    |    |  \   /     \   /     \   /  |    |    |
|    |L_11|   \ /   o y \ / x o   \ /   |L_00|    |
|    o---------o    |    o    |    o---------o    |
|         |     \ x @   / \   @ y /     |         |
|         |      \     /   \     /      |         |
|         |       \   /     \   /       |         |
|         |L_10    \ /   o   \ /    L_01|         |
|         o---------o    |    o---------o         |
|                    \   @   /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        o                        |
|                                                 |
o-------------------------------------------------o
Figure 15.  Higher Order Universe [L_uv] c [[x, y]]

Jon Awbrey

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