[Inquiry] Re: Really Useful Logic

[Jon Awbrey]

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RUL.  Note 5

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Slogan 1.  Logic is qualitative statistics.

There are several good reasons to consider a functional view of predicates,
that is, to regard a predicate name "P" as denoting a function P : X -> B,
where X is a relevant domain and where B is any convenient 2-element set
whose values can be used to mark the distinctions that are drawn in X.

One of the best reasons for doing this is that it helps us to maintain
the needful connections between logic and statistics, that is to say,
to preserve the ligaments between deductive and inductive reasoning
that are needed to articulate any reasonably scientific world view.

One of the best ways to do this is to maintain a dual interpretation
for predicate names.  In other words, we use the same predicate name
"P" in a "polymorphous" way, reading it under one light as denoting
a function of type P : X -> B, as a proposition about the elements
of the domain X, and reading it under another light as denoting
a function of the type P : X -> R, as a frequency distribution
or a probability density over the sample space X.  This helps
us to exploit the analogies between the two function spaces,
{X -> B} and {X -> R}, as far as they go.  In practice, it
is fairly easy to handle the residual differences between
the qualitative and the quantitative readings as if they
were purely contextual or interpretive differences, as
matters of accent, so to speak.

One very common way of finding or making bridges between the
qualitative logical view and the quantitative statistical view
of the same underlying scene is through the choice of thresholds.

We can see a concrete instance of this in the Green-Hot Website example.
We left that example in a somewhat vaguely defined condition, with two
sets of predicates, {G_1, H_1} and {G_2, H_2}, none of which pairs of
measures were specified with any degree of computational exactness.

The intent of the predicate Hot : X -> B was stated like so:

"A website is Hot iff it has a very high frequency of hits."

In order to give this vague intention a computational definition
we would have to specify a particular arrangement of hit counters
to measure the frequency of hits in a uniform way across the whole
domain X, and then we would have to pick a threshold frequency value
to define what we mean by "high".

Figure 6 depicts one such frequency distribution,
where "High" is defined as "Greater Than 10,000".

o-------------------------------------------------o
|                                                 |
|       Hits                                      |
|  10^8 o               o                         |
|  10^7 |              / \                        |
|  10^6 |           o-o   \                       |
|  10^5 |          /       \                      |
|  10^4 |- - - - -o- - - - -o- - - - - - ->       |
|  10^3 |        /'         '\                    |
|  10^2 |  o----o '         ' \                   |
|  10^1 | /       '         '  o----o             |
|  10^0 |/  ~Hot  '   Hot   '  ~Hot  \            |
|  0000 o---------o---------o---------o---> X     |
|                                                 |
o-------------------------------------------------o
Figure 6.  Distribution of Hits over URL's

Figure 7 shows the result of applying the corresponding
threshold operation to the above frequency distribution,
in effect, counting everything below threshold as equal
to zero, and everything above threshold as equal to the
value of that threshold.  The result is a step-function
approximation to the original frequency distribution.

o-------------------------------------------------o
|                                                 |
|       Hits                                      |
|  10^4 o         o---------o                     |
|       |         |         |                     |
|       |         |         |                     |
|       |         |         |                     |
|       |         |         |                     |
|  0000 o---------o---------o---------o---> X     |
|                                                 |
o-------------------------------------------------o
Figure 7.  Step Function Approximation

Figure 8 shows the result of "normalizing" the previous approximation
to give a top value of 1 and a bottom value of 0, producing a function
that has the form of a predicate H : X -> B.

o-------------------------------------------------o
|                                                 |
|       Hot                                       |
|     1 o         o---------o                     |
|       |         |         |                     |
|       |         |         |                     |
|       |         |         |                     |
|       |         |         |                     |
|     0 o---------o---------o---------o---> X     |
|                                                 |
o-------------------------------------------------o
Figure 8.  Predicate Hot : X -> B

Now, all of this is pretty standard stuff, and it probably seems pretty trivial.
In a way, it is, at least, it ought to be.  And if it were not, we should hope
to make it so.  But what is not so standard and not so trivial to implement in
practice is the idea that the syntax and the software that we use for coping
with predicates P : X -> B and the syntax and the software that we use for
dealing with distributions Q : X -> R ought to make it as easy as possible
to move back and forth between these two needfully related applications.

Jon Awbrey

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