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p: Most Swedes are tall.

Assume that the queries are:

qp How many Swedes are not tall

q2: How many are short

q3: What is the average height of Swedes.

In our earlier discussion of this example, we found that p translates into a generalized constraint on the count density function, h. Thus,

where a and b are the lower and upper bounds on the height of Swedes. Precisiands of q1, q2 and q3 may be expressed as

Considering q1, we note that Consequently,

which may be rewritten as

where 1-most plays the role of the antonym of most (Fig. 8). Considering q2, we have to compute

/b

ah( u )p,tall( u )du is most.

Applying the extension principle, we arrive at the desired answer to the query:

subject to

and

Likewise, for q3 we have as the answer

subject to and

As an illustration of application of protoformal deduction to an instance of this example, consider

p: Most Swedes are tall q: How many Swedes are short?

We start with the protoforms of p and q (see earlier example):

Most Swedes are tall-? — У. Count(G[A is R]) is Q,

n

T Swedes are short-? У. Count(G[A is 5]) is T, where

n

An applicable deduction rule in symbolic form is The computational part of this rule is expressed as

where subject to

What we see is that computation of the answer to the query, q, reduces to the solution of a variational problem, as it does in the earlier discussion of this example—a discussion in which protoformal deduction was not employed.

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