B8. SCALES
1. Basics
A scale is a set of entities with a partial ordering among them.
Thus, it is a composite entity where the components are the members of
the set and one of the relations is the partial ordering. It is of
course possible for particular scales to have other properties and
other relations.
(forall (s) (1)
(if (scale s)
(and (compositeEntity s)
(exists (s1 e x y s2 s3)
(and (componentsOf s1 s)
(partialOrdering e x y s)
(subset s2 s3)(relationsOf s3 s)
(forall (e1)
(if (member e1 s2)
(exists (x1 y1)
(and (member x1 s1)(member y1 s1)
(subst2 x1 y1 e1 x y e))))))))))
The fifth line says there is a set s1 of components or elements of the
scale. Line 6 says there is a partial ordering e. Lines 7 says there
is a subset s2 of the relations of s, and lines 8 through 12 say that
the members of s2 are instances of the partial ordering.
In more conventional notation, we can think of e as a lambda
expression and x and y as its two bound variables. However, since the
"subst2" predicate we defined in Chapter B3 works equally well on types
and tokens, we don't need to specify that x and y are variables, or
types, or typical elements, or anything else.
Note that we allow other relations in the scale besides the partial
ordering ones; s2 is a subset of the relations of s. It can be the
entire set, so that the ordering relations are the only relations. It
is also possible for s2 to be the empty set, giving us the
uninteresting limiting case of a scale with no ordering relations
among its elements.
We can state the conditions on the arguments of "partialOrdering" as
follows:
(forall (e x y s) (2)
(if (partialOrdering e x y s)
(and (scale s)(arg* x e)(arg* y e))))
That is, s is a scale and e is a relation between x and y.
We will use "inScale" as an abbreviation for being a component of a
scale.
(forall (y s) (3)
(iff (inScale y s)
(and (scale s)(componentOf y s))))
We will frequently refer to the components of a scale as being
"points" on that scale.
It is generally more convenient to speak directly of the partial
ordering relation among elements. We can define a "less than"
relation as follows, using the predicate name "lts" to indicate that
it is relative to a particular scale s.
(forall (e x1 y1 s) (4)
(iff (lts' e1 x1 y1 s)
(exists (e x y e1)
(and (partialOrdering e x y s)
(subst2 x y e x1 y1 e1)))))
Then the standard properties of partial orderings can be defined in
terms of the predicate "lts". The partial ordering is antireflexive.
(forall (x s) (not (lts x x s))) (5)
It is antisymmetric.
(forall (x y s) (if (lts x y s)(not (lts y x s)))) (6)
It is transitive.
(forall (x y z s) (7)
(if (and (lts x y s)(lts y z s))
(lts x z s)))
We can define a "greater than" relation in the obvious way.
(forall (x y s) (8)
(iff (gts y x s)(lts x y s)))
We can define "less than or equal" and "greater than or equal"
relations.
(forall (x y s) (9)
(iff (leqs x y s)
(or (lts x y s)(equal x y))))
(forall (x y s) (10)
(iff (geqs x y s)
(or (gts x y s)(equal x y))))
The antisymmetric property for "leqs" follows from (6) and (9).
(forall (x y s) (11)
(if (and (leqs x y s)(leqs y x s))
(equal x y)))
In Axiom (4) we did not state any restrictions on the arguments x and
y of the "lts" relation. It may seem that they should be points on,
or components of, the scale. But in fact we will just as often be
making comparisons between entities that are "at" points on the scale.
Suppose someone gets up at 7 a.m. and goes to work at 8 a.m. It will
be true that 7 a.m. is before 8 a.m., but it will also be true that
the getting up is before the going to work, and that the getting up is
before 8 a.m., and that 7 a.m. is before the going to work.
Therefore, we will allow the arguments of "lts" to be either points on
the scale or entities that are at points on the scale.
(forall (x y s) (12)
(if (lts x y s)
(and (or (inScale x s)
(exists (x1) (and (at x x1 s)(inScale x1 s))))
(or (inScale y s)
(exists (y1) (and (at y y1 s)(inScale y1 s))))
The "at" relation preserves the partial ordering.
(forall (x1 x2 y1 y2 s) (13)
(and (if (at x1 y1 s)
(iff (lts x1 y2 s)(lts y1 y2 s)))
(if (at x2 y2 s)
(iff (lts y1 x2 s)(lts y1 y2 s)))
(if (and (at x1 y1 s)(at x2 y2 s))
(iff (lts x1 x2 s)(lts y1 y2 s)))))
The top of a scale is the highest point in the scale.
(forall (x s) (14)
(iff (top x s)
(forall (y) (if (inScale y s)(leqs y x s)))))
The bottom of a scale is the lowest point in the scale.
(forall (x s) (15)
(iff (bottom x s)
(forall (y) (if (inScale y s)(leqs x y s)))))
Since the scale is only a partial ordering, it may not have a top
and/or a bottom.
A subscale of a scale has as its components a subset of the components
of the scale and its partial ordering relation is the partial ordering
of the scale restricted to that subset.
(forall (s1 s) (16)
(iff (subscale s1 s)
(and (forall (x) (if (inScale x s1)(inScale x s)))
(forall (x y)
(iff (lts x y s1)
(and (inScale x s1)(inScale y s1)
(lts x y s)))))))
The reverse of a scale is one in which the partial ordering is
reversed.
(forall (s1 s) (17)
(iff (reverse s1 s)
(iff (lts x y s)(lts y x s1))))
Two scales are disjoint if their sets of components are disjoint.
(forall (s1 s2) (18)
(if (and (scale s1)(scale s2))
(iff (disjoint s1 s2)
(exists (s3 s4)
(and (componentsOf s3 s1)(componentsOf s4 s2)
(disjoint s3 s4))))))
A total ordering is a partial ordering in which of any two elements,
one is either less than, equal to, or greater than the other.
(forall (e x y s) (19)
(if (lts' e x y s)
(iff (totalOrdering e x y s)
(forall (x1 y1)
(if (and (inScale x1 s)(inScale y1 s))
(or (lts x1 y1 s)(equal x1 y1)
(lts y1 x1 s)))))))
A scale is potentially a ground, and very frequently we speak of
entities being located "at" some point on a scale.
(forall (s) (if (scale s)(ground s))) (20)
2. Scale-to-Scale Functions
We can extend the notion of "function" to scales by saying that for
a scale to be the domain or range of a function is for its set of
components to be the domain or range.
(forall (f s1 s2) (21)
(iff (function f s1 s2)
(exists (s3 s4)
(and (if (set s1)(equal s3 s1))
(if (scale s1)(componentsOf s3 s1))
(if (set s2)(equal s4 s2))
(if (scale s2)(componentsOf s4 s2))
(function0 f s3 s4)))))
In lines 4 through 7, s3 and s4 are assigned to the appropriate set,
and in the last line "function" on sets and scales both is defined as
"function0" on the corresponding sets.
A scale-to-scale function is monotone-increasing if the mapping
preserves the domain scale's "less than" ordering.
(forall (f s1 s2) (22)
(if (and (function f s1 s2)(scale s1)(scale s2))
(iff (monotoneIncreasing f)
(forall (x1 x2 y1 y2)
(if (and (map f x1 y1)(map f x2 y2)(lts x1 x2 s1))
(lts y1 y2 s2))))))
In Chapter B6, we defined functions as "onto". A function f is "into"
a set or scale s3 if there is a subset or subscale s2 for which f is a
function "onto".
(forall (f s1 s3) (23)
(iff (functionInto f s1 s3)
(exists (s2) (and (or (subset s2 s3)(subscale s2 s3))
(function f s1 s2)))))
3. Constructing Scales
We will have frequent occasion to define particular scales. This is
done by specifying the set of entities that are the components of the
scale, and the relation that is the partial ordering of scale.
(forall (s s1 e) (24)
(iff (scaleDefinedBy s s1 e)
(and (scale s)(componentsOf s1 s)
(exists (x y) (partialOrdering e x y s)))))
For example, we can say that a sequence is a scale whose partial
ordering is the "beforeInSeq" relation. The bottom element is the
first element.
(forall (s s1 e x y) (25)
(if (and (sequence s)(componentsOf s1 s)
(beforeInSeq' e x y s))
(and (scaleDefinedBy s s1 e)
(forall (z)(if (first z s)(bottom z s))))))
Similarly, a set of sets under the "subset" relation is a scale. If
it contains the null set, that is the bottom element.
(forall (s1 e x y) (26)
(if (and (forall (s2)(if (member s2 s1)(set s2)))
(subset' e x y))
(exists (s)
(and (scaleDefinedBy s s1 e)
(forall (z) (if (null z)(bottom z s)))))))
We will see many scales defined in this fashion.
We will often be faced with the problem of characterizing scales where
we cannot define them precisely but wish to place some constraints on
their interpretations. Frequently, the entities we are placing at
points on the scale are associated with sets, and we would like the
ordering on the scale to be consistent with the subset relation on the
associated sets. For example, in Chapter B11 we analyze the notion of
Task A being "more difficult than" Task B. Something is difficult to
the extent that there is a set of obstructions that tend to prevent it
from happening. We would like to capture the idea that the more
obstructions there are, the more difficult the task is. At the very
least, the difficulty ordering for tasks should be consistent with the
subset ordering on their associated sets of obstructions. If Task B
has all the obstructions of Task A and then some, Task B is at least
as difficult as Task A.
In the following definition, s is a scale, and e is an abstract
relation between a set (line 4) and the kind of entities that are
placed at points on the scale (line 5). In lines 6-10, s1 is the set
associated with x1, and s2 is the set associated with x2. If s1 is a
subset of s2, then x1 is less than or equal to x2 on the scale s.
(forall (s e) (27)
(iff (subsetConsistent s e)
(and (scale s)(eventuality e)
(forall (s0)(if (argn s0 1 e)(set s0)))
(forall (x)(if (argn x 2 e)(exists (y)(at x y s))))
(forall (e1 e2 s1 s2 x1 x2)
(if (and (instance e1 e)(argn s1 1 e1)
(argn x1 2 e1)
(instance e2 e)(argn s2 1 e2)
(argn x2 2 e2)
(subset s1 s2))
(leqs x1 x2 s))))))
Suppose we have two scales with the same set of components. Then we
can define a composite scale that is consistent with the two original
scales. For example, suppose the set is points in the United States,
in the first scale the partial ordering (in this case total) is
"northOf", and in the second scale the partial ordering is "eastOf".
Then in the composite scale the partial ordering is at least
consistent with the "northAndEastOf" relation. We may in addition
impose further structure on the composite scale, for example, by
saying that the "northOf" relation takes precedence, giving us a kind
of lexicographic ordering.
The loose constraints on a composite scale are as follows:
(forall (s s1 s2) (28)
(if (compositeScale s s1 s2)
(and (exists (s3)
(and (componentsOf s3 s1)(componentsOf s3 s2)
(componentsOf s3 s)))
(forall (x y)
(if (and (lts x y s1)(leqs x y s2))
(lts x y s)))
(forall (x y)
(if (and (leqs x y s1)(lts x y s2))
(lts x y s))))))
That is, the same set is the set of components of the two original
scales and the composite scale (lines 3-5). If an entity x is less
than an entity y on one of the original scales and less than or equal
to y on the other, then it is less than y on the composite scale
(lines 6-11).
We will have frequent occasion to define composite scales.
4. Qualitative Structure on Scales
There is a range of structures we can impose on scales. These map
complex scales into simpler scales. For example, in much work in
qualitative physics the actual measurement of some parameter may be
anything on the real line, but this is mapped into one of three values
-- positive, zero, and negative. Where the parameter is vertical
velocity, this means we are only interested in whether something is
going up, staying at the same elevation, or going down.
An even simpler scale is one with only two points, where one is less
than the other. Any predication defines such a scale, where the two
points are the predication being false and the predication being true,
and where the former is less than the latter.
When we axiomatize arithmetic in the next chapter, it will also be a
scale, and of course it imposes a very common kind of structure on
other scales, namely, measurements of quantities. In addition, we
will define a "half order of magnitude" structure for scales that is
frequently useful in making approximate judgments.
Here we will define another sort of structure on scales, one reflected
in language. What we have defined so far is adequate for
characterizing the comparative and superlative forms of adjectives --
"taller" and "tallest" -- but not for the absolute form of adjectives
-- "tall". In natural language and in qualitative reasoning we often
characterize something as being in the high or low region of a scale, or
somewhere in the middle, with no more precise characterization of its
location. We will call these regions the "Hi", "Md" and "Lo" regions
of the scale. Each of these predicates is a relation between a scale
s and one of its subscales s1 -- (Hi s1 s).
The Hi, Md and Lo regions of a scale are subscales.
(forall (s1 s) (if (Hi s1 s)(subscale s1 s))) (29)
(forall (s1 s) (if (Md s1 s)(subscale s1 s))) (30)
(forall (s1 s) (if (Lo s1 s)(subscale s1 s))) (31)
There are certain purely scalar properties of these regions. The top
of the scale is in the Hi region of the scale, and in fact is the top
of the Hi region.
(forall (s1 s x) (32)
(if (and (Hi s1 s)(top x s))
(top x s1)))
Similarly, the bottom of a scale is the bottom of its Lo region.
(forall (s1 s x) (33)
(if (and (Lo s1 s)(bottom x s))
(bottom x s1)))
The bottom of the Hi region and the top of the Lo region, as well as
the top and bottom of the Md region, will rarely be known exactly, and
the Hi and Md region will generally overlap, as well as the Lo and Md
regions. Nevertheless, we can say that if a point is in the Lo
region, then it is less than all the points in the Hi region.
(forall (s s1 s2 x y) (34)
(if (and (Hi s1 s)(Lo s2 s)(inScale x s1)(inScale y s2))
(lts y x s)))
It is often useful to go from the absolute form of an adjective to its
underlying scale, for example, from "tall" to the height scale. We
will use the predicate "scaleFor" for this relation.
(forall (s e) (35)
(iff (scaleFor s e)
(exists (s1)
(and (Hi s1 s)
(forall (e1 x y)
(if (and (at' e1 x y s1)(argn x 1 e))
(iff (Rexist e)(Rexist e1))))))))
For example, suppose we have "(tall' e x)", that is, e is the property
of x's being tall. Then s is the height scale, s1 is the Hi region of
the height scale, and whenever we have a relation e1 of x being at a
point in that Hi region, then e1 holds exactly when e holds. That is,
some entity x is tall exactly when x is at a point in the Hi region of
the height scale. The height scale is the "scaleFor" the property
"tall". In line 6 we specify that x must be the 1st argument of e,
because if there are multiple arguments, we need to say which one is
the relevant argument placed on the scale.
There are two primary external theories that a theory of the
qualitative structure on scales should link to. The first is a theory
of distributions and the second is a theory of functionality.
The real distribution of a collection of entities along some scale is
often Gaussian. But Gaussian curves are not something that people can
reason with very well in their everyday life. We will not develop a
commonsense theory of distributions here. But one can imagine going
about it by approximating the Gaussian by means of a step function.
The coarsest approximation would have three steps -- the highest
around the mean and two lower steps to approximate the two tails.
Perhaps the middle step goes from one standard deviation before to one
standard deviation after the mean. Given such a theory, we could
stipulate that, defeasibly, when an entity falls into the region
corresponding to the leftmost step, it is in the Lo region of the
scale, and when it falls into the region corresponding to the
rightmost step, it is in the Hi region. Exactly how such step
functions would approximate the Gaussian would have to be worked out.
A theory of functionality is precisely what we work out in our theory
of goals in Part C of this book. Often when we say that an entity is
tall, we mean that it is tall _enough_ for something or _too_ tall for
something. Discovering that something is recognizing the connection
between qualitative scalar judgments and functionality.
More specifically, we can say that, defeasibly, if something is in the
Hi region of a scale, then that property plays a causal or enabling
role in some agent's goal being achieved or not being achieved. We
can state this as follows:
(forall (e x y s1 s) (36)
(if (and (at' e x y s1)(Hi s1 s)(etc))
(exists (c a g)
(and (member e c)(goal g a)
(or (causalComplex c g)
(and (not' g1 g)(causalComplex c g1)))))))
That is, if e is the property of x being in the Hi region of some
scale, then defeasibly e is part of a causal complex that will bring
about some agent's goal or its negation. Causal complexes are
explicated in Chapter B11.
This axiom does not tell us what the goal is. That has to be
determined from context. But it does alert us to the possible
relevance of such a goal.
Predicates Introduced in this Chapter
(scale s): s is a scale.
(partialOrdering e x y s): e is a partial ordering on the components
of s, where x is less than y.
(inScale y s): y is a component of the scale s.
(lts x y s): x is less than y in the partial ordering
for scale s.
(gts x y s): x is greater than y in the partial ordering
for scale s.
(leqs x y s): x is less than or equal to y in the partial
ordering for scale s.
(geqs x y s): x is greater than or equal to y in the
partial ordering for scale s.
(top x s): x is the highest element in the scale s.
(bottom x s): x is the lowest element in the scale s.
(subscale s1 s): s1 is a subscale of scale s.
(reverse s1 s): s1 is the reverse of scale s.
(disjoint s1 s2): The component sets of scales s1 and s2 are
disjoint.
(totalOrdering e x y s): e, the partial ordering on the components
of s, where x is less than y, is in fact
total.
(function f s1 s2): f is a function from a set or scale s1 onto
a set or scale s2.
(monotoneIncreasing f): Function f is monotone-increasing
scale-to-scale function preserving the
scales' "less than" ordering.
(functionInto f s1 s2): f is a function from a set or scale s1 into
a set or scale s2.
(scaleDefinedBy s s1 e): s is the scale with components s1 and
partial ordering defined by relation e.
(subsetConsistent s e): s is a scale whose ordering is consistent
with the subset ordering among sets
associated by the relation e with
entities placed at points in s.
(compositeScale s s1 s2): s is a composite scale with the same
components as scales s1 and s2 and a
partial ordering consistent with the
partial orderings of s1 and s2.
(Hi s1 s): s1 is the high region of scale s.
(Md s1 s): s1 is the middle region of scale s.
(Lo s1 s): s1 is the low region of scale s.
(scaleFor s e): The property e corresponds to being in the
Hi region of scale s.
In addition, we used the two following predicates that have not yet
been defined, the first from Chapter B11, and the second from Chapter
C??.
(causalComplex c e): The collection c of eventualities is a
causal complex for effect e.
(goal g a): Eventuality type g is a goal for agent a.
We also used the constants 1 and 2.