Hedges: A Study In Meaning Criteria And The Logic Of Fuzzy Concepts

TL;DR: Students of language, especially psychologists and linguistic philosophers, have long been attuned to the fact that natural language concepts have vague boundaries and fuzzy edges and that, consequently, natural language sentences will very often be neither true, nor false, nor nonsensical.

Abstract: Logicians have, by and large, engaged in the convenient fiction that sentences of natural languages (at least declarative sentences) are either true or false or, at worst, lack a truth value, or have a third value often interpreted as ‘nonsense’. And most contemporary linguists who have thought seriously about semantics, especially formal semantics, have largely shared this fiction, primarily for lack of a sensible alternative. Yet students of language, especially psychologists and linguistic philosophers, have long been attuned to the fact that natural language concepts have vague boundaries and fuzzy edges and that, consequently, natural language sentences will very often be neither true, nor false, nor nonsensical, but rather true to a certain extent and false to a certain extent, true in certain respects and false in other respects.

Summary

2. F u z z y LOGIC

• That is, in most cases membership functions will assign values between zero and one to individuals on the basis of some property or properties of those individuals.
• The finite number of perceived distinctions would then result from 'low level' perceptual factors, though perhaps the number of perceived distinctions and their distribution would depend on the shape of the underlying curve and various contextual 5'3" 5'5" 5'7" 5'9" 5'11" 6'1" 6'3" Height Fig. 3 . factors.

H E I G H T

• This can be seen clearly in Figure 5 , where both curves are given.
• Then 'This wall is red or not red' will be true to degree 0.6, according to the given semantics.
• Another fact about the fuzzy propositional logic given above is worth noting.
• Suppose the authors have a one-place predicate TALL.

ITALL(j)I =#TALL (])

• This raises the question of whether there are such things as statements which are necessarily true to a degree.
• The one type of possible example that comes to mind is an arithmetic statement that contains an approximation.

3. H E D G E S

• Let us begin with a hedge that looks superficially to be simple: sort of.
• But above intermediate height the values for 'sort of tall' drop off sharply.
• In fact, (6b) presupposes that Esther Williams is not literally a fish and asserts that she has certain other characteristic properties of a fish.
• Yet in sentences with regular, such as (6b), ( 7b) and (Sb), the truth value of the sentences as a whole depends not upon the literal meaning of the predicates involved, but strictly upon their connotations!.
• The distinction between {14a) and (14b) indicates that the authors must distinguish between those properties capable of conferring some degree of category membership and those properties which happen to be characteristic of category members, but do not confer category membership to any degree at all.

4. F U Z Z Y LOGIC W I T H HEDGES

• Let each predicate F be assigned two values, a vector value ][Fll and an absolute value [FI.
• The authors define the k-tuple (#~1,--., #6~) as the vector value of the predicate F and call each element of the k-tuple a "meaning component'.
• These will, when applied to the vector value of a predicate F, pick out the appropriate meaning components and form a new function, which itself will be a membership function for a fuzzy set.
• So far, the authors have looked only at hedges whose truth conditions depend on vector values.
• He does not intend those exact numbers to be taken seriously.

I / J / I WYTAL

• Whatever the shortcomings of the valuations in ( 6), I think there is something basically right about Zadeh's idea, and if there is, then there is a rather remarkable consequence: algebraic functions play a role in natural language semantics!.
• Two possible ways of describing this were suggested above.
• (1) Restrict fuzzy logic to some finite number of values.
• Second, the number and location of the perceived values seem to depend on the shape of the curve.

5. SOME I N A D E Q U A C I E S OF THE TREATMENT OF H E D G E S I N S E C T I O N 4

• Hedges have barely begun to be studied and I have discussed only a handful.
• I have no doubt that the apparatus needed to handle the rest of them will have to be far more sophisticated.
• Moreover I think that four types of criteria is far to few, though I have not done further investigation.

5.1. Dependence upon Context

• The valuations for hedges given in Section 4 were independent of context.
• But whether a given TV set is technically a piece of furniture will vary with the situation.
• In fact, it would not be surprising to find that which criteria were considered primary and which secondary depended on context.
• Take contexts where current sex is what matters, for example, job applications, sexual encounters, examinations for venereal disease, choice of rest room, etc.
• In such situations, one could imagine that (2b) might be true and (2a) not.

5.2. Modifiers that Affect the Number of Criteria Considered

• Under Zadeh's proposals for the definition of words like VERY and SORT OF, such modifiers affect only the absolute values of the predicates modified.
• First, one can, for the f~ed number of criteria considered in judging mere similarity, require that the values assigned to the various criteria be closer.
• B. Richard Nixon and Warren G. Harding are very similar.
• Secondly, in judging the degree of truth of (2b) versus (2a), one may take fewer criteria into account.
• These considerations show that an adequate account of the meanings of VERY and SORT OF cannot be given simply in terms of how they affect the absolute values of the predicates they modify; one must take into account the way they change the consideration of vector values.

5.3. Some Hedges Must Be Assigned Vector Values

• In the treatment given in Section 4, all of the hedges were assigned only absolute values.
• Given that Sam did not measure up according to the primary criteria, one might accurately say (la), though perhaps nothing stronger.
• One of the things that VERY does, when applied to STRICTLY SPEAKING, is further restrict the number of categories considereal most important: this can be viewed as changing the weights assigned to various criteria at the upper end of the spectrum.
• The same is true of LOOSELY SPEAK-ING, as expressions like VERY LOOSELY SPEAKING show.
• Another thing suggested by these facts is that there may not be a strict division between primary and secondary criteria; rather there may be a continuum of weighted criteria, with different hedges picking out different cut-off points in different situations.

5.4. Perhaps Values ShouM Not Be Linearly Ordered, But only Partially Ordered

• What these hedges seem to do is say there are certain criteria which, if given ~e a t weight, would make the statement true.
• In some respects, Nixon has helped the country.
• But very often, sentences without such hedges are meant to be taken in the same way.
• The situation, of course, gets worse with VERY VERY TALL, VERY VERY VERY TALL, etc., since according to Zadeh's treatment, they all hit the value I at the same place as TALL.
• The reason for the latter suggestion is that as one iterates occurrences of VERY the curve is not shifted further and further to the right, but rather reaches a point of diminishing returns.

5.6. Restrictions on the Occurrence of Modifiers

• (1) Technically, I said that Harry was a bastard.
• That is, TECHNICALLY in (1) seems to be cancelling the implicature that if you say something, you mean it.
• Essentially, the sergeant did order the private to close the window.
• Obviously hedges interact with felicity conditions for utterances and with rules of conversation.

C.

• The question as to what such primitives are in natural language is a fundamental question about the nature of the human mind.
• The question has, of course, been raised innumerable times before, but to my knowledge the possibility that the primitives themselves might be fuzzy has not been discussed.

6.4. Semantics is Not Independent of Pragmatics

• The study of the hedge REGULAR by Bolinger 1972 reveals that sentences with REGULAR assert connotations, not any aspect of literal meaning.
• Since the truth conditions of sentences with REGULAR depend only on connotations, it follows that if connotations are part of pragmatics, then semantics is not independent of pragmatics.
• Since connotations are closely tied to the real-world situation, it seems reasonable to maintain the traditional view that connotations are part of pragmatic information.

6.5. Algebraic Functions Play a Role in the Semantics of Certain Hedges

• Hedges like SORT OF, RATHER, PRETTY, and VERY change distribution curves in a regular way.
• Zadeh has proposed that such changes can be described by simple combinations of a small number of algebraic functions.
• Whether or not Zadeh's proposals are correct in all detail, it seems like something of the sort is necessary.

Perceptual Finiteness Depends on an Underlying Continuum of Values

• Since people can perceive, for each category, only a finite number of gradations in any given context, one might be tempted to suggest that fuzzy logic be limited to a relatively small finite number of values.
• But the study of hedges like SORT OF, VERY, PRETTY, and RATHER, whose effect seems to be characterizable at least in part by algebraic functions, indicates that the number and distribution of perceived values is a surface matter, determined by the shape of underlying continuous functions.
• For this reason, it seems best not to restrict fuzzy logic to any fixed finite number of values.
• Instead, it seems preferable to attempt to account for the perceptual phenomena by trying to fi~mare out how, in a perceptual model, the shape of underlying continuous functions determines the number and distribution of perceived values.

6.7. The Logic of Hedges Requires Serious Semantic Analysis for All Predicates

• In a fuzzy predicate logic with hedges, the notion of a valuation is fundamentally more complex than the corresponding notion in other logics developed to date.
• Nothing is said about whether it has to have wings or a beak, whether it typically flies, what its body structure is, how it reproduces, whether it has feathers, etc.
• In a fuzzy predicate logic with hedges, all these matters must be taken into account in every valuation for the predicate BIRD.
• The Kripke semantics for modal logics is based on the notion of a'possible world', that is, a complete and consistent assignment of truth values to every proposition, in other words, a classical (two-valued) valuation.
• To get Zadeh values, take 1 -Scott's values.

V~(~P) iff not V~_~(P)

• The point is to show how the connectives of fuzzy propositional logic can be defined in terms of the connectives of classical propositional logic and sequences of two-valued valuations.
• The literature on many-valued logics contains extensions of their ' ~' to intermediate values, which are motivated on purely formal grounds.
• But even if the authors restrict every sentence either to making complete sense or being complete nonsense, the supervaluation approach has a serious defect.
• Or the authors could pick 0.5 as a designated value and say that they had a presupposition failure only when the value of P fell below 0.5, that is, when P was more false than true.
• When the authors get down to (3f), however, the degree of truth of (3b) gets lower, and it makes less sense to say (a).

A F N T v i T N F F F F F T I T T T N F N N N [ T N N T F N T F T N F

• There are certain general principles governing the determination of values in these systems.
• If one (or both) of the component sentences has the highest value in the hierarchy, the conjunction has that value.
• The authors keep principles I, II, and III and the hierarchies in (8).
• The authors can now give directions for computing the values for Bochvar-style and Lukasiewicz-style conjunctions and disjunctions in fuzzy presuppositional logics.
• (2) is an interesting curiosity, an intuitively obvious truth of Euclidean geometry which cannot be deduced from Euclid's postulates.

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Title
Hedges: A Study In Meaning Criteria And The Logic Of Fuzzy Concepts
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Lakoff, George
Publication Date
1973
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GEORGE LAKOFF
HEDGES: A STUDY 1N MEANING CRITERIA AND THE
LOGIC OF FUZZY CONCEPTS*
1. DEGREES OF TRUTH
Logicians have, by and large, engaged in the convenient fiction that sen-
tences of natural languages (at least declarative sentences) are either true
or false or, at worst, lack a truth value, or have a third value often inter-
preted as 'nonsense'. And most contemporary linguists who have thought
seriously about semantics, especially formal semantics, have largely
shared this fiction, primarily for lack of a sensible alternative. Yet students
of language, especially psychologists and linguistic philosophers, have
long been attuned to the fact that natural language concepts have vague
boundaries and fuzzy edges and that, consequently, natural language sen-
tences will very often be neither true, nor false, nor nonsensical, but rather
true to a certain extent and false to a certain extent, true in certain re-
spects and false in other respects.
It is common for logicians to give truth conditions for predicates in
terms of classical set theory. 'John is tall' (or 'TALL(j)') is defined to be
true just in case the individual denoted by 'John' (or 'j') is in the set of tall
men. Putting aside the problem that tallness is really a relative concept
(tallness for a pygmy and tallness for a basketball player are obviously
different) 1, suppose we fix a population relative to which we want to define
tallness. In contemporary America, how tall do you have to be to be tall?
5'8"? 5'9"? 5'10"? 5'11"? 6'? 6'2"? Obviously there is no single fixed an-
swer. How old do you have to be to be middle-aged? 35? 37? 39? 40? 42?
45? 50? Again the concept is fuzzy. Clearly any attempt to limit truth
conditions for natural language sentences to true, false and "nonsense'
will distort the natural language concepts by portraying them as having
sharply defined rather than fuzzily defined boundaries.
Work dealing with such questions has been done in psychology. To
take a recent example, Eleanor Rosch Heider (1971) took up the question
of whether people perceive category membership as a clearcut issue or a
matter of degree. For example, do people think of members of a given
Journal of Philosophical Logic
2 (1973) 458-508.
All Rights Reserved
Copyright 9
1973
by D. Reidel Publishing Company, Dordrecht-Holland

HEDGES" A STUDY IN MEANING CRITERIA
459
species as being simply birds or nonbirds, or do people consider them
birds to a certain degree? Heider's results consistently showed the latter.
She asked subjects to rank birds as to the degree of their birdiness, that is,
the degree to which they matched the ideal of a bird. If category mem-
bership were simply a yes-or-no matter, one would have expected the
subjects either to balk at the task or to produce random results. Instead,
a fairly well-defined hierarchy of 'birdiness' emerged.
(1) Birdiness hierarchy
robins
eagles
chickens, ducks, geese
penguins, pelicans
bats
Robins are typical of birds. Eagles, being predators, are less typical.
Chickens, ducks, and geese somewhat less so. Penguins and pelicans less
still. Bats hardly at all. And cows not at all.
A study of vegetableness yielded a similar hierarchical result:
(2) Vegetableness hierarchy
carrots, asparagus
celery
onion
parsley
pickle
Further experiments by Heider showed a distinction between central
members of a category and peripheral members. She surmised that if sub-
jects had to respond 'true' or 'false' to sentences of the form 'A (member)
is a (category)' - for example, 'A chicken is bird' - the response time would
be faster if the member was a central member (a good example of the
category) than if it was a peripheral member (a not very good example of
the category). On the assumption that central members are learned earlier
than peripheral members, she surmised that children would make more
errors on the peripheral members than would adults. (3) lists some of the
examples of central and peripheral category members that emerged from

460
the study:
GEORGE LAKOFF
(3) Category Central Members Peripheral Members
Toy ball, doll swing, skates
bird robin, sparrow chicken, duck
fruit peru, banana strawberry, prune
sickness cancer, measles rheumatism, tickets
metal copper, aluminum magnesium, platinum
crime rape, robbery treason, fraud
sport baseball, basketball fishing, diving
vehicle car, bus tank, carriage
body part arm, leg lips, skin
I think Heider's work shows dearly that category membership is not
simply a yes-or-no matter, but rather a matter of degree. Different people
may have different category rankings depending on their experience or
their knowledge or their beliefs, but the fact of hierarchical ranking seems
to me to be indisputable. Robins simply are more typical of birds than
chickens and chickens are more typical of birds than penguins, though
all are birds to some extent. Suppose now that instead of asking about
category membership we ask instead about the truth of sentences that
assert category membership. If an X is a member of a category Y only to
a certain degree, then the sentence 'An X is a Y' should be true only to
that degree, rather than being clearly true or false. My feeling is that this
is correct, as (4) indicates.
(4) Degree of truth (corresponding to degree of category membership)
a. A robin is a bird.
b. A chicken is a bird.
c. A penguin is a bird.
d. A bat is a bird.
e. A cow is a bird.
(true)
(less true than a)
(less true than b)
(false, or at least very far from true)
(absolutely false)
Most speakers I have checked with bear out this judgement, though some
seem to collapse the cases in (4a-c), and don't distinguish among them.
My guess is that they in general judge the truth of sentences like those in
(4) according to the truth of corresponding sentences like those in (5).
(5) a. A robin is more of a bird than anything else. (True)
b. A chicken is more of a bird than anything else. (True)

HEDGES: A STUDY IN MEANING CRITERIA
461
c. A penguin is more of a bird than anything else. (True)
d. A bat is more of a bird than anything else. (False)
e. A cow is more of a bird than anything else. (False)
That is, some speakers seem to turn relative judgments of category mem-
bership into absolute judgments by assigning the member in question to
the category in which it has the highest degree of membership. As we
shall see below, speakers who judge the sentences in (4) to have a pattern
like those in (5) do make the distinctions shown in (4), but then collapse
them to the pattern in (5).
2. Fuzzy LOGIC
Although the phenomena discussed above are beyond the bounds of clas-
sical set theory and the logics based on it, there is a well-developed set
theory capable of dealing with degrees of set membership, namely, fuzzy
set theory as developed by Zadeh (1965). The central idea is basically
simple: Instead of just being in the set or not, an individual is in the set
to a certain degree, say some real number between zero and one.
(1) Zadeh's Fuzzy Sets
In a universe of discourse X = {x}, a fuzzy set A is a set of
ordered pairs {(x, #a(x))}, where pa(x) is understood as the degree of
membership of x in A. PA(X) is usually taken to have values in the real
interval [0, 1 ], though the values can also be taken to be members of any
distributive complemented lattice.
Union: /~auB = max(~ua, PB).
Complement:
~tA, = 1 -- IrA
Intersection: /~A,B = rnin (/z a,/~B)
Subset: A ~ B iff /~n(x) ~<
Ira(X),
for all x in X.
A fuzzy relation R" is a fuzzy subset of XL
In most of the cases of fuzzy sets that we will be interested in, the
membership function is not primitive. That is, in most cases membership
functions will assign values between zero and one to individuals on the
basis of some property or properties of those individuals. Take tallness
for example. How
tall
one is considered to be depends upon what one's
height
is (plus various contextual factors) - and height is given in terms of