Synthese(Author’s finally accepted version of manuscript)

The final publication is available at Springer via

http://dx.doi.org/ [ DOI :10.1007/s11229-015-0950-7]

Normativity, probability, and meta-vagueness
Masaki Ichinose
Professor/ PhD
Department of Philosophy, Graduate School of Humanities and Sociology
The University of Tokyo
7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-0033, JAPAN
Telephone Number:+81-3-5841-3739

Fax Number:+81-3-5841-3739

Email Address:

ichinose@l.u-tokyo.ac.jp

Abstract:
This paper engages with a specific problem concerning the relationship between descriptive andnormative claims. Namely, if we understand that descriptive claims frequently contain normative
assertions, and vice versa, how then do we interpret the traditionally rigid distinction that is madebetween the two, as ‘Hume’s law’ or Moore’s ‘naturalistic fallacy’ argument offered.

In particular,

Kripke’s interpretation of Wittgenstein’s ‘rule-following paradox’ is specially focused upon in orderto re-consider the rigid distinction.

As such, the paper argues that if descriptive and normative

claims are not mutually exclusive, then we need a new framework with which to understand thisrelationship. In this regard, the paper borrows from concerns with vagueness, particularly using a
degree-theoretic approach in terms of subjective probability, in an attempt to graphically figure out
these differences.

Consequently, the paper tentatively proposes the hyperbola model in which

degrees of normativity and degrees of descriptivity could be expressed and measured.

It is hoped,

as a result, that this tentative proposal will contribute to deepening the debate on vagueness ingeneral.
Keywords:
Normativity, descriptivity, probability, vagueness, rule-following, Kripke0Normativity, probability, and meta-vagueness1

Normativity matters

It is intuitively obvious that a matter of fact which happened to occur once has no implication
for what might happen next. If I happened to sit down on a particular seat at a particular
restaurant, for example, there is nothing that requires me to sit in the same seat the next time I
go to that restaurant. However, if the same action were repeated to the point of accumulation,a habit or a tendency to repeat that action could arise and, under certain circumstances, thehabit might urge us to continue repeating the same event or action. This seems to be inherent
in human nature. For example, if I have sat in exactly the same seat at the same restaurant
every day for a month, I feel obliged to maintain the custom. It’s even possible that the ownerof the restaurant might feel obliged to save the seat for me when I typically come.
This phenomenon is applicable to many territorial issues between countries concerning
small islands. If some fishermen of a particular country happened to stop at an inhabitedisland dozens of times in the past, their country might claim a right of possession of the island,
although other countries could claim the same right because their fishermen have also stopped
at the island in the past. Of course, the answer to this dilemma relies on political or legal
definitions of territory and prior occupation. However, the problem itself serves as anotherexample of an initial, accidental matter of fact (stopping at the island) being repeated often
enough such that it might become a kind of force that urges us to maintain the fact. In otherwords, a sentence describing a fact – ‘We have occupied this island’ – seems to give evidential
support to a derivative normative claim – ‘We ought to occupy this island’.
However, facts that can be described in an indicative manner cannot always give
evidential support to the normative claims that may be derived from them. Rather, and as aresult of experience, it may seem ‘natural’ to distinguish normative claims from descriptive
facts, even if the facts are iterated repeatedly. Suppose, for instance, that a student is routinely
beaten by a fellow student. Could we then justify the claim that the beating of fellow students
ought to be permitted, based on the fact that it has occurred repeatedly in the past? Intuitively,
we are inclined to say no; facts, even if they have occurred many times before, do notnecessarily justify normative claims. This example corresponds with the traditional distinction
between is and ought to, between descriptive and normative claims. David Hume underlined
the distinction in a formulation often referred to as ‘Hume’s Law’: ‘Of a sudden I am
surpriz’d to find, that instead of the usual copulations of propositions, is, and is not, I meetwith no proposition that is not connected with an ought, or an ought not … this ought, orought not, expresses some new relation or affirmation, ‘tis necessary that it shou’d beobserv’d and explain’d’ (Hume 2000, T3.1.1.27). Following Hume’s formulation is the‘naturalistic fallacy’ argument, as presented by G. E. Moore (see Moore 1903) in his refutationof J. S. Mill’s view that we can derive the normative ‘desirable’ from the descriptive ‘desired’(Mill 1987, p. 307). The distinction is certainly also bolstered by modest commonsense. Even1

if many people violate a traffic law as a matter of fact, it is imperative that we still ought toobserve the law, just as a student ought not to be beaten regardless of whether the student has
been beaten before.
Nonetheless, we hesitate to declare that the distinction between descriptive andnormative claims always hold true. The previous example, concerning territorial issues, may
validate this hesitation. There are, however, problematic cases which invoke the connection
between descriptive facts and normative claims far more strongly. I’ll mention two cases, bothof which have been maintained for more than a thousand years. The first is the practice inmany cultures of burying dead people in the ground. On the surface, this seems to be a mere
matter of fact. However, many feel reluctant to abolish this tradition, as we, consciously or
unconsciously, accept the normative sentence ‘We ought to bury the dead’ because of an
overwhelmingly long accumulation of facts. The second case is concerned with the succession
lineage of the Imperial throne in Japan. For the last 125 successions, the Japanese monarchyhas been headed by the male line of the Imperial Family. Apart from a legal stipulation
requiring this, the succession lineage can also be seen as a historical matter of fact. However,
how should successions be managed in the future? For instance, could the current generation
change the tradition such that a successor could be adopted from the female line, especiallyconsidering that the maintenance of the male line is but a historical fact? This is acontroversial issue in Japan, generating numerous debates. However, it seems to me that for
many Japanese people, the change in tradition is seen as somehow beyond their discretionary
power. Can we assume that this generation has some particular authority, especially incomparison to other generations, that would allow them to impose a change? In contemplatingsuch an act, one is frequently overwhelmed by the sheer weight of the historical facts. In thiscase, the descriptive fact, ‘the Japanese Imperial Throne has continually been succeeded by
the male line of the Imperial Family’, seems to strongly imply a normative claim, ‘theJapanese Imperial Throne ought to be succeeded by the male line of the Imperial Family’,irrespective of the present legal concerns. As such, if we questioned when the descriptive fact
became a reason for the normative claim, the example of the succession of the Imperial throne
in Japan appears to be one case in which the distinction between descriptive and normative
claims is not clear-cut.
2Meta-vague predicates

This article’s primary aim is to explore the possibility of a more appropriate model (than the
dichotomous distinction) with which to understand the tangled relationship between
descriptive and normative claims. It does so by outlining a hypothetical model that takes into
account subjective probability. In developing such an argument, I will examine the classicalrule-following paradox, or what has become known in contemporary philosophy as the
Kripkenstein paradox, as a means of engaging with this relationship. The ‘following the rule’phenomenon is employed as a central example because it is a useful means of highlighting the2relationship between descriptive and normative claims.
Before I do so, however, I would like to draw the reader’s attention to two concerns.
Firstly, while I represent normative claims as altogether having the same quality, I am fully
aware that there are different forms of normativity, found, for instance, in the differences
between logical, legal, moral concerns, and the Kantian transcendental type of normativity. In
spite of this, I understand that there is a common characteristic that defines all of these
normative positions, including the Kantian one. Namely, there is always some kind of
sanction that may be imposed if one were to violate the ‘ought to’ dictum. This point will befurther explored below.
Secondly, the argument will be developed presupposing that the complex relationshipbetween descriptive and normative claims can be treated, ultimately, as a phenomenon ofvagueness. Of course, in this instance, the claim is not the same as those problems usuallyassociated with vague predicates, for example, ‘child’, ‘red’, ‘tall’, and so on. In all likelihood,‘descriptive’ and ‘normative’ claims belong to a different order of predicates than the ones
cited here. Indeed, as far as it can be said that the phrase ‘something is red’ is descriptive andthat ‘something is good’ is normative, ‘descriptive’ and ‘normative’ can be characterised as
examples of meta-vague predicates (even though the term ‘meta-vague’ is most appropriately
applied to the meta-distinction between ‘vague’ and ‘sharp’1). Even if this is the case, however,meta-vague predicates do share something intrinsic with the vagueness of other vague
predicates – the predicates admit borderline cases. This requires clarification. For example,
Hilary Putnam once argued there are ‘concepts that defy simple classification as “descriptive
or normative” – concepts like the concept of cruelty’ (Putnam 2002, p. 24)2. The killing of a

1 By ‘meta-vagueness’ I mean the vagueness of predicates which are predicated of sentences
rather than subjects. This corresponds to a semiotic distinction between object-language andmeta-language. Suppose A to be a sentence (that may include a vague predicate like red, child,or good, and so on), and P to be a predicate. Thus, if ‘A is P’ is vague, then the vaguenesshere could be called ‘meta-vagueness’, as this is the vagueness of the sentence. Please notethat meta-vagueness should be differentiated from higher-order vagueness on the boundary
between penumbra and definitely true/false cases. Regarding the vagueness of the predicate‘vague’, Sorensen (1985) strongly suggests that any argument involving a vague predicate,including the predicate ‘vague’ itself, must be ‘a self-refuting thesis’ (Sorensen 1985, p. 136).
I do not believe that the same pessimistic conclusion should be drawn in the case of a
descriptive/normative distinction, as will be shown by my argument as a whole. In any case,the vagueness of ‘vague’ must be investigated further.
2
Richard Dietz, one of my colleagues, suggests that ‘child’ is more illuminating example
than ‘cruel’ as having an inferential role both in descriptive reasoning and in normative
reasoning. I fully agree that ‘child’ is another, maybe more helpful, example of this point. For
the time being, however, I follow Putnam’s example, ‘cruel’, in order to situate my argument3

human that takes fourteen minutes, even under the auspices of a death penalty, is usuallyconsidered cruel3. However, is this affirmation simply concerned with descriptive facts? If we
accept or reject this as cruel, do we, implicitly or explicitly, accept the further normative claimthat we ought to continue or abolish the death penalty? Something like cruelty’s ambivalence,that it can be both descriptive and normative, may also be reflected in the example of theterritorial issues. It seems that when we recognise our fishermen’s stopping at the inhabitedisland in the past dozens of times as a descriptive fact, we tend, implicitly or explicitly, tocommit to a normative claim to support our right to its possession. These examples suggestthat there are numerous instances in which the boundary between descriptive and normative
claims is vague4.
Moreover, a sorites paradox may also arise in analysing these meta-vague predicates.
Continuing with the Japanese succession example, the present Emperor is the 125th on thethrone, a product of 1500 years of history. Faced with such a long and indisputable history, thedescriptive fact of male succession is likely to become normative. However, suppose that thepresent Emperor were only the 10th in a lineage stretching back a mere 150 years. Even this(less imposing) descriptive fact may be enough to compel some people to accept it as
normative. However, what if we were to continually shorten the period of Imperial rule byincreasingly incremental steps? Should it be accepted that the present Imperial system ought
to be retained, even if it began only yesterday? At what point does a descriptive claim becomea normative one? Is it possible to quantify this slippage? It is obvious that a sorites paradox
can arise here, and we might formalize it as follows:

more easily in the current debate on the relation between descriptivity and normativity.
Actually, ‘cruel’ could play an inferential role in descriptive reasoning as well as in normative
reasoning, in that, for example, a dog cruelly treated is damaged.
3
In Japan where the death penalty (by hanging) is still retained under the constitutional
restriction of not imposing cruel punishment, there has been the debate on whether killing by
hanging is cruel or not. It takes about fourteen minutes to end a convict’s life by hanging,but the court judged that it is not cruel by saying that convicts lose their consciousness
instantly after being hanged. Of course, it is still a controversial issue.
4
Dietz also raised a question about whether components of descriptivity and normativity in
my examples only coexist as individual constituents (like different colours in one and thesame picture) so that this does not lead to vagueness of the distinction between them. To thisquestion, I will answer that descriptivity and normativity cannot separately coexist in the samesentence without being interrelated (unlike in the case of colours) because of theircountervailing asymmetry which I will discuss later in section 7 by investigating how weshould react if the relevant sentence conflicted with the matter of fact. Rather what I want tofocus on is the relational fact that the more descriptive, the less normative, and vice versa.
4

(Pre1) ‘The Empire has ruled for 1 minute’ is purely descriptive, having no normative force.
(Pre2) ‘The Empire has ruled for 2 minutes’ is purely descriptive, having no normative force.
··(Con) ‘The Empire has ruled for 1500 years’ is purely descriptive, having no normative
force.5
The initial premises seem to be perfectly acceptable, while the conclusion is clearly
unacceptable, at least from many Japanese people’s point of view. The differentiation at the
beginning and end of the trajectory suggests that there are vague boundaries between ‘purely
descriptive’ and ‘normative’. A structurally similar argument can also be built using theexample of cruelty. Cutting a human being or an animal in two would undoubtedly be seen as
cruel, and therefore we ought not to do it. In this case, the descriptive claim is directly linked
to a normative one. However, what would happen if the organism that was to be cut in two
was much smaller? Should we accept that the cutting in two of a microorganism ought not to
be done? Perhaps it would be described as cruel by some, but we feel no need to derive anormative prescription from the action. However, following the supposition above, we would
be required to admit such a normative claim – and the sorites paradox emerges again. With
this in mind, it is hoped that the argument below will provide a new perspective on thedebates concerning problems of vagueness in general, as well as the debate concerning therelationship between descriptive and normative claims more specifically.
3

Reengaging with the rule-following paradox

Re-examining the rule-following paradox, particularly as discussed by Kripke (1982), may
provide the foundations for developing a new understanding of the problematic relationshipbetween descriptive and normative claims. The rule-following paradox is, moreover,frequently described in relation to Goodman’s (1983) grue paradox, as both are supposed toshare a similar structure. Outlining this common structure provides a useful entry point into
the debate.
Both paradoxes emerge as a result of the tension between specific forms of data, andthe regularity or iteration of that data. The relationship seems to be direct, and yet a

5 Of course, this formulation is made by simplifying the structure of the argument. Precisely
speaking, the Imperial ruling is established according to various elements including historicaland mythological contexts in addition to temporal length. However, this may not be a flaw inthe argument, because such simplification is usually made in constructing the sorites paradox
concerning typically vague predicates like ‘hot’ or ‘child’. What matters is whether the
structure of the argument admits the sorites paradox or not.
5

one-to-one relationship cannot be established. Certainly both paradoxes seem to share asimilar structure. Such a structural understanding can however cause us to overlook a crucial
difference between them. There is, of course, a superficial difference, in that the one isconcerned with meaning while the other with induction. However, meaning and induction
actually complement each other. Induction can be carried out, and exists, through the meaningof the predicates used in the data set, while the meaning of words is invariably inferred
inductively via their past usage and connotations. However, it seems to me that a further
distinction is possible, one which requires a more substantive review of the Kripkenstein
rule-following paradox. Wittgenstein’s own argument will be temporarily ignored, as its
analysis is beyond the limits of this study.
Kripke imagines that ‘68 + 57’ is a computation that I have never performed, having
only ever carried out addition with numbers less than 57. The supposition is perfectly possible.
Answering correctly, I will obtain the answer ‘125’. It is correct, because ‘plus’, as the word
has been intentionally used in the past, denotes a function which, when applied to the symbols
I have called ‘68’ and ‘57’, yields the value ‘125’. The logic of addition has been procedurally
implemented. However, a sceptical challenge could be posed. Namely, what if, when I used
the term ‘plus’ in the past, the answer I intended to produce from the equation ‘68 + 57’ was‘5’? My intention that ‘68 + 57’ should turn out to denote ‘125’, based upon my use of thesymbol ‘+’, cannot rely on the fact that I explicitly gave myself instructions to produce theresult of 125, because by the very hypothesis, I did no such thing. Certainly, I may think I
ought to apply the same function or rule that I have applied so many times before. However,
who is to say what function this is? I gave myself only a finite number of examples (all ofwhich were smaller than 57) in instantiating this function (Kripke 1982, pp. 8–9). Perhaps inthe past I used ‘plus’ and ‘+’ to denote a function called ‘quus’, defined like this:x○+ y = x + y, if x, y < 57
= 5 otherwise.
Who is to say that this is not the function I previously meant by ‘+’? Consequently, if I answer‘68 + 57 = 125’, can I say I am merely applying a previously used rule, even though otherpossibilities may exist, such as ‘quus’? As Kripke explains,
How can I justify my present application of such a rule, when a sceptic could easilyinterpret it so as to yield any of an indefinite number of other results? It seems that myapplication of it is an unjustified stab in the dark. I apply the rule blindly. (Kripke 1982,
p. 17. Emphasis added.)
Obviously, this remark shows that Kripke understands the problem to be a matter ofjustification. Consequently, the mentioning of any number of descriptive facts will notundermine the sceptical challenge, precisely because a justification is a normative claim. This6

point is highlighted in his next remark:When I respond in one way rather than another to such a problem as ‘68+57’, I canhave no justification for one response rather than another … there is no fact about methat distinguishes between my meaning plus and my meaning quus. Indeed, there is nofact about me that distinguishes between my meaning a definite function by ‘plus’ …
and my meaning nothing at all. (Kripke 1982, p. 21)
Following this view, Kripke utterly rejects the resulting ‘dispositional account’ as a
misconception of the sceptic’s problem. In this vein he notes,
Suppose I do mean addition by ‘+’. What is the relation of this supposition to thequestion of how I will respond to the problem ‘68 + 57’? The dispositionalist gives a
descriptive account of this relation: if ‘+’ meant addition, then I will answer ‘125’. But
this is not the proper account of the relation, which is normative, not descriptive. The
point is not that, if I meant addition by ‘+’, I will answer ‘125’, but that, if I intend toaccord with my past meaning ‘+’, I should answer ‘125’. (Kripke 1982, p. 37)
It follows from this remark that Kripke’s formulation ultimately depends upon a dichotomous
distinction between descriptive fact and normative justification. At least in this context, thedistinction works as a necessary condition for his arguments. Without the distinction, Kripke’s
argument does not work at all. In this sense, Wittgenstein’s paradox, as interpreted by Kripke,
is an extended variant of Moore’s naturalistic fallacy argument. Consequently, if this is correct,
we are unknowingly implicated in a naturalistic fallacy whenever we follow any rule solely
because of custom or facts.6

6

It was pointed out by an anonymous reviewer that the absence of a sharp distinction is nottantamount to the absence of a distinction – it is just a vague distinction. So why, the reviewer
asks, is Kripke’s formulation not compromised given the existence of a vague distinction?
This observation is absolutely true. Actually, my aim is not to criticize Kripke’s argument, but
to elucidate the relation between two meta-predicates, that is, descriptivity and normativity.
Therefore, if Kripke in reality admitted the vague boundary between two meta-predicates, myargument would result in bringing to light a hidden structure of Kripke’s argument. For thetime being, however, Kripke so obviously develops his argument at least in this context byappealing to the sharp contrast between descriptivity and normativity that I provisionallypresuppose he uses the sharp contrast between two meta-predicates as a necessary condition
for his argument.

7

4

Mutual permeability rather than penumbra

From a contemporary standpoint, how might one interpret Kripke’s argument? Evidently, it
does not seem convincing to adopt, along with Hume and Moore, a clear-cut dichotomy
between descriptive and normative claims. Moreover, numerous arguments for theinterconnectedness, rather than mutual exclusivity, of descriptive and normative claims havesince been made. Using Putnam’s example of ‘cruelty’ again:‘Cruel’ simply ignores the supposed fact/value dichotomy and cheerfully allows itself
to be used sometimes for a normative purpose and sometimes as a descriptive term …
such concepts are often referred to as ‘thick’ ethical concepts. That the thick ethicalconcepts are counterexamples to the idea that there exists an absolute fact/value
dichotomy has long been pointed out. (Putnam 2002, p. 35)
Needless to say, ‘the absolute fact/value dichotomy’ Putnam highlights is structurally the sameas the division between descriptive and normative claims under discussion. Considering this,we may understand the Kripkenstein argument as critically problematic, as this formulation of
the rule-following paradox depends, entirely or at least crucially, on a clear-cut dichotomy
between descriptive and normative claims.
Equally however, a complete abandonment of the distinction does not seem reasonable,
the result of which would be a kind of bland monism. This is because, as was mentioned in
the first section, it seems that there are instances in which there is a clear difference between
descriptive and normative claims. That a student physically beat another is basically a matterof description and historical fact, whereas inflicting violence on someone is normatively laden(and problematic), irrespective of whether or not it has occurred before (although if weinterpret the physical event of a beating by others as ‘violence’, a similar normative concern
to that of ‘cruelty’ arises).
It follows that there may exist degrees of difference between descriptive andnormative claims, something Kripke’s rule-following paradox might not explicitly take into
account. Indeed, the notion of a ‘degree’ originates from the meta-vague distinction between
descriptive and normative claims. In so doing, what kind or measurement of ‘degree’ may beintroduced? In answering this, it is necessary to carefully re-examine the concepts ofnormativity and descriptivity that make up the Kripkenstein paradox.
Firstly, with regard to normativity, I begin with re-confirming that there is a distinction
between the descriptive facts and normative claims that structures our actual lives. For
instance, if I drive a car at 60km/h, thus violating a traffic law that stipulates that the speed
limit is 50km/h, I am still driving at a speed of 60km/h, as a matter of descriptive fact.
However, this fact does not negate the normative claim that I ought to drive at or slower thana speed of 50km/h. Here it might be said that there is a sharp distinction between what is andwhat ought to be. This is, however, not always the case.
8

Indeed, there may be numerous borderline cases, both practical and theoretical, in
which the distinction between is and ought to can become vague. For instance, consider thefollowing famous proposition, known as Hempel’s raven (Hempel 1965, p. 12ff):(i)

All ravens are black.

In Hempel’s paradox, proposition (i) is treated as a claim that can be empirically confirmed or
falsified, resulting in it being regarded as a descriptive fact. However, apart from Hempel’s
argument concerning non-black, non-raven objects (a yellow banana, for example), yetanother understanding is possible. How should we react, for instance, if we found a pinkraven? One possible reaction to the discovery is to assert that proposition (i) has now beenfalsified, and that we must conclude that not all ravens are black. However, it is also plausible
to say that the pink bird is not actually a raven at all, but a new species. In this instance, theproperty of blackness continues to belong to the definition of the original species of raven.
Consequently,(ii)

All ravens ought to be black.

It seems then that this kind of vagueness between descriptive and normative claims isfrequently found in ordinary language. Actually, Friedrich Waismann, one of the main
members of the Vienna Circle which is usually known for what is called ‘verificationism’,proposed the open texture of empirical concepts, according to which ‘the fact that in manycases there is no such thing as a conclusive verification is connected with the fact that most of
our empirical concepts are not delimited in all possible directions’ (Waismann 1951, pp.
119–120). His open-texture argument is akin to my point about the concept of ‘raven’. 7 From
my point of view, we could say that we treat empirical concepts as purely descriptive when we
do not have to take extraordinary possibilities (like the appearance of pink raven) into account,whereas we are in a position to choose and declare a particular possibility as an only option ina normative way when we have to consider many possible treatments despite dealing withissues in an actual context. Waismann’s open texture argument was initially concerned withtheoretical knowledge, but was later developed in jurisprudence by H. L. A Hart. Hart argues
that precedent or legislation works well in ordinary cases, but the open texture of things will
become apparent when their application is in question (Hart 1961, p.124). One example Hart
provides is the ‘rule that no vehicle may be taken into the park’ (Hart 1961, p.125). This
general rule ordinarily works well. However, if we took an unenvisaged cases like ‘a toy
motor-car electrically propelled’ (Hart 1961, p.126), we would be involved in an open texture,7

The linkage between my argument and Waismann’s open texture argument was pointed out
to me by an anonymous reviewer, for whose insightful observation I am most grateful.
9where we would have to explicitly make a normative decision. As Hart’s remark suggests, theopen texture finally reaches normative judgement in spite of being proposed initially in thecontext of theoretical knowledge. This line of thought supports my argument on the vaguenessbetween propositions (i) and (ii).
Another theoretical example is that of the ontological status of viruses. Are they livingorganisms or not? Should this question be empirically confirmed/falsified or regarded as amatter of definition, a concern limited to logical normativity? Answers to this question would
fluctuate depending upon contexts and answerer. In this instance, the difference between
whether a virus is or ought to be alive is vague. How should we understand such borderline
cases, a product of the meta-vagueness of the distinction between descriptive and normative
claims? One possible path is to adopt a form of supervaluationism, or to use a ‘truth-value
gap’ approach by introducing the notion of a penumbra. The idea can be illustrated thusly:
Diagram 1

Super-false

Penumbra

Super-true

In this case, what matters in the evaluation of the truth is whether ‘sentence X is descriptive’is true or not, or conversely, whether ‘claim X is normative’ is true or not. However, this idea
can still be problematic. It is true that the sentence ‘I am driving at a speed of 60km/h’ isdescriptive in any context in which it corresponds to the physical event. Thus, ‘the sentence is
descriptive’ is also super-true in any precisifications (disregarding higher-order vagueness).
However, could we understand borderline cases that may fall within the penumbra of thisdiagram? For instance, what of the case of cruelty in cutting a water flea in two? Consider the
next sentence:(iii)

Cutting a water flea in two is cruel.

This sentence (iii) is probably true in a descriptive sense (when viewed through a microscope),
but it also carries some (even though a very little) normative implication in that, qua cruelty,‘we ought not to do that’. In other words, what matters here is not whether sentence (iii) is
descriptive or normative (i.e., whether ‘sentence (iii) is descriptive’ is neither true nor false[corresponding to the truth-value gap]), but how sentence (iii) is both descriptive andnormative (i.e., how ‘sentence (iii) is descriptive’ is both true and false [corresponding to thetruth-value glut])’. Thus, it is the degree of interaction that is of concern here; that is, the10extent of descriptivity and of normativity is in question. In this situation it is possible to seethat descriptivity and normativity are mutually permeable, rather than following a penumbral
pattern.
5

Degree by probability

Thus, in sentence (iii) both the implicit descriptive and normative claims interact, further
supporting the idea that we should understand the interaction of the meta-vague predicates interms of degrees. The degree of normativity in sentence (iii) is most likely less than the degreeof normativity inherent in the next sentence (iv):(iv)

Cutting a human being in two is cruel.

In contrast, the degree of descriptivity in sentence (iii) may be more prominent than insentence (iv), as we can observe the associated phenomenon corresponding to sentence (iii)more calmly than the case of sentence (iv), while the degree of normativity in (iv) is greaterthan that of (iii). How can this be understood? One possible answer is that the degrees ofdescriptivity and normativity complement one another. If we follow the standard custom ofdefining a ‘degree’ as a value found between the parameters of 0 and 1, we could plot theabove circumstances in the form of a basic linear function (where the horizontal axis is thedegree of normativity and the vertical axis that of descriptivity):
Diagram 2
Descriptivity

Normativity

11

This formulation may be valid, although perhaps too simple. Firstly, if the ‘degree’ displayed
is to be measured in terms of probability (as will be done below), it satisfies Kolmogorov’s
axiom (see Howson and Urbach 1993, pp. 30–31). Secondly, and following from this, thesignificance of the Kripkenstein argument is strengthened by interpreting the notion of a ‘rule’
as being located at the point where the degree of normativity is 1 (correct) or 0 (incorrect).
Each of these two points requires further discussion.
Firstly, it is necessary to discuss how the difference between a normative anddescriptive claim can be measured in degrees. As has been mentioned, the primary difference
can be shown in the contrast between what ought to be and what is. If ought to is interpreted
generally (so that it includes all of the logical, social, moral, and legal normativities created bylanguage), there seems to be a common axiom – a feeling of compulsory duty or force implicitin the acceptance of a normative statement. Perhaps, and historically speaking, this viewcorresponds to the Humean understanding of ‘necessity’ as ‘a determination of mind’ (seeHume 2000, T1.1.3.14.1). However, how might we measure this force? The theoreticalformulation of a system of degrees is required. Consequently, I propose that the notion of(mainly interpersonal) subjective probability be used (or what might be termed ‘degrees ofbelief’) in the measurement of degrees of compulsory force. In the following example I draw
upon a case of legal normativity, primarily because the case conspicuously highlights thecharacter of normative force. A criminologist, Deryck Beyleveld, once proposed a utilitariansystem with which to measure and evaluate the effect or deterrence of different punishmentson specific criminal activities. His idea, broadly, was that the offence rate will inversely varywith both the likelihood of a sanction being imposed for committing a specific offence, andthe severity of the sanction (Beyleveld 1979, p. 217).
Put differently, Beyleverld’s idea can be understood as the claim that the psychologicalforce that deters us from violating legal norms can be measured in terms of what might becalled the ‘expected severity of sanctions’. The (mainly interpersonal,) subjective severity ofthe expected sanction (abbreviated as SS) is multiplied by the (mainly interpersonal,)
subjective, conditional probability of the sanction being imposed should the norm be violated(PS). The generalised application of this idea may serve as an example of how the degree of
normativity concerning a statement or action can be measured, by emphasising the probabilityor severity of the associate sanction.8 As a result, the degree of normativity (abbreviated as
DN) can be expressed as a function of the expected severity of the sanction that would be
imposed should a particular norm be violated. In this example, the subjective severity of the
expected sanction is defined between the variables of 0 and 1, where 1 is the severest sanction

8 Philip Pettit has also developed a framework for understanding social norms by appealing
to the notion of sanction. He offers the concept of the ‘the intangible hand’, through whichpeople may suffer a sanction as a result of their own actions. This sanction motivates people
to observe social norms (Pettit 1999, p. 225).
12

possible, such as being executed or killed. If we take normativity to be essentially connectedwith rationality (as is usually thought), it does not seem strange to judge the degree of
normativity and consequently, the degree of rationality, from negative reactions to instances of
their violation. Indeed, Joseph Raz has clearly highlighted this using the notion of ‘blame’ as a
sanction: ‘Common are standards which tie irrationality to blame. One’s beliefs are irrational
when one is blameworthy for having them’ (Raz 2000, p. 43). Thus, we can represent this idea
in the following equation, supposing ‘A’ to be a relevant sentence or expression:
Degree of normativity
DN(A) = SS(A) × PS(A)The intrinsic characteristics and mutual permeability of descriptive and normative claims
become immediately apparent when understanding their difference by degrees in this way9.
First, it can be seen that normative claims involve descriptive concerns, because DN isconstituted by a degree of belief, which itself must ultimately be based on descriptive facts.10
Even when we actively observe normative rules, we must also passively accept someunavoidable (descriptive) facts, without which the interactive practice of observing normative
claims cannot begin at all (see Hookway 2000, p. 73, note 8). In order to consciously observe
legal rules, for instance, we must (through experience) be aware of their existence andapplication, and have specific understandings, fears, and concerns with the sanctions that arederivative of the rules should they be violated. We must, in short, know what a legal andcriminal justice system is in order to obey it. All of these processes should then be counted as
the product of the internalisation of innumerable descriptive facts.
From this, a crucial point can be drawn – circumstances simultaneously correlating toa degree of normativity represented by 1 and a degree of descriptivity correlating to 0 (i.e.,[degree of normativity, degree of descriptivity] = [1, 0]) are essentially impossible. This
indicates that the formulation of their difference as a simple linear function (as in diagram 2)
is not viable. Normative and descriptive claims are not dichotomously exclusive, nor can theysatisfy Kolmogorov’s axiom, even if they are frequently and rightly seen as in contrast to one
another.
Secondly, as has been shown, the possibility remains that normative rules can beviolated. Yet, this does not entail that DN(A) = 1 for any A is logically impossible. If theseverest sanction (being killed) is always imposed in the case of a particular action or

9Regarding the degree of descriptivity, I will discuss it further in section 7 below.
10 As far as the degree of difference between a normative and descriptive claim can be
defined in terms of degrees of belief, the Bayesian approach (see Howson and Urbach 1993)
may allow for dynamic interactions or updates, such as by interfacing with the social
environment. This would require further exploration.
13

statement, the DN could possibly be 1 (presupposing that death is the most feared sanction).
However, that does not mean we cannot violate the norm, at least physically, although weshould then resign ourselves to the corresponding sanction. Peter Railton, for instance, has
introduced this idea with regard to the freedom to violate rules or laws:
The must here is not the must of something irresistible – the moral law is normatively,
not actually, ‘inviolable’ … the normative domain must be a domain of freedom aswell as ‘bindingness’. (Railton 2000, p. 3)
Actually, for instance, it is considered a legal norm to observe the traffic law. Nonetheless, it
can be broken, but then we must accept that there will probably be a consequence or sanction.
The same is true of logical norms or rules like the law of non-contradiction. We ought tofollow the norm to communicate with others, but it can be violated. However, our violation of
the norm would cause other people to doubt our intelligence, which might negatively
influence their evaluation of our personality. This should be a kind of social sanction11.
This second point could lead to an interesting problem concerning the Kripkensteinargument. Kripke frequently presumes that the ‘plus’ function, rather than the ‘quus’ function,
should be normatively justified. However, it need not be so. The ‘quus’ function may hold true
simpliciter, in which our counterintuitive impression of the ‘quus’ function may be explainedor understood in terms of the low degree of normativity it holds in relation to the (far moreestablished) ‘plus’ function. As far as this theoretical sense, which presupposes a degree ofdifference, is concerned, Kripke’s argument is viable. The ‘plus’ function should be more(although not entirely) justifiable than the ‘quus’ function simply because of difference intheir degrees of normativity. Even so, we may choose to ignore the ‘plus’ function and adopt
the ‘quus’ function, although in such a case we should accept some kinds of sanction (mostprobably a social sanction, such as being shunned). In other words, there is no true
justification in a strict sense. My argument merely shows that ‘plus’ sounds relatively morejustifiable than ‘quus’ based on empirical grounds using our subjective probability. If theenvironment around us were drastically different, it would be possible for ‘quus’ to sound11

I gave a talk on one part of my idea about DN in connection with the problem of causation
by absence on 3 April 2015 at the 89th Annual Meeting of the American Philosophical
Association Pacific Division, which was held in Vancouver of Canada. Ernest Sosa kindly
gave me some comments, in which he asked how my idea of DN could treat the case ofpeople lacking an ability to obey a norm. That is a quite significant question. For the time
being, as long as I formulate that both SS and PS are measured from a mainly interpersonal
point of view, SS should be evaluated with a consideration about people lacking an ability toobey a norm. But, this requires further clarification in detail, which is one of my futuretasks.
14

more justifiable than ‘plus’. As a result, the notion of justification has to be applied veryloosely when the DN is near 1, depending on our assignment of subjective probability. Thiskind of justification might be expressed as ‘being the most forced to be accepted’.
6

Pure normativity collapsed

The argument that has been made thus far suggests that the rule-following paradox may leadto an analytic stalemate so long as the clear-cut dichotomy between descriptive and normative
claims is maintained despite the numerous of objections and counterexamples that have been
mentioned. Consequently, we should be aware that there are no purely normative claims (suchas would be needed to entirely justify only the ‘plus’ function) that can exist completely
independently of any descriptive facts. In tracing out the parameters of this stalemate, it isnecessary to revisit the Kripkenstein argument once more, to explore the consequences ofseeking purely normative claims. In this example, the Kripkenstein argument will be applied
to itself.
Could we say that we ought to deduce the consequent ‘Q’ from the premise ‘P ⊃ Q’
and ‘P’ according to modus ponens? This would of course be valid in normal circumstances.
However, the Kripkenstein paradox casts doubt on this statement’s validity because eventhough we may have used the logical connectives ‘⊃’ and ‘and’ in the past (and so long as
they maintain logical validity), the possibility remains that in reality we have used otherlogical functions, such as ‘⊃*’ and ‘and*’. As a result, we might derive a different consequent,
such as ‘~Q’, which may be entailed by premises ‘P ⊃ Q’ and ‘P’. This example corresponds
to the contrast between ‘plus’ and ‘quus’. Then, how should we understand the normative
justification, if even logical normativity, which is truly the most fundamental, can be brought
into doubt? This is a desperate state of affairs. As long as we accept the Kripkensteinargument in a literal sense and admit a ‘quus-like’ possibility on a par with our commonsense
ideas like ‘plus’, we have no choice but to be involved in such a hopelessly chaotic situation.
As a result, as far as we take the Kripkenstein paradox seriously, presupposing that the notionof justification applies only in a strict sense, it would be completely impossible to justify
anything.
In contrast to the Kripkenstein paradox, Goodman’s grue paradox may seem moreplausible because although he is concerned with issues of justification the concepts are notapplied so strictly. In Goodman’s argument, the concept of justification remains connected todescriptive facts or our practices of inference (which can themselves be presumed to be
descriptive facts). As he argues,
Rules and particular inferences … are justified by being brought into agreement witheach other. A rule is amended if it yields an inference we are unwilling to accept; aninference is rejected if it violates a rule we are unwilling to amend. The process ofjustification is the delicate one of making mutual adjustments between rules and15

accepted inferences; and in the agreement achieved lies the only justification neededfor either. (Goodman 1983, p. 64)
Unlike the Kripkenstein paradox, Goodman’s grue paradox does not at all depend upon aclear-cut dichotomy between descriptive and normative claims, although, of course, a broad
distinction between the two categories is presupposed. Goodman, rather, emphasises the‘mutual adjustment’ possible between the rules and inferences. This also supports the idea of a‘mutual permeability’, as previously pointed out. This is the intrinsic difference between the
Kripkenstein and Goodman paradoxes.
Moreover, Goodman’s formulation of the grue paradox has little to do with purely
normative claims. By way of example, even if all the emeralds we have hereto seen are green,
this does not imply that we ought to expect that the next emerald will be green. The same istrue of the predicate ‘grue’. What is important in Goodman’s paradox is which propositional
hypothesis is more probable regarding the emerald’s colour, and not whether we ought toconclude if it is true or not. Accordingly, Ian Hacking has described Goodman’s claims as a‘pragmatic vision of justification’ (Hacking 1993, p. 274). Indeed, it is obvious that
Goodman’s primarily focus is on inductive rather than normative justifications. As a result,the following of the rule, without admitting any exceptions (as in logical normativity), is
irrelevant from the outset. It is a matter of degree.12
7

Descriptivity and the degree of unchangeability

Descriptive facts also need to be examined, particularly in relation to the Kripkensteinargument. As has been shown, the Kripkenstein argument concerning the rule-following
paradox, at least on the surface, presupposes a clear-cut distinction between descriptive facts
and their normative justification. However, in addition to showing that this distinction is itself
not well founded, it seems that (in the paradox) descriptive facts may themselves be dubious.
In a word, descriptive facts presented in the Kripkenstein paradox seem to be brute facts (asSearle once called them, in contrast to institutional facts (Searle 1969, pp. 50–53)) that can inprinciple be accurately and uniquely identified in an objective way. Their use, however, may
be doubted, as can be simply shown. Any descriptive fact is only made meaningful through its
linguistic articulation. As a result, those facts depend upon the meaning of words. Thosewords, however, are employed according to a normative framework that determines how theyought to be used, by such conventions as grammar and spelling. Indeed, Railton has declaredthat ‘Spelling is a normative concept’ (Railton 2000, p. 4). As a result, and at a fundamental
level, descriptive facts are intrinsically inseparable from issues of normativity, unless theexistence of brute facts, without any linguistic mediation, can be established (which seems to
12 Sober (1994) provides an actual example of analysing the grue paradox in terms ofprobability.
16

be a naïve form of ontological realism).
Returning to the example of ‘plus’, Kripke argues that ‘there is no fact about me that
distinguishes between my meaning plus and my meaning quus’ (Kripke 1982, p. 21).
Nevertheless, he does acknowledge that there remains the (often hidden) fact that I still meant
to say something, irrespective of whether it was ‘plus’ or ‘quus’. However, should we not also
take into account the possibility that the verb ‘mean’ is itself also involved in therule-following paradox? Do we not need to consider, for example, the possibility that I may
have meant ‘nean’ (which may denote antonyms for objects in a sentence that has never beenused)? If such a possibility is to be seriously considered, the claim may fall into
self-referential, infinite regress. Consequently, this possibility suggests that the Kripkensteinparadox may never be completed as, after all, the most basic word, ‘mean’, falls into theparadox, requiring the paradox to continue. As a result, it must be impossible to formalize theparadox itself in a fixed way.
This concern could also apply to the solution that Kripke provides for therule-following paradox. As is well known, Kripke suggests the solution from a communitarianpoint of view:
Any individual who claims to have mastered the concept of addition will be judged by
the community to have done so if his particular responses agree with those of thecommunity in enough cases … (and if his ‘wrong’ answers are not often bizarrely
wrong, as in ‘5’ for ‘68 + 57’, but seem to agree with ours in procedure, even when hemakes a ‘computational mistake’). (Kripke 1982, pp. 91–92)
In response to this remark, two questions naturally arise. How can we confirm ‘if his
particular responses agree with those of the community’? Furthermore, how can or should we
accept that the individual ‘has mastered the concept of addition’? In point of fact, Kripke
himself asks this question, and explains the concern by making an appeal to sensations:
Since … the adult’s confirmation whether he agrees with the child’s avowal is basedon the adult’s observation of the child’s behavior and circumstances, the fact that such
behavior and circumstances characteristic of pain exist is essential in this case to theworking of Wittgenstein’s sceptical solution … If the individual generally makes his
avowals under the right such circumstances, others will say of him that he has
mastered the appropriate expression. (Kripke 1982, p. 100)
This may be an interesting way of solving the rule-following paradox. However, it seems toalso evade the core problem highlighted by the paradox, and thus does not offer ananalytically useful solution. Indeed, this solution appeals to the process of observation (whichis perhaps a fact in itself), which intrinsically includes many predicates about perceptions,such as ‘seeing’, ‘hearing’, and so on, as long as Kripke focuses on the linguistic aspect of17

observation by remarking ‘others will say of him’. Without those predicates of perception, thesolution would not be attainable. What should we make of these predicates? Would thepossibility of the rule-following paradox arise again in the use of the predicates? It seems thatthe paradox must permeate what we suppose to be brute facts themselves, once we haveaccepted a meaningful version of the paradox. As a result, so far as the Kripkenstein paradox
is concerned, there really are no objective ways of escaping it apart from taking somenormativity into account.13
To conclude the argument above, the Kripkenstein paradox would be
self-contradictory because it (at least on the surface) presupposes a dichotomous, mutually
exclusive distinction between descriptive and normative claims without considering thedegrees of difference (or similarity) implicit in such claims. The Kripkenstein argument
structurally discusses ‘pure normativity’ completely independent of descriptivity, so that it
falls into serious difficulty of formulating itself in a fixed way. However, normativity anddescriptivity, in reality, permeate each other to varying degrees, such that a ‘pure’ form ofeither is impossible. What is of concern, therefore, is how those degrees should be measured.
This question has already been discussed with reference to the degree of normativity (DN).
What, however, of the degree of descriptivity (DD)? It seems that the degree of difference
between normative and descriptive claims is composed of a peculiar, countervailingasymmetry. There is a contrast between them, as I have mentioned, even though a clear-cut,
dichotomous distinction should be denied. It seems that, generally speaking, we can interpretthese two meta-predicates as bound by a correspondence between expressions (such as
sentences, rules, and laws) and phenomena (such as states of affairs, utterances, or activities).
The asymmetrical relationship could be understood by considering how one would deal withthe difference that arises out of a conflict between the two resulting moments:
In case there is a conflict between expressions and phenomena,

13 That descriptive facts must involve some kind of normative claim can also be seen inother cases. For instance, if we connect the notion of normativity with that of rationality, as
Raz discussed, we have to consider normativity as seriously concerning the rationality of
every epistemic belief. As Raz notes, ‘When studying reasons we study normative aspects of
the world … Our ability to reason is central to our rationality in all its manifestations, that isregarding reasons for belief, action, emotion, or anything else’ (Raz 2000, p. 43). If weunderstand the notion of normativity as a kind of guide or map drawn from our active
deliberations on theoretical questions, as well as practical ones (such as Hookway developed),then normativity would conceptually permeate any description: ‘We can examine thenormative standards that guide us when we try to carry out theoretical deliberations andconduct inquiries. These norms will guide us in formulating cognitive goals and selecting
methods of deliberation or inquiry to employ in pursuit of them’ (Hookway 2000, pp. 60–61).
18

Descriptivity: Expressions should be corrected, while phenomena are unchangeable.
Normativity: Phenomena should be sanctioned, while expressions are retained.
The idea of contrast or asymmetry may roughly correspond with that of ‘direction of fit’, asJohn Searle once argued. Searle posits that ‘belief’ has the word-to-world direction of fit,
while ‘desire’ or ‘intention’ has the world-to-word direction of fit (see Searle 1979, p. 3ff).
These correspond to descriptivity and normativity respectively. If this formulation is correct,
then it is not difficult to measure the degree of descriptivity (DD). In other words, we canmeasure DD by referring to the subjective probability concerning the ‘unchangeability’ of thephenomenon under question. For example, suppose that a particularly heavy storm washedover Japan on a particular day in 2013. We may firmly believe that this phenomenon has
already occurred, so that it cannot be undone or changed. It has simply occurred. As a result, if
someone claims that a heavy storm did not wash over Japan at that specific time and place,
one can and should correct them. In this case, value 1 can be applied to the DD of the sentence,‘the heavy storm washed over Japan on that day in 2013’, although even here there are
implicit normative elements, especially with regard to the usage of the concept of ‘Japan’ and‘storm’, not to mention spelling and grammar conventions. These are simply unavoidable,
especially considering that normativity and descriptivity both permeate and are compatible
with one another. These points are applied even to cases concerning the future phenomena, as
the ‘unchangeability’ in question is based on our subjective probability, where we will assign
probability 1 to future events which we are certain will occur (e.g., the sunrise tomorrow). Of
course, purely contingent future events will be assigned lower degrees of unchangeability.
Moreover, in re-examining the borderline example of ‘cruel’, ‘cutting a human beingin two is cruel’ is intended to be a descriptive statement as long as we highlight the physicalaspects associated with the phenomenon. As far as those aspects are concerned, in case there
is a conflict between expressions and phenomena (e.g., a case wherein we say ‘cutting ahuman being in two is NOT cruel’), the statement should be corrected, whereas thephenomenon is itself unchanged. However, the statement is not simply descriptive. Unlike inthe example of the storm, the status of the two meta-predicates is more likely to fluctuate in
the cruelty case. In other words, this case is no less normative than it is descriptive. If
someone performed an action of cutting a human flesh in two, the act could itself be judged
and blamed. As a result, the phenomenon or action (cutting human flesh) is what is sanctioned.
Here the normative rule, ‘we ought not to cut a human being in two’ is meant by ‘expression’in ‘the case where there is a conflict between expressions and phenomena’.
In any case, and in following these considerations, we can propose the next formula in
the form of a conditional probability, signified by DD (supposing ‘A’ to be a relevant sentenceor expression as in the case of DN, and PrA to be the phenomena relevant to ‘A’ ):Degree of descriptivity
DD(A) = P (PrA is unchangeable | A differs from PrA)
19

As a result, we have arrived at a place where we can understand the difference between is andought to more appropriately, by degrees of difference.
8

The possibility of an appropriate model

How then should we re-formulate the relationship between DN and DD to make anappropriate model of the relationship? My argument thus far could be summarised in thefollowing two requirements.
1) The asymmetry between descriptivity and normativity must be reflected in an
appropriate model, although a clear-cut, dichotomous distinction between those twometa-predicates should be denied. In other words, the appropriate model must reflect
such a situation that the more DD becomes, the less DN becomes, and vice versa.
2) Neither pure descriptivity nor pure normativity can obtain. Namely, neither value of[DD, DN] = [1, 0] nor that of [DD, DN] = [0, 1] is possible.
Probably, the simplest model that satisfies those two requirements would be a linear function
expressed by, for instance, DD + DN = a ( > 1). However, this possibility does not seemplausible, for if we accepted it, then we should admit that the sum of DD and DN is always
more than 1. This is not suitable as a model constituted by degrees between 0 and 1 ofparameters (i.e., DD and DN) asymmetrical to each other and not realistic either. Recall
sentence (iii) in section 4 above, that is, ‘Cutting a water flea in two is cruel’. Its DD would bemuch less than 1, as, if someone claims that cutting a water flea in two is not cruel, we are notfully convinced that sentence (iii) is still true (i.e., absolutely unchangeable). Furthermore, theDN of the sentence (iii) is rightly assessed as very low, perhaps near 0, although not exactly
the same as 0. Actually, we can unwittingly cut water fleas when manipulating water in nature,
but nobody is likely to blame us. Thus, in the case of sentence (iii), the sum of DD and DN isless than 1.
The next possible candidate that satisfies those two requirements is, probably, a
hyperbola model, which could be formulaically understood thusly:
DN × DD = k (k is a constant, 0 < k < 1)This hyperbola model seems to meet those two requirements in a more refined way than thelinear function model, as we can smoothly place cases like sentence (iii) in the hyperbola. The
model could be graphically illustrated thus (hypothetically presupposing that cases with adegree of less than zero do not exist):
Diagram 3
20

Degree of descriptivity

Degree of normativity

We can easily recognize that there are some parts of this model where the sum of DD and DN
is less than 1, corresponding to the case of sentence (iii). The diagram illustrates two curves,
representing the difference values of k which correspond to (although not precisely stand for)the degree of truth (or, in other words, the degree of acceptability). For example, logical truths
have the maximum value of k (here in the diagram 3 I tentatively presume that the maximum
value of k is 0.04), whereas vague sentences like ‘a man whose height is 169 cm is tall’ have alower k value. Consequently, if we were to temporarily adopt this model, a tentative answer
could be given to Kripke’s distinction between the ‘plus’ function and the ‘quus’ function. In
point of fact, the ‘plus’ function is located close to the point where DN = 1, with almost thehighest value of k (where its DD is not absolute zero), while the ‘quus’ function has a much
lower DN and much lower value of k (even where its DD is relatively high). Within thescheme about DN, DD, and k, the ‘plus’ function is relatively more justifiable than the ‘quus’
function, as Kripke argues, even though we should not expect its true justification in a strict
sense. Similar answers can be extended to some of the difficulties mentioned before such as
the contrast between ‘⊃’ and ‘⊃*’ or between ‘mean’ and ‘nean’ through considering thedifference of DN of each.
In addition, if we adopted the hyperbola model, we might deduce an interestingsuggestion about the problem of vagueness in general. The hyperbola model seems to match
the paraconsistent dialetheism (based upon the truth-value glut approach) concerning the
problem of vagueness and the sorites paradox, especially with regard to the end zones around
DD = 1 or DN = 1 in diagram 3, no matter how much value k is assigned, as the sum of DD
and DN is more than 1 around those zones. However, if k is assigned a very low value, thesum of DD and DN could be less than 1 in the middle zone, where the hyperbola model could21

match the paracomplete theory (based upon the truth-value gap approach) at least around thezone. In suggesting these, I am following the characteristics given by Field with regard toparaconsistent dialetheism and paracomplete theory (Field 2008, p. 364).14 Although myargument is, for the time being, only concerned with meta-vagueness, I hope that this view
will find a general applicability in broader debates concerning vagueness, suggesting thatparaconsistency could be compatible with paracompleteness.
Admittedly, the hyperbola model, strictly speaking, is far from perfect, though it works
as a springboard for discussion. For one thing, it is not at all clear how to determine the valueof the constant k in relation to DD and DN,15 even though DD and DN can be precisely
measured in terms of subjective probability as argued thus far. This lack of clarity probably
originates from my first strategy of incorporating three distinct kinds of degrees (DD, DN, andk) into a two-dimensional diagram. This point could be illustrated by comparing the next
sentence (v) with sentences (iii) and (iv) above.
(v)

Cutting a teddy bear in two is cruel.

It is instantly apparent that it is vague whether sentence (v) is true or not. This vaguenessshould correspond to how to determine k in my hyperbola model above. Obviously, thevagueness represented above is on a different level from that of arguing about whether thesentence (v) is descriptive or normative. This difference is exactly that between vagueness andmeta-vagueness emphasized in section 2 above. Thus it is not entirely reasonable to represent
this vagueness in the same plane with DD and DN. However, simultaneously, it still seems
that the vagueness represented by sentence (v) is somehow interwoven with DD and DN.
Consequently, in order to correctly analyse the situation, it would be helpful to develop athree-dimensional formalisation, which I am not able to carry out right now.
Secondly, and more fundamentally, I am not entirely convinced that the actual
relationship between DD and DN could be traced as a kind of function in a mathematical

14 According to Field, paraconsistent dialetheism takes P(A) + P(¬A) to be greater than orequal to 1, while paracomplete theory takes P(A) + P(¬A) to be less than or equal to 1 (Field
2008, p. 364). I regard P(A) and P(¬A) as corresponding to DD and DN on the ground that
there must be asymmetry between DD and DN. Of course, this is just a possibility, requiring
more meticulous research. That is another subject for another paper.
15 This crucial drawback was correctly pointed out by my colleague, Richard Dietz. I
appreciate his acute observation. Actually, an anonymous reviewer also raised a question
concerning the value of k. Should there be any constraints on the possible values k may take? I
have no firm view on this question. As for the diagram, I presume that the maximum value fork in the diagram 3 should be 0.04. At the end of the day, I regard this question as empirical innature.
22formalization. It seems we have to take into account the possibility of conducting empirical,
statistical research tied up with experimental philosophy or psychology which might result infinding some irregular or unformalizable relationship between DD and DN actually obtains.
This needs further investigation.
Nevertheless, I do not think that my argument thus far is insignificant. For instance,
my claim on the mutual permeability of descriptivity and normativity could lead to a defenceof Mill’s famous view, resisting Moore’s naturalistic fallacy argument by claiming that thenotion of ‘desired’ seems to be based upon a tacit agreement of normative values – such as‘desirable’ – as well as descriptive facts. Indeed, Raz has noted, ‘Since we desire only what
we think of as worth desiring, our desires are among our responses to perceived reasons’ (Raz
2000, p. 42). This implication will be quite helpful, for example, in re-examining how anaturalistic analysis of moral issues should be evaluated. Additionally, my introduction of theconcepts of DD and DN and how to measure them will compel the discussion forward toward
a more thorough analysis of the intertwined relation between descriptive modes like ‘desired’
and normative modes like ‘desirable’. My ideas on such concepts of degrees would beanalytically applicable to many problems in philosophy. Indeed, the contrast between
descriptivity and normativity is crucial to practical and political issues (e.g., territorial issuesor legislation about the succession of thrones) as well as philosophical discussions. Thus I
hope my view on DD and DN can inform and refine practical discussions in various fields.
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