ORI GIN AL ARTICLE
Incarnating Kripke’s Skepticism About Meaning
Eisuke Sakakibara
Received: 15 December 2010 / Accepted: 11 February 2012 / Published online: 6 April 2012
� Springer Science+Business Media B.V. 2012
Abstract Although Kripke’s skepticism leads to the conclusion that meaning does
not exist, his argument relies upon the supposition that more than one interpretation
of words is consistent with linguistic evidence. Relying solely on metaphors, he
assumes that there is a multiplicity of possible interpretations without providing any
strict proof. In his book The Taming of the True, Neil Tennant pointed out that there
are serious obstacles to this thesis and concluded that the skeptic’s nonstandard
interpretations are ‘‘will o’ wisps.’’ In this paper, contra Tennant, I demonstrate how
to construct alternative interpretations of the language of algebra. These constructed
interpretations avoid Tennant’s objections and are shown to be interdefinable with
the standard interpretation. Kripke’s skepticism is, as it were, an incarnate demon.
In contrast, it is currently uncertain whether the same technique is generally
applicable to the construction of an alternative interpretation of natural language.
However, the reinterpretation of those aspects of natural language that directly
relate to numbers seems to be a promising candidate for the development of non-
standard interpretations of natural language.
1 Introduction
Among recent discussions on the philosophy of language, Kripke’s skeptical
argument against the reality of meaning is one of the most disquieting and
controversial. Since this problem is related to diverse themes in contemporary
philosophy, a great many philosophers from different backgrounds have been
involved in this heated debate. The accumulation of literature concerned with this
issue is so immense, and the remarks found in them are so intricately intertwined
E. Sakakibara (&)
National Center of Neurology and Psychiatry, 4-1-1 Ogawa-Higashi, Kodaira, Tokyo
187-8551, Japan
e-mail: [email protected]
123
Erkenn (2013) 78:277–291
DOI 10.1007/s10670-012-9367-6
that visitors unfamiliar with this issue could easily become confused. This paper
examines the entrance to this huge labyrinth.
In his book Wittgenstein on Rules and Private Language, Kripke first introduces
a peculiar function named ‘‘quus.’’ Suppose I had never added a number equal to 57
or greater; Kripke asserts that, no matter how many instances of calculation are cited
as evidence, the possibility that I meant by ‘‘?’’ the following quus func-
tion � cannot be excluded (Kripke 1982, p. 7f.).
a� b ¼ a + b a; b\57ð Þ5 ðotherwiseÞ:
�
After discussing the underdetermination of meaning by the past behavior, Kripke
goes on to examine whether other candidates, such as our dispositions and mental
pictures, could fix the interpretation. I abstain from tracing the path of Kripke’s
entire argument; instead, I investigate deeper into this first line of thought by
focusing on Neil Tennant’s doubts about the multiplicity of possible interpretations.
In his book The Taming of the True, Tennant articulates an objection against
Kripke’s skeptical argument (Tennant 1997, pp. 100–115). The outline of his
argument is roughly as follows: He begins by pointing out that, although Kripke’s
skepticism aims to establish an anti-realistic view of meaning, his argument relies
upon the lemma that there are alternative interpretations of words compatible with
all existing evidence, and that assign a different truth values to some of the as yet
unstated sentences from the standard interpretation.
If a sentence stated in the past is incompatible with all interpretations but the
standard one, the standard interpretation would be preferable to alternative ones.
Such asymmetry would thus render the skeptical argument unpersuasive. The latter
condition ensures that the alternative interpretations are different from the standard
one enough to make difference to the correct behavior in the future. It is evident that
interpretations isomorphic to standard interpretations invest all sentences, whether
or not they have been stated, with the same truth values. But those interpretations
are not what Kripke needed.1
Tennant, then, casts doubt on the very existence of such nonstandard
interpretations. He stresses the fact that if the interpretation of a word is altered,
the interpretation of other words should also be altered, like a domino toppling, to
make up for the inconsistency; thus, if ‘‘?’’ is interpreted to mean quus function, the
word ‘‘addition’’ should be interpreted as quaddition, ‘‘sum’’ as quum, and
‘‘counting’’ as quounting (Ibid., pp. 104–107; Kripke 1982, p. 15f.). Nevertheless,
are not sentences with universal quantifier symbol counterexamples of skepticism’s
alternative hypothesis?2 Let us reconsider the instance that Kripke himself provided.
1 Regarding this point, Kripke’s skepticism is clearly different from the problem discussed by Hilary
Putnam in his ‘‘Models and Reality’’ (Putnam 1980). The logical question underlying Kripke’s problem is
whether there is a model for a set of sentences different from the set of sentences that are true in the
standard interpretation: suppose I had not added a number equal to 57 or greater, and A is a set of
algebraic sentences having been stated so far and regarded as true, the question would be whether or not
A[{‘‘57 ? 68 = 5’’} is a set of non-contradictory sentences.2 Supplementary explanations might be necessary for citing quantified sentences as evidence. First, it is
true that, except for a minority of people who are good at mathematics and logic, most people have never
278 E. Sakakibara
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If I had used ‘‘?’’ to indicate the quus function, would not the sentence expressing
the associative law, namely ‘‘ VxVyVz (x ? y) ? z = x ? (y ? z)’’ be false? For,
‘‘(23 ? 34) ? (-10) = 23 ? (34 ? (-10))’’ is false if ‘‘?’’ means quus.
However, the skeptic would continue by asserting that ‘‘V’’ is nothing but another
symbol that has been used, however large, a finite number of times in the past, and
that a similar line of argument could also be developed regarding ‘‘V.’’ Kripke
prepares a counterargument against the objection based on the associative law: the
truth of what is called ‘‘associative law’’ will be preserved if we reconsider the
symbol ‘‘Vx’’ to mean\for every x that is less than some number h[ (Kripke 1982,
p. 16f, footnote 12). For instance, the ‘‘associative law’’ is saved if we put h = 28.5.
However, interpreting ‘‘Vx’’ as a bounded universal, in turn, makes the following
sentence false: ‘‘:Ax(x [ @ ^ Vy(y [ @ ? y B x)),’’ where the symbol ‘‘@’’
represents the set of natural numbers. In the standard interpretation, this sentence
expresses that there is no largest natural number. However, since there is always a
largest natural number that is less than a specified upper bound, the alternative
interpretation of the sentence is false.
To construct a consistent alternative interpretation, Tennant stresses that the
interpretation of words must be reshaped globally. Kripke and his followers have
never indicated how to perform such global reshaping; instead, they suggest that we
cunningly adjust candidate alternative interpretations whenever an opponent
produces new counterexamples. However, there is no guarantee that we will
eventually arrive at a stable alternative interpretation. In view of this, after
exemplifying three additional obstacles to such global reinterpretation, Tennant
denounces the skeptic’s alternative interpretations as ‘‘will o’ wisps’’ (Tennant
1997, p. 101).
Kripke has confidence in the existence of alternative interpretations. He considers
giving a consistent interpretation for a finite set of sentences to be analogous to
finding a rule in a finite sequence of numbers (Kripke 1982, p. 18). However, an
analogy is too unstable a foundation upon which to build a philosophical
Footnote 2 continued
stated such quantified sentences. However, that does not allow us to disregard them. If it were the case
that quantified sentences leave no room for a nonstandard interpretation, meaning would be indeterminate
only for laypeople, and experienced mathematicians and logicians would enjoy full-blown determinate
meaning. If this were the conclusion, skepticism would be almost dead. For skepticism to retain its power,
it must hold that even if all the sentences having been stated by someone were cited as evidence, the
interpretation would be underdetermined. Second, although Kripke’s skepticism is often understood as a
mere application of Goodman’s argument to linguistics (see for example Allen 1989), a clear distinction
between Kripke’s problem and that of Goodman are found regarding whether or not sentences with
universal quantifier symbols are counted as evidence. The hinge of Goodman’s ‘‘new riddle of induction’’
is that it is logically indeterminate in terms of the way we should generalize singular statements obtained
from our observations so far (see Goodman 1983, pp. 72–81). Therefore, from Goodman’s viewpoint,
quantified sentences such as ‘‘All emeralds are green’’ are the conclusions of induction rather than
evidence that support them. On the other hand, evidence that supports a certain interpretation of words
includes all the sentences that have been stated and regarded as true, whether or not they are quantified.
Since human creatures not only perform concrete calculations but also discuss general theorems of
algebra, Kripke, and not Goodman, must tackle the problems brought about by those quantified sentences.
Incarnating Kripke’s Skepticism 279
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demonstration, since we can draw another analogy that indicates the contrary. To be
sure, the total number of sentences that have been stated is finite, while the total
number of words that have been used in the history of humankind is far less. We are
faced with a dilemma here. On the one hand, according to the principle of
compositionality, the reinterpretation of sentences is possible only through the
reinterpretation of some of the words contained in those sentences. On the other
hand, the reinterpretation must meet the condition that the truth value of all the
sentences that have ever been stated must remain unchanged. Then, is not the
question ‘‘In how many ways can we give alternative interpretations?’’ analogous to
the question ‘‘How large would the degree of freedom for solutions be if there were
simultaneous equations, the number of which were far greater than that of
variables’’? If this analogy holds, the fact that there is an interpretation, namely the
standard one, seems to be miraculous in the first place, and the existence of still
another interpretation is no longer self-evident. When two analogies collide with
each other, an argument based upon analogy takes us nowhere. We need strict proof.
Tennant insists that the burden of proof is on the skeptic. To be sure, burden shifting
is unproductive during philosophical discussions. Yet, complaining about the burden
shifting is just another unproductive discourse. I am going to respond, albeit partially,
to his challenge head-on. This study principally aims to construct a nonstandard
interpretation of the language of algebra. For reasons of convenience, I restrict the
domain of discourse to real numbers. However, the method employed here is
sufficiently general in that it is instinctively clear that the method can be applied to
more advanced algebra dealing with imaginary numbers, matrices, and so forth. The
construction will be demonstrated in Sect. 2, and the three obstacles raised by Tennant
will be illustrated and answered in Sect. 3. Kripke’s skepticism is, as it were, not an
elusive will o’ wisp but an incarnate demon. This incarnation makes the skeptical
argument more urgent and may lead to revelations about its fundamental nature.
In Sect. 4, I demonstrate that the newly proposed nonstandard interpretation is
interdefinable with the standard interpretation. In other words, I show that if a
nonstandard interpretation proposed in Sect. 2 is definable from the standard
interpretation, the standard interpretation can also be defined from the nonstandard
interpretation. In ‘‘The New Riddle of Induction,’’ Goodman stresses that ‘‘grue’’
and ‘‘bleen’’ are interdefinable with ‘‘green’’ and ‘‘blue’’ (Goodman 1983, p. 79f).
Moreover, both directions of definition are symmetrical in that their definientia
contain reference to time. Goodman does so in order to sweep away the suspicion
that employing novel notions such as ‘‘grue’’ and ‘‘bleen’’ in induction is
illegitimate because, in contrast to ‘‘green’’ and ‘‘blue,’’ those notions refer to time
and are therefore not ‘‘purely qualitative.’’ In contrast, definitions between the
standard interpretation and nonstandard interpretations are almost, though not
exactly, symmetrical. Interdefinability with the standard interpretation is not the
requirement for the skeptic’s alternative interpretations. Yet, if they are interde-
finable with the standard interpretation, some approaches to rebut the skeptical
argument will be blocked.
In Sect. 5, I briefly discuss the possibility of alternative interpretations of natural
languages, borrowing Kripke’s uses of ‘‘grue’’ as an instance. Although the general
applicability of the techniques developed in previous sections to natural languages is
280 E. Sakakibara
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beyond the scope of this study, the reinterpretation of those aspects of natural
language that directly relate to numbers seems to be a promising candidate for the
development of nonstandard interpretations.
2 Interpretation Q
Let interpretation C be the standard interpretation of the language of algebra. The
skeptic’s alternative interpretation must (1) aberrantly interpret one or more of the
algebraic symbols employed in the past, (2) preserve the truth value of all the
sentences that have been stated so far, and (3) assign a different truth value to some
of the as yet unstated sentences from interpretation C. In this Section, I demonstrate
how to construct a satisfying interpretation, referred to as interpretation Q, from
interpretation C in two steps.
In the first step, the interpretation which assigns aberrant denotations to numerals,
named interpretation N, is proposed. Since it is shown that interpretation N satisfies (2)
and (3) only when it does not satisfy (1), interpretation N cannot be the desired
alternative interpretation. In the second step, interpretation N is transformed into
interpretation Q, which assigns aberrant meanings to function and predicate symbols.
Those meanings are altered in a consistent manner by applying the operation called
Qfication. Finally, it is shown that interpretation Q satisfies all three conditions.
2.1 Step One: From Interpretation C to Interpretation N
In the first step, let us consider interpretation N, which interprets numerals
aberrantly. In interpretation N, following the famous example of the pupil who
appears in Wittgenstein’s Philosophical Investigations (Wittgenstein 2001, para-
graph 185), numbers larger than 1000 increment half as fast as their notations do,
viz. ‘‘1’’ means 1, ‘‘2’’ means 2, ‘‘1000’’ means 1000, but ‘‘1002’’ means 1001,
‘‘1004’’ means 1002, and ‘‘3000’’ means 2000. Note that the notation of
intermediary numbers follows a similar path (e.g., ‘‘1003’’ means 1001.5).
Let q be the following function:
q xð Þ ¼ xðx� 1000Þ2� x� 1000ðx [ 1000Þ:
�ð2:1Þ
q is a bijective function and the inverse function q-1 exists. Interpretation N is
defined as follows:
Interpretation N is the interpretation in which the canonical representation of
the number x in decimal notation is taken to mean q-1(x), and symbols other
than numerals are interpreted as in interpretation C.
I will say the ‘‘bending point’’ of function q is 1000, following Simon Blackburn
(Blackburn 1984, p. 290f). Interpretations N and C provide the same denotation to
numerals equal to or below the bending point, and yet, assign different denotations
to numerals beyond the bending point. Those who adopt interpretation N, if asked to
continue the series ‘‘2, 4, 6, …’’ by adding 2 serially in the same way, will answer
Incarnating Kripke’s Skepticism 281
123
‘‘8, 10, …, 998, 1000, 1004, 1008, ….’’ In interpretation N, all the following
sentences are true:
‘‘30� 30 ¼ 900; ’’‘‘700þ 400 ¼ 1200; ’’
‘‘30� 40 ¼ 1400; ’’‘‘120� 10 ¼ 60� 20; ’’
‘‘1500þ 500þ 700ð Þ ¼ 3900;’’
‘‘For every x, y, and z, (x ? y) ? z = x ? (y ? z),’’ and
‘‘If p is a prime and a is an integer coprime to p, ap-1 : 1 (mod p).’’
(Fermat’s little theorem)
The important point is that the extensionality of the language of algebra warrants
that interpretations C and N give the same truth value to a sentence, however
complicated it may be, as long as it does not contain a numeral that represents a
number larger than the bending point. We then have the following proposition:
Proposition C–N: A sentence of algebra that does not contain a numeral which
represents a number larger than the bending point is true in interpretation N if
and only if it is true in interpretation C; when a sentence contains such
numerals, that correspondence is not generally sustained.
In the above-mentioned example, the bending point where interpretation C and N
split is 1000. Yet, the bending point can be altered to an arbitrary number by
modifying the definition of q. One possibility is to set the bending point to be the
largest number mentioned so far. In that case, interpretation N preserves the truth
value of every sentence in algebra that has been stated thus far, while it assigns a
truth value different from interpretation C to sentences that contain numerals that
have yet to be mentioned. In other words, interpretation N satisfies conditions (2)
and (3) of the requirement. However those conditions are satisfied only when the
difference between interpretations N and C is confined to the denotation of numerals
that have never been used so far, and condition (1) is not satisfied. Hence,
interpretation N cannot be regarded as an adequate example of what Kripke has
been seeking. The existence of a nonstandard interpretation such as N reflects the
trivial fact that you can freely interpret the numerals that have never been used in
the past because no sentence in the past conditions their denotation.
2.2 Step Two: From Interpretation N to Interpretation Q
I now refine interpretation N. The desired interpretation alters the denotations of the
symbols that have been employed repeatedly, but continues to have the same
character as interpretation N. Let us return to the case in which (2.1) holds. For
example, ‘‘1500 ? (500 ? 700) = 3900’’ is false in interpretation C, but true in
interpretation N because the following is true:
282 E. Sakakibara
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q�1 1500ð Þ þ q�1 500ð Þ þ q�1 700ð Þ� �
¼ q�1 3900ð Þ: ð2:2Þ
Can we make the same sentence true when we interpret numerals in the standard
manner, but adjust the interpretation of ‘‘?’’ and ‘‘=’’ instead? The basic idea for
doing this is to detach q-1 from the interpretation of numerals and attach it to the
interpretations of function symbols and predicate symbols. Let us call the required
modification of interpretations Qfication, and let the Qfied function f and predicate P
be written as Qf and QP, respectively. Then, Q? and Q= have to be defined in such
a way that the following holds:
1500 Q+ 500 Q+ 700ð ÞQ= 3900: ð2:3Þ
As a first approximation, let us consider Q? and Q= to correspond to each shaded
part of the formula below:
However, this arrangement is inconsistent because inner Q? and outer Q? are
treated differently. Hence, let us insert q-1q, which is equal to the identity
function, into the spaces indicated by the black arrows above, and redistribute
qs and q-1s.
This time, inner and outer Q? become the same function. Q? and Q= are
defined as follows:
a Q+ b ¼defq q�1 að Þ þ q�1 bð Þ� �
;
a Q= b,def
q�1 að Þ ¼ q�1 bð Þ:
By interpreting ‘‘?’’ to mean Q? and ‘‘=’’ to mean Q= , ‘‘1500 ? (500 ?
700) = 3900’’ becomes true.
The procedure of Qfication can be systematically formulated. First, numerical
functions, predicates, and logical constants have their inputs and outputs and are
considered as functions in a broad sense. The difference between them resides in the
kind of inputs that they accept and the kind of outputs that they produce. The inputs
and outputs of numerical functions are both numbers. The inputs of predicates are
numbers, whereas the outputs of predicates are formulae, as are both the input and
the output of logical constants. Then, the Qfication of the arbitrary n-ary function (in
a broad sense) F is formulated as follows:
The procedure of Qfication: For every natural number k (1 B k B n), if the
k-th input of F is a number xk, then replace xk with q-1(xk), and if the output
of F is a number y, then replace y with q(y).
The point of the procedure is to shift q-1, which belongs to the meaning of
numerals in interpretation N, to the meaning of function symbols. Replacing the
Incarnating Kripke’s Skepticism 283
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output y with q(y) if y is a number is necessary to cancel q-1, which belongs to
another function that takes the output of the former function as an input. According
to the above-mentioned definition, the Qfication of arbitrary n-ary numerical
function f is formulated as follows:
Qf x1; x2; . . .; xnð Þ ¼defq f q�1 x1ð Þ; q�1 x2ð Þ; . . .; q�1 xnð Þ� �� �
: ð2:4Þ
For example,
a Q� b ¼defq q�1 að Þ � q�1 bð Þ� �
;
Qsin xð Þ ¼defq sin q�1 xð Þ
� �� �:
The Qfication of n-ary predicate P is formulated as follows:
QP x1; x2; . . .; xnð Þ,def
P q�1 x1ð Þ; q�1 x2ð Þ; . . .; q�1 xnð Þ� �
: ð2:5Þ
For example,
a Q\ b,def
q�1ðaÞ\q�1ðbÞ;
a Q2 @,def
q�1 að Þ 2 @:
The operator symbol ‘‘[,’’ together with the symbol of a set on its right, is regarded
as a one-place predicate symbol (e.g., ‘‘[ @’’ is a one-place predicate symbol that
denotes \* is a natural number[). Since logical constants are functions of which
neither the input nor output is a number, Qfication does not alter the meaning of
logical constants. Note that Qfication does not alter the meaning of the equal sign
either, for q-1(x) = q-1(y) , x = y. In the following, ‘‘Q’’ will be omitted from
logical symbols and equal signs for simplicity. The procedure of Qfication is
applicable to still another type of functions. This is discussed in Sect. 3.
At length, the interpretation Q is characterized as follows:
Interpretation Q is the interpretation in which every function symbol denotes
the Qfied version of what it denotes in interpretation C, and the denotation of
numerals are identical with interpretation C.
For instance, in interpretation Q, ‘‘Vx(x [ 2 ? x ? x \ x 9 x)’’ means Vx(x
Q[ 2 ? x Q? x Q\ x Q9 x).
The most important aspect is that function q is the isomorphism between
interpretations N and Q.3 The isomorphism theorems state that every sentence of
3 Let SN and SQ be the denotations of a symbol S in interpretations N and Q, respectively. We have the
following three conditions:
(i) From the definition of interpretation N, for every numeral n,
q nNð Þ ¼ nQ: (2.6)
(ii) From (2.4), for every n-ary function symbol f and for every series of numbers u1; u2; . . .; un;
fQ u1; u2; . . .; unð Þ ¼ q fN q�1 u1ð Þ; q�1 u2ð Þ; . . .; q�1 unð Þð Þ� �
:
By substituting q(ux) for ux for every natural number x (1 B x B n), we have
q fN u1; u2; . . .; unð Þ� �
¼ fQ q u1ð Þ; q u2ð Þ; . . .; q unð Þð Þ: (2.7)
284 E. Sakakibara
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algebra is true in interpretation Q if and only if it is true in interpretation N. From
this fact and proposition C–N, we have the below proposition:
Proposition C-Q: A sentence of algebra that does not contain a numeral that
represents a number larger than the bending point is true in interpretation Q if
and only if it is true in interpretation C; when a sentence contains such
numerals, that correspondence is not generally sustained.
In the above-mentioned example, the bending point where interpretations C and Q
split was 1000. But the bending point can be altered to an arbitrary number by
modifying the definition of q. One possibility is to take the bending point to be the
largest number that has ever been stated in the history of the world. The
interpretation Q with such a bending point preserves the truth value of any algebraic
sentence that has been stated so far. And yet, the interpretation of function symbols
and predicate symbols are completely different. It is when a sentence containing a
numeral that has never been employed by humankind appears that those who adopt
interpretation Q and those who adopt interpretation C begin to behave differently.
Such a sentence will be stated in the future. This is the crack in the edifice of which
the skeptic takes advantage.
3 Tennant’s Three Obstacles
In this section, I illustrate three obstacles exemplified by Tennant and show that the
alternative interpretations constructed in the previous section are impervious to his
objections. To reformulate Tennant’s objections, a function that takes a set as its
argument and quantification over predicates is necessary. Therefore, this section is
also a brief illustration of how to apply the notion of Qfication to such an extension.
Tennant first mentions the difficulties that arise when the universal quantifier
symbol is reinterpreted (Tennant 1997, pp. 108–110). He describes a couple of
difficulties, but the principal one is that if the universal quantifier symbol is
interpreted to mean some bounded universal, as in Kripke’s footnote 12, the
descriptions to specify how quantification is bounded should vary as the universe of
discourse varies. This indicates that the universal quantification turns into a sortal
sensitive notion. For instance, when the universe of discourse is numbers, ‘‘Vx’’ is
interpreted, say, to mean \ for every x that is less than some number h[. However,
when quantification is ranged over terms rather than numbers, ‘‘Vx’’ must be
interpreted to mean, say, \ for every x that is a member of some set of terms S[.
Footnote 3 continued
(iii) From (2.5), for every n-ary predicate symbol and for every series of numbers, u1; u2; . . .; un;
PQ u1; u2; . . .; unð Þ,PN q�1 u1ð Þ; q�1 u2ð Þ; . . .; q�1 unð Þð Þ:By substituting q(ux) for ux for every natural number x (1 B x B n), we have
PN u1; u2; . . .; unð Þ,PQ q u1ð Þ; q u2ð Þ; . . .; q unð Þð Þ: (2.8)
(2.6), (2.7), and (2.8) comprise the necessary and sufficient condition for q to be an isomorphism
between interpretation N and Q.
Incarnating Kripke’s Skepticism 285
123
Therefore, ‘‘V’’ should be differently interpreted from context to context. This seems
to destroy the uniformity of interpretation.
His second objection begins by pointing out the fact that for every natural
number n, there are exactly n ? 1 addition sums of natural numbers with the result
n (Ibid., pp. 110–112). He described it as a metamathematical condition. Yet, since
the number of addition sums is equated with the number of ordered pairs of
summands, this fact is formulated more simply as the following:
‘‘8n n 2 @ ! Num a,bð Þjaþ b ¼ n ^ a,b 2 @f gð Þ ¼ nþ 1ð Þ:’’ ð3:1Þ
Here, Num(S) represents the number of elements (in other words, the cardinality) of
the set S. Needless to say, (3.1) is true in interpretation C. However, as Tennant
points out, (3.1) would be false if it were interpreted, as Kripke does, with ‘‘?’’
meaning quus; this is because there are more than six ordered pairs of ‘‘summands’’
with the result five. In fact, there is an infinitely many such pairs, including (57, 68).
Hence, the hypothesis that ‘‘?’’ means quus is not as well supported by evidence as
is the standard hypothesis.
Third, Tennant draws our attention to the fact that if there are exactly n F’s and
exactly m G’s, and nothing is both F and G; there are exactly n ? m things that are
F or G (Ibid., pp. 112–114). As Tennant correctly recognizes, this is a logical truth
that can be formulated with multiple quantification sans numerals or addition signs.
Then, does not this logical fact fix the meaning of ‘‘?,’’ for sentences such as the
following
‘‘8F8G :9x F xð Þ ^ G xð Þð Þ ! Num xjF xð Þf gð Þ þ Num xjG xð Þf gð Þð¼ Num xjF xð Þ _ G xð Þf gð ÞÞ’’ ð3:2Þ
must be true in any candidate interpretation?
Tennant does not think that those three examples are the only obstacles to
Kripke’s skeptical argument; he also does not think that they are proved to be
insoluble in principle. Instead, his strategy is to impress the difficulty with which the
skeptics, having once interpreted one word aberrantly, are to reinterpret adjacent
words one after another, as well as how bizarre these necessary reinterpretations
become. I have already demonstrated how to construct the skeptic’s alternative
interpretation Q. In the following, it is shown that interpretation Q deftly avoids
three obstacles raised by Tennant.
Tennant’s first objection is directed toward the nonstandard interpretations that
alter the meaning of logical symbols, especially the universal quantifier symbol.
Since interpretation Q does not manipulate the meaning of logical symbols, the first
obstacle is irrelevant to interpretation Q.
Let us then tackle the second problem. Is (3.1) true in interpretation Q? At first
sight, this does not seem to hold. Consider the case in which q is defined as in (2.1).
If we put n = 1004, there are only 1003 ordered pairs of ‘‘natural numbers,’’ the
‘‘sum’’ of which is 1004; to be concrete, they are (0, 1004), (1, 1002), (2, 1000), …,
(999, 3), (1000, 2), (1002, 1), and (1004, 0). Note that 1001 and 1003 are not
‘‘natural numbers’’ in interpretation Q because ‘‘1001 [ @’’ and ‘‘1003 [ @’’ are not
286 E. Sakakibara
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true in interpretation Q. On the other hand, ‘‘1004 ? 1’’ means 1004 Q? 1 and is
equal to 1006 in interpretation Q.
The gap exists because we have overlooked the fact that the operator Num must
also be Qfied. The procedure of Qfication tells us how to Qfy the operator Num.
Since Num is a function the input of which is a set and the output of which is a
number, QNum is defined as follows:
QNum Sð Þ ¼defq NumðSÞð Þ: ð3:3Þ
Since q(1003) = 1006, the gap is filled. In general, the ‘‘number’’ of ordered pairs
of ‘‘natural numbers’’ having the ‘‘sum’’ of n is q(q-1(n) ? 1), which is always
equal to n ‘‘plus’’ 1.
Similarly, the third problem is resolved. ‘‘?’’ must mean the plus function to
make (3.2) true, as long as ‘‘Num’’ is interpreted in a standard way. However, Num
is also Qfied in interpretation Q. It is easily found that if ‘‘Num’’ means QNum, ‘‘?’’
must mean Q? in order for (3.2) to be true.
The second and third problems raised by Tennant are thus avoided by
reinterpreting ‘‘Num.’’ Moreover, the way ‘‘Num’’ is reinterpreted is along the
procedure of Qfication. Therefore, it is neither ad hoc nor bizarre.
4 Interdefinability
Interdefinability with the standard interpretation is not a requirement for the
skeptic’s nonstandard interpretation. Nonetheless, if this condition is satisfied, a
certain line of objections against Kripkean skepticism is blocked. Such an objection
attempts to exclude nonstandard interpretations, relying on the principle that among
the candidate meanings, symbols have the logically simplest or the most logically
prior meanings, unless there are special reasons to suppose otherwise. This
objection is discussed critically by Kripke himself (Kripke 1982, p. 37f). The
interdefinability buttresses his position, for it overturns the very assumption that
there is asymmetry in simplicity or logical priority among candidate interpretations.
Therefore, proving interdefinability is meaningful in that it clarifies to which sorts of
objections skeptical arguments are immune.
Quus function is definable in terms of the plus function (in combination with
disjunction and inequality relation), whereas the plus function is indefinable in
terms of the quus function, since the range of the quus function is less than 114.
Kripke suggests that the meaning of ‘‘V’’ may be altered to bounded universal
quantification. Yet, this suggestion also disregards the interdefinability between the
conventional interpretation and novel interpretation because (unbounded) universal
quantification is not definable from bounded universal quantification.
In this section, I demonstrate that Qfied functions and predicates proposed in
Sect. 2 are interdefinable with the original functions and predicates; however, before
doing that, let us first deepen our understanding of function q.
An infinite number of variations of interpretation Q can be constructed by
modifying the definition of q. The only constraint imposed on q is that it must be
Incarnating Kripke’s Skepticism 287
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bijective for there to exist an inverse function of q. If you want the correspondence
between interpretation C and interpretation Q to hold within a certain range, you
have only to arrange the definition of q so that q(x) = x holds within the same
range. Furthermore, the minimal requirement to preserve the truth value of every
sentence that has been stated so far is that q(x) = x holds if the numeral that denotes
number x has been used at some point in human history. Hence, q(x) = x need not
hold for every intermediary number x as long as the numeral that denotes x has
never been used before. The function q, as long as it is bijective, need not be
definable by combining known functions. However, the function q that meets the
requirement can, in principle, be defined in infinitely many ways in terms of known
functions, since the number of numerals that has been used by someone so far is,
however large it may be, finite. It is for this condition of function q that giving a
consistent interpretation for a finite set of sentences is analogous to finding a rule in
a finite sequence of numbers. Consequently, Kripke’s analogy is appropriate. I
stress, however, that a strict proof is necessary to reach this conclusion.
Because Qfied functions and predicates are defined in terms of original functions
and predicates in combination with function q, Qfied functions and predicates are
not definable if function q itself is not definable in terms of known functions and
predicates. Therefore, let us focus our attention on the cases where q is definable by
combining known functions and predicates and where q(x) = x holds as long as x is
equal to or less than the bending point, as is found in (2.1).
By solving (2.4) and (2.5) for f and P, respectively, we have the following:
f x1; x2; . . .; xnð Þ ¼ q�1 Qf q x1ð Þ; q x2ð Þ; . . .; q xnð Þð Þð Þ; and ð4:1ÞP x1; x2; . . .; xnð Þ,QP q x1ð Þ; q x2ð Þ; . . .; q xnð Þð Þ: ð4:2Þ
(4.1) and (4.2) tell us that original functions and predicates are definable in terms of
Qfied functions and predicates in combination with function q. Therefore, the in-
terdefinability depends on whether q is definable in terms of Qfied functions and
predicates. In the rest of this section, I prove that this is indeed the case.
First, from the procedure of Qfication,
Qq xð Þ ¼ q q q�1 xð Þ� �� �
¼ q xð Þ: ð4:3Þ
Thus, function q has the interesting feature that it is identical with the Qfication of
itself. Therefore, the problem is whether Qq is definable in terms of Qfied known
functions and predicates. The affirmative answer to it is derived from the presup-
position that q is definable in terms of original functions and predicates with a subtle
artifice. To illustrate this, let q be defined as in (2.1). From (2.1),
‘‘8x q xð Þ ¼ x ^ x� 1000ð Þ _ q xð Þ ¼ 2� x� 1000 ^ x [ 1000ð Þð Þ’’ is true in
interpretation C. Because this sentence contains no numeral that denotes a number
larger than 1000, proposition C-Q tells us that it is also true in interpretation Q.
Therefore, we have
8x Qq xð Þ ¼ x ^ x Q� 1000ð Þ _ Qq xð Þ ¼ 2 Q� x Q�1000 ^ x Q[ 1000ð Þð Þ: ð4:4Þ
From (4.3) and (4.4), we have
288 E. Sakakibara
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q xð Þ ¼ Qq xð Þ ¼ x ðx Q� 1000Þ2 Q� x Q� 1000ðx Q[ 1000Þ:
�ð4:5Þ
If a numeral that denotes a number larger than the bending point is contained in the
sentence defining q, you have only to rewrite the definition in such a way that the
sentence no longer contains such numerals (e.g., rewrite ‘‘q(x) = 2 9 (x ? 100) -
1200’’ as ‘‘q(x) = 2 9 (x ? 100) - 600 - 600’’). It is obvious that this maneuver
is applicable to any case in so far as q is defined in terms of any known functions
and predicates. It therefore follows that q is definable in terms of Qfied functions
and predicates whenever q is definable in terms of original functions and predicates.
From (2.4), (2.5), (4.1), (4.2), and the argument above, it is demonstrated that Qfied
functions and predicates are interdefinable with original functions and predicates in
so far as q is definable in terms of original functions and predicates. Moreover,
comparing (2.4) and (2.5) with (4.1) and (4.2), both directions of definition are
almost symmetrical with the exception that the positions of q and q-1 are reversed.
That is, from the logical viewpoint, neither of them is simpler or more fundamental
than the other.
5 Are There Alternative Interpretations of Natural Languages?
In this section, the possibility of alternative interpretations of natural languages is
briefly discussed. To begin with, let us draw on the example of ‘‘grue’’ provided by
Kripke. He points out that the notion of grue, which was coined by Goodman to
investigate ‘‘the new riddle of induction,’’ can also be taken as an example that
illustrates skepticism regarding meaning. Here, the predicate ‘‘grue’’ applies to all
things examined before t and green and to those not examined before t and blue,
where t is an arbitrary but fixed time in the future. The manner in which the skeptic
proceeds with the argument is similar to that in the case of quus: It is true that
people have uttered statements such as ‘‘This emerald is green,’’ ‘‘That grass is
green,’’ and ‘‘That sky is not green,’’ and so forth. Yet, no matter how much
evidence of this kind is accumulated, the possibility that what was meant by
‘‘green’’ was not green but instead was grue cannot be excluded because every piece
of evidence that supports the standard hypothesis also supports an infinite number of
nonstandard hypotheses, including the hypothesis that ‘‘green’’ meant grue.
Nevertheless, for the skeptic’s alternative hypothesis to carry some conviction,
the hypothesis must preserve the truth value of all sentences ever stated, rather than
only singular statements. Included among these sentences are statements such as
‘‘All emeralds are green,’’ ‘‘Light with a wavelength of 530 nm is green,’’ and ‘‘If
we mix green light and red light, we get yellow light.’’ Therefore, to construct a
plausible hypothesis, the meanings of ‘‘all,’’ ‘‘green,’’ ‘‘red,’’ ‘‘emerald,’’ ‘‘wave-
length,’’ etc. have to be coordinated to satisfy all those conditions.
The readers of this paper would anticipate that the same technique employed for
the language of algebra is also applicable to natural languages. To obtain the
alternative interpretation of a natural language through this approach, the language
must in the first place be equipped with the systematic semantics on which to apply
Incarnating Kripke’s Skepticism 289
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the technique. Although the systematic semantics of natural languages have been
investigated by logicians and philosophers such as R. Montague and D. Davidson,
further inquiry into whether or not their semantics are appropriate for our purpose is
beyond the scope of this study.
However, it is suggested from previous sections that alternative interpretations of
words that are adjacent to numbers are obtained among natural languages. The
procedure of Qfication works here again. Let me sketch such a reinterpretation by
describing a few examples:
The statement ‘‘the sum of the number of apples and the number of oranges is
larger than the number of the students’’ is, by Qfying the notion of number,
largeness, and sum as in Sects. 2 and 3, reinterpreted as the Qsum of the Qnumber of
apples and the Qnumber of oranges is Qlarger than the Qnumber of the students.
Units are regarded as functions from numbers to measurements. Hence, the
Qfication of the unit cm (centimeter) is defined as follows:
x is y Qcm,def
x is q�1ðyÞ cm:
‘‘The height of the refrigerator is 185 cm’’ is thus reinterpreted as the height of the
refrigerator is 185 Qcm.
Lastly, to denote is a binary relation \ x denote y [ , where y is sometimes a
number. Therefore, the Qfication of this relation is
x Qdenote y,def
x denote q�1ðyÞðy is a numberÞx denote y(otherwise).
�
When q is defined as in (2.1), ‘‘‘1200’ denotes 1400’’ is true in the alternative
interpretation, for q-1 (1400) = 1200 and ‘‘‘1200’ denotes 1200’’ is true in the
standard interpretation. Although it has not been proven that this strategy will
always work, it is a promising approach to obtain a nonstandard interpretation of
natural language.
6 Conclusion
In this paper, I described how to construct alternative interpretations of the language
of algebra, which is indispensable for the formulation of Kripke’s skepticism of
meaning. In addition, I proved that our alternative interpretations have the following
features:
1. They systematically alter the meaning of those algebraic symbols which have
been frequently employed by many people;
2. They preserve the truth value of every sentence that has been stated thus far;
3. They assign abnormal truth values to some of as yet unstated sentences;
4. They do not manipulate the meaning of logical symbols including the equality
sign;
5. They avoid the objections raised by Neil Tennant; and
290 E. Sakakibara
123
6. They are interdefinable, in a nearly symmetrical way, with the standard
interpretation.
The general applicability of the technique to natural languages is unresolved;
however, it seems that a kind of nonstandard interpretation of natural language is
obtained by Qfying every notion that is related to numbers according to the
procedure of Qfication.
I did not touch on the relationship between the meaning of a word and the
linguistic dispositions of a speaker or the mental pictures a speaker has, which is the
central issue of the first part of Wittgenstein on Rules and Private Language, nor
have I mentioned the skeptical solution of the rule-following paradox. The present
paper is devoted exclusively to inspecting the validity of the introductory part of
Kripke’s argumentation. Therefore, if the validity of this paper is confirmed, the
validity of his whole argument is left open. However, by demonstrating alternative
interpretations of the language of algebra, I believe that a foundation for further
discussion acceptable to both Kripke’s followers and challangers is provided.
Acknowledgments I would like to express my deepest gratitude to Prof. T. Iida from Nihon University
who provided enlightening comments and suggestions. I am also indebt to anonymous reviewers whose
meticulous comments were an enormous help to me.
References
Allen, B. (1989). Gruesome arithmetic: Kripke’s sceptic replies. Dialogue, 28, 257–264.
Blackburn, S. (1984). The individual strike back. Synthese, 58(3), 281–301.
Goodman, N. (1983). Fact, fiction, and forecast (4th ed.). Cambridge: Harvard University Press.
Kripke, S. A. (1982). Wittgenstein on rules and private language. Cambridge: Harvard University Press.
Putnam, H. (1980). Models and reality. The Journal of Symbolic Logic, 45(3), 464–482.
Tennant, N. (1997). The taming of the true. New York: Oxford University Press.
Wittgenstein, L. (2001). Philosophical investigations (3rd ed.). (trans: Anscombe, G. E. M.). Oxford:
Blackwell.
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