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ORI GIN AL ARTICLE

Incarnating Kripkes 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 Kripkes skepticism leads to the conclusion that meaning doesnot 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 thereare serious obstacles to this thesis and concluded that the skeptics nonstandard

interpretations are will o wisps. In this paper, contra Tennant, I demonstrate howto construct alternative interpretations of the language of algebra. These constructed

interpretations avoid Tennants objections and are shown to be interdefinable with

the standard interpretation. Kripkes 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, Kripkes 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: sakakibaraeisuke@gmail.com

123

Erkenn (2013) 78:277291

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 introducesa 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 a + b a; b\57 5 otherwise:

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 Kripkes

entire argument; instead, I investigate deeper into this first line of thought by

focusing on Neil Tennants doubts about the multiplicity of possible interpretations.

In his book The Taming of the True, Tennant articulates an objection againstKripkes skeptical argument (Tennant 1997, pp. 100115). The outline of his

argument is roughly as follows: He begins by pointing out that, although Kripkes

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. 104107; Kripke 1982, p. 15f.). Nevertheless,

are not sentences with universal quantifier symbol counterexamples of skepticisms

alternative hypothesis?2 Let us reconsider the instance that Kripke himself provided.

1 Regarding this point, Kripkes skepticism is clearly different from the problem discussed by Hilary

Putnam in his Models and Reality (Putnam 1980). The logical question underlying Kripkes 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

123

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 anothersymbol 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. Kripkeprepares 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 followingsentence 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 skeptics 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 Kripkes skepticism is often understood as a

mere application of Goodmans argument to linguistics (see for example Allen 1989), a clear distinction

between Kripkes problem and that of Goodman are found regarding whether or not sentences with

universal quantifier symbols are counted as evidence. The hinge of Goodmans 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. 7281). Therefore, from Goodmans 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 Kripkes Skepticism 279

123

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 thestandard 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. Kripkes 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 skeptics 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 Kripkes 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

skeptics 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 Wittgensteins 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 x xx 10002 x 1000x [ 1000:

2:1

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 Kripkes Skepticism 281

123

8, 10, , 998, 1000, 1004, 1008, . In interpretation N, all the followingsentences are true:

30 30 900; 700 400 1200;

30 40 1400; 120 10 60 20;

1500 500 700 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).(Fermats 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 CN: 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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q1 1500 q1 500 q1 700 q1 3900 : 2:2Can 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 Pbe 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+ 700 Q= 3900: 2:3As 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 def q q1 a q1 b ;a Q= b,

defq1 a q1 b :

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 Kripkes Skepticism 283

123

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; . . .; xn def q f q1 x1 ; q1 x2 ; . . .; q1 xn

: 2:4For example,

a Q b def q q1 a q1 b ;Qsin x def q sin q1 x :

The Qfication of n-ary predicate P is formulated as follows:

QP x1; x2; . . .; xn ,def

P q1 x1 ; q1 x2 ; . . .; q1 xn

: 2:5

For example,

a Q\ b,def

q1a\q1b;

a Q2 @,def

q1 a 2 @:

The operator symbol [, together with the symbol of a set on its right, is regardedas a one-place predicate symbol (e.g., [ @ is a one-place predicate symbol thatdenotes \* is a natural number[). Since logical constants are functions of whichneither 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 fromlogical 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(xQ[ 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 nN 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; . . .; un q fN q1 u1 ; q1 u2 ; . . .; q1 un

:

By substituting q(ux) for ux for every natural number x (1 B x B n), we have

q fN u1; u2; . . .; un fQ q u1 ; q u2 ; . . .; q un : (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 CN, 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 Tennants 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 Tennants 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. 108110). 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 Kripkes 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 isinterpreted, 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 beinterpreted 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; . . .; un ,PN q1 u1 ; q1 u2 ; . . .; q1 un :By substituting q(ux) for ux for every natural number x (1 B x B n), we have

PN u1; u2; . . .; un ,PQ q u1 ; q u2 ; . . .; q un : (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 Kripkes Skepticism 285

123

Therefore, V should be differently interpreted from context to context. This seemsto 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. 110112). 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,b ja b n ^ a,b 2 @f g n 1 : 3:1Here, 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 Fs and

exactly m Gs, and nothing is both F and G; there are exactly n ? m things that are

F or G (Ibid., pp. 112114). 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 x ^ G x ! Num xjF x f g Num xjG x f g Num xjF x _ G x f g 3:2

must be true in any candidate interpretation?

Tennant does not think that those three examples are the only obstacles to

Kripkes 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 theskeptics, 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 skeptics alternative

interpretation Q. In the following, it is shown that interpretation Q deftly avoids

three obstacles raised by Tennant.

Tennants 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 S def q NumS : 3:3Since 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

skeptics 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 universalquantification. 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 Kripkes Skepticism 287

123

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, Kripkes 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; . . .; xn q1 Qf q x1 ; q x2 ; . . .; q xn ; and 4:1P x1; x2; . . .; xn ,QP q x1 ; q x2 ; . . .; q xn : 4:2

(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 x q q q1 x q x : 4:3Thus, 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 x x ^ x 1000 _ q x 2 x 1000 ^ x [ 1000 is true ininterpretation 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 x x ^ x Q 1000 _ Qq x 2 Q x Q1000 ^ x Q[ 1000 : 4:4From (4.3) and (4.4), we have

288 E. Sakakibara

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q x Qq x x x Q 10002 Q x Q 1000x Q[ 1000:

4:5

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 skeptics alternative hypothesis to carry some conviction,

the hypothesis must preserve the truth value of all sentences ever stated, rather thanonly 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 Kripkes 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 q1y 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 anumber. Therefore, the Qfication of this relation is

x Qdenote y,def

x denote q1yy is a numberx 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 Kripkes 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

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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, norhave 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

Kripkes 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 Kripkes followers and challangers is provided.

Acknowledgments I would like to express my deepest gratitude to Prof. T. Iida from Nihon Universitywho 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: Kripkes sceptic replies. Dialogue, 28, 257264.Blackburn, S. (1984). The individual strike back. Synthese, 58(3), 281301.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), 464482.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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Incarnating Kripkes Skepticism About MeaningAbstractIntroductionInterpretation QStep One: From Interpretation C to Interpretation NStep Two: From Interpretation N to Interpretation Q

Tennants Three ObstaclesInterdefinabilityAre There Alternative Interpretations of Natural Languages?ConclusionAcknowledgmentsReferences