Incarnating Kripke’s Skepticism About Meaning

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<ul><li><p>ORI GIN AL ARTICLE</p><p>Incarnating Kripkes Skepticism About Meaning</p><p>Eisuke Sakakibara</p><p>Received: 15 December 2010 / Accepted: 11 February 2012 / Published online: 6 April 2012</p><p> Springer Science+Business Media B.V. 2012</p><p>Abstract Although Kripkes skepticism leads to the conclusion that meaning doesnot exist, his argument relies upon the supposition that more than one interpretation</p><p>of words is consistent with linguistic evidence. Relying solely on metaphors, he</p><p>assumes that there is a multiplicity of possible interpretations without providing any</p><p>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</p><p>interpretations are will o wisps. In this paper, contra Tennant, I demonstrate howto construct alternative interpretations of the language of algebra. These constructed</p><p>interpretations avoid Tennants objections and are shown to be interdefinable with</p><p>the standard interpretation. Kripkes skepticism is, as it were, an incarnate demon.</p><p>In contrast, it is currently uncertain whether the same technique is generally</p><p>applicable to the construction of an alternative interpretation of natural language.</p><p>However, the reinterpretation of those aspects of natural language that directly</p><p>relate to numbers seems to be a promising candidate for the development of non-</p><p>standard interpretations of natural language.</p><p>1 Introduction</p><p>Among recent discussions on the philosophy of language, Kripkes skeptical</p><p>argument against the reality of meaning is one of the most disquieting and</p><p>controversial. Since this problem is related to diverse themes in contemporary</p><p>philosophy, a great many philosophers from different backgrounds have been</p><p>involved in this heated debate. The accumulation of literature concerned with this</p><p>issue is so immense, and the remarks found in them are so intricately intertwined</p><p>E. Sakakibara (&amp;)National Center of Neurology and Psychiatry, 4-1-1 Ogawa-Higashi, Kodaira, Tokyo</p><p>187-8551, Japan</p><p>e-mail:</p><p>123</p><p>Erkenn (2013) 78:277291</p><p>DOI 10.1007/s10670-012-9367-6</p></li><li><p>that visitors unfamiliar with this issue could easily become confused. This paper</p><p>examines the entrance to this huge labyrinth.</p><p>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</p><p>or greater; Kripke asserts that, no matter how many instances of calculation are cited</p><p>as evidence, the possibility that I meant by ? the following quus func-</p><p>tion cannot be excluded (Kripke 1982, p. 7f.).</p><p>a b a + b a; b\57 5 otherwise:</p><p>After discussing the underdetermination of meaning by the past behavior, Kripke</p><p>goes on to examine whether other candidates, such as our dispositions and mental</p><p>pictures, could fix the interpretation. I abstain from tracing the path of Kripkes</p><p>entire argument; instead, I investigate deeper into this first line of thought by</p><p>focusing on Neil Tennants doubts about the multiplicity of possible interpretations.</p><p>In his book The Taming of the True, Tennant articulates an objection againstKripkes skeptical argument (Tennant 1997, pp. 100115). The outline of his</p><p>argument is roughly as follows: He begins by pointing out that, although Kripkes</p><p>skepticism aims to establish an anti-realistic view of meaning, his argument relies</p><p>upon the lemma that there are alternative interpretations of words compatible with</p><p>all existing evidence, and that assign a different truth values to some of the as yet</p><p>unstated sentences from the standard interpretation.</p><p>If a sentence stated in the past is incompatible with all interpretations but the</p><p>standard one, the standard interpretation would be preferable to alternative ones.</p><p>Such asymmetry would thus render the skeptical argument unpersuasive. The latter</p><p>condition ensures that the alternative interpretations are different from the standard</p><p>one enough to make difference to the correct behavior in the future. It is evident that</p><p>interpretations isomorphic to standard interpretations invest all sentences, whether</p><p>or not they have been stated, with the same truth values. But those interpretations</p><p>are not what Kripke needed.1</p><p>Tennant, then, casts doubt on the very existence of such nonstandard</p><p>interpretations. He stresses the fact that if the interpretation of a word is altered,</p><p>the interpretation of other words should also be altered, like a domino toppling, to</p><p>make up for the inconsistency; thus, if ? is interpreted to mean quus function, the</p><p>word addition should be interpreted as quaddition, sum as quum, and</p><p>counting as quounting (Ibid., pp. 104107; Kripke 1982, p. 15f.). Nevertheless,</p><p>are not sentences with universal quantifier symbol counterexamples of skepticisms</p><p>alternative hypothesis?2 Let us reconsider the instance that Kripke himself provided.</p><p>1 Regarding this point, Kripkes skepticism is clearly different from the problem discussed by Hilary</p><p>Putnam in his Models and Reality (Putnam 1980). The logical question underlying Kripkes problem is</p><p>whether there is a model for a set of sentences different from the set of sentences that are true in the</p><p>standard interpretation: suppose I had not added a number equal to 57 or greater, and A is a set of</p><p>algebraic sentences having been stated so far and regarded as true, the question would be whether or not</p><p>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</p><p>true that, except for a minority of people who are good at mathematics and logic, most people have never</p><p>278 E. Sakakibara</p><p>123</p></li><li><p>If I had used ? to indicate the quus function, would not the sentence expressing</p><p>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.</p><p>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</p><p>that a similar line of argument could also be developed regarding V. Kripkeprepares a counterargument against the objection based on the associative law: the</p><p>truth of what is called associative law will be preserved if we reconsider the</p><p>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.</p><p>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</p><p>expresses that there is no largest natural number. However, since there is always a</p><p>largest natural number that is less than a specified upper bound, the alternative</p><p>interpretation of the sentence is false.</p><p>To construct a consistent alternative interpretation, Tennant stresses that the</p><p>interpretation of words must be reshaped globally. Kripke and his followers have</p><p>never indicated how to perform such global reshaping; instead, they suggest that we</p><p>cunningly adjust candidate alternative interpretations whenever an opponent</p><p>produces new counterexamples. However, there is no guarantee that we will</p><p>eventually arrive at a stable alternative interpretation. In view of this, after</p><p>exemplifying three additional obstacles to such global reinterpretation, Tennant</p><p>denounces the skeptics alternative interpretations as will o wisps (Tennant</p><p>1997, p. 101).</p><p>Kripke has confidence in the existence of alternative interpretations. He considers</p><p>giving a consistent interpretation for a finite set of sentences to be analogous to</p><p>finding a rule in a finite sequence of numbers (Kripke 1982, p. 18). However, an</p><p>analogy is too unstable a foundation upon which to build a philosophical</p><p>Footnote 2 continued</p><p>stated such quantified sentences. However, that does not allow us to disregard them. If it were the case</p><p>that quantified sentences leave no room for a nonstandard interpretation, meaning would be indeterminate</p><p>only for laypeople, and experienced mathematicians and logicians would enjoy full-blown determinate</p><p>meaning. If this were the conclusion, skepticism would be almost dead. For skepticism to retain its power,</p><p>it must hold that even if all the sentences having been stated by someone were cited as evidence, the</p><p>interpretation would be underdetermined. Second, although Kripkes skepticism is often understood as a</p><p>mere application of Goodmans argument to linguistics (see for example Allen 1989), a clear distinction</p><p>between Kripkes problem and that of Goodman are found regarding whether or not sentences with</p><p>universal quantifier symbols are counted as evidence. The hinge of Goodmans new riddle of induction</p><p>is that it is logically indeterminate in terms of the way we should generalize singular statements obtained</p><p>from our observations so far (see Goodman 1983, pp. 7281). Therefore, from Goodmans viewpoint,</p><p>quantified sentences such as All emeralds are green are the conclusions of induction rather than</p><p>evidence that support them. On the other hand, evidence that supports a certain interpretation of words</p><p>includes all the sentences that have been stated and regarded as true, whether or not they are quantified.</p><p>Since human creatures not only perform concrete calculations but also discuss general theorems of</p><p>algebra, Kripke, and not Goodman, must tackle the problems brought about by those quantified sentences.</p><p>Incarnating Kripkes Skepticism 279</p><p>123</p></li><li><p>demonstration, since we can draw another analogy that indicates the contrary. To be</p><p>sure, the total number of sentences that have been stated is finite, while the total</p><p>number of words that have been used in the history of humankind is far less. We are</p><p>faced with a dilemma here. On the one hand, according to the principle of</p><p>compositionality, the reinterpretation of sentences is possible only through the</p><p>reinterpretation of some of the words contained in those sentences. On the other</p><p>hand, the reinterpretation must meet the condition that the truth value of all the</p><p>sentences that have ever been stated must remain unchanged. Then, is not the</p><p>question In how many ways can we give alternative interpretations? analogous to</p><p>the question How large would the degree of freedom for solutions be if there were</p><p>simultaneous equations, the number of which were far greater than that of</p><p>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</p><p>another interpretation is no longer self-evident. When two analogies collide with</p><p>each other, an argument based upon analogy takes us nowhere. We need strict proof.</p><p>Tennant insists that the burden of proof is on the skeptic. To be sure, burden shifting</p><p>is unproductive during philosophical discussions. Yet, complaining about the burden</p><p>shifting is just another unproductive discourse. I am going to respond, albeit partially,</p><p>to his challenge head-on. This study principally aims to construct a nonstandard</p><p>interpretation of the language of algebra. For reasons of convenience, I restrict the</p><p>domain of discourse to real numbers. However, the method employed here is</p><p>sufficiently general in that it is instinctively clear that the method can be applied to</p><p>more advanced algebra dealing with imaginary numbers, matrices, and so forth. The</p><p>construction will be demonstrated in Sect. 2, and the three obstacles raised by Tennant</p><p>will be illustrated and answered in Sect. 3. Kripkes skepticism is, as it were, not an</p><p>elusive will o wisp but an incarnate demon. This incarnation makes the skeptical</p><p>argument more urgent and may lead to revelations about its fundamental nature.</p><p>In Sect. 4, I demonstrate that the newly proposed nonstandard interpretation is</p><p>interdefinable with the standard interpretation. In other words, I show that if a</p><p>nonstandard interpretation proposed in Sect. 2 is definable from the standard</p><p>interpretation, the standard interpretation can also be defined from the nonstandard</p><p>interpretation. In The New Riddle of Induction, Goodman stresses that grue</p><p>and bleen are interdefinable with green and blue (Goodman 1983, p. 79f).</p><p>Moreover, both directions of definition are symmetrical in that their definientia</p><p>contain reference to time. Goodman does so in order to sweep away the suspicion</p><p>that employing novel notions such as grue and bleen in induction is</p><p>illegitimate because, in contrast to green and blue, those notions refer to time</p><p>and are therefore not purely qualitative. In contrast, definitions between the</p><p>standard interpretation and nonstandard interpretations are almost, though not</p><p>exactly, symmetrical. Interdefinability with the standard interpretation is not the</p><p>requirement for the skeptics alternative interpretations. Yet, if they are interde-</p><p>finable with the standard interpretation, some approaches to rebut the skeptical</p><p>argument will be blocked.</p><p>In Sect. 5, I briefly discuss the possibility of alternative interpretations of natural</p><p>languages, borrowing Kripkes uses of grue as an instance. Although the general</p><p>applicability of the techniques developed in previous sections to natural languages is</p><p>280 E. Sakakibara</p><p>123</p></li><li><p>beyond the scope of this study, the reinterpretation of those aspects of natural</p><p>language that directly relate to numbers seems to be a promising candidate for the</p><p>development of nonstandard interpretations.</p><p>2 Interpretation Q</p><p>Let interpretation C be the standard interpretation of the language of algebra. The</p><p>skeptics alternative interpretation must (1) aberrantly interpret one or more of the</p><p>algebraic symbols employed in the past, (2) preserve the truth value of all the</p><p>sentences that have been stated so far, and (3) assign a different truth value to some</p><p>of the as yet unstated sentences from interpretation C. In this Section, I demonstrate</p><p>how to construct a satisfying interpretation, referred to as interpretation Q, from</p><p>interpretation C in two steps.</p><p>In the first step, the interpretation which assigns aberrant denotations to numerals,</p><p>named interpretation N, is proposed. Since it is shown that interpretation N satisfies (2)</p><p>and (3) only when it does not satisfy (1), interpretation N cannot be the desired</p><p>alternative interpretation. In the second step, interpretation N is transformed into</p><p>interpretation Q, which assigns aberrant meanings to function and predicate symbols.</p><p>Those meanings are altered in a consistent manner by applying the operation called</p><p>Qfication. Finally, it is shown that interpretation Q satisfies all three conditions.</p><p>2.1 Step One: From Interpretation C to Interpretation N</p><p>In the first step, let us consider interpretation N, which interprets numerals</p><p>aberrantly. In interpretation N, following the famous example of the pupil who</p><p>appears in Wittgensteins Philosophical Investigations (Wittgenstein 2001, para-graph 185), numbers larger than 1000 increment half as fast as their notations do,</p><p>viz. 1 means 1, 2 means...</p></li></ul>


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