## 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 nonstandard interpretations of natural language.

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## Notes

- 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 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.

- 3.
Let S

^{N}and S^{Q}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,

\( {\text{q}}\left( {{\text{n}}^{\text{N}} } \right) = {\text{n}}^{\text{Q}}. \)(2.6)

(ii) From (2.4), for every n-ary function symbol f and for every series of numbers \( {{\text{u}}_{1} ,{\text{u}}_{2} , \ldots,{\text{u}}_{\text{n}} }, \)

\( {\text{f}}^{\text{Q}} \left( {{\text{u}}_{1} ,{\text{u}}_{2} , \ldots ,{\text{u}}_{\text{n}} } \right) = {\text{q}}\left( {{\text{f}}^{\text{N}} \left( {{\text{q}}^{ - 1} \left( {{\text{u}}_{1} } \right),{\text{q}}^{ - 1} \left( {{\text{u}}_{2} } \right), \ldots ,{\text{q}}^{ - 1} \left( {{\text{u}}_{\text{n}} } \right)} \right)} \right). \)

By substituting q(u

_{x}) for u_{x}for every natural number x (1 ≤ x ≤ n), we have\( {\text{q}}\left( {{\text{f}}^{\text{N}} \left( {{\text{u}}_{1} ,{\text{u}}_{2} , \ldots ,{\text{u}}_{\text{n}} } \right)} \right) = {\text{f}}^{\text{Q}} \left( {{\text{q}}\left( {{\text{u}}_{1} } \right),{\text{q}}\left( {{\text{u}}_{2} } \right), \ldots ,{\text{q}}\left( {{\text{u}}_{\text{n}} } \right)} \right). \)(2.7)

(iii) From (2.5), for every n-ary predicate symbol and for every series of numbers, \( {{\text{u}}_{1} ,{\text{u}}_{2} , \ldots,{\text{u}}_{\text{n}} }, \)

\( {\text{P}}^{\text{Q}} \left( {{\text{u}}_{1} ,{\text{u}}_{2} , \ldots ,{\text{u}}_{\text{n}} } \right)\mathop \Leftrightarrow \limits_{{}} {\text{P}}^{\text{N}} \left( {{\text{q}}^{ - 1} \left( {{\text{u}}_{1} } \right),{\text{q}}^{ - 1} \left( {{\text{u}}_{2} } \right), \ldots ,{\text{q}}^{ - 1} \left( {{\text{u}}_{\text{n}} } \right)} \right). \)

By substituting q(u

_{x}) for u_{x}for every natural number x (1 ≤ x ≤ n), we have\( {\text{P}}^{\text{N}} \left( {{\text{u}}_{1} ,{\text{u}}_{2} , \ldots ,{\text{u}}_{\text{n}} } \right)\mathop \Leftrightarrow \limits_{{}} {\text{P}}^{\text{Q}} \left( {{\text{q}}\left( {{\text{u}}_{ 1} } \right),{\text{q}}\left( {{\text{u}}_{ 2} } \right), \ldots ,{\text{q}}\left( {{\text{u}}_{\text{n}} } \right)} \right). \)(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.

## 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.

## 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.

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Sakakibara, E. Incarnating Kripke’s Skepticism About Meaning.
*Erkenn* **78, **277–291 (2013). https://doi.org/10.1007/s10670-012-9367-6

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### Keywords

- Natural Language
- Alternative Interpretation
- Function Symbol
- Predicate Symbol
- Universal Quantifier