Abstract
This chapter reviews Kripke’s original description of Wittgenstein’s paradox and its solution. Quite a bit of critical commentary has engaged with Kripke’s interpretation; but I’m not concerned with whether Kripke’s interpretation of Wittgenstein is right; I’m only concerned with Kripke’s puzzle, as he presents it. Two distinctions discussed in the chapter are Kripke’s description of straight and sceptical solutions, and that between grounding facts and correspondence facts. Three key discussions occur in the chapter. First, there is a discussion of Kripke’s three conditions on a straight solution to the paradox, the infinitude requirement, the justification requirement, and the mistake requirement. Second, there is a discussion of why dispositional approaches to the paradox don’t work. Third, there is a discussion of why a straight sociological solution doesn’t work. I engage as well with some of the critical commentary on rule following that has arisen subsequently to Kripke’s book.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Some of the tacit knowledge required to successfully do this is labeled, by Gelman and Gallistel (1986, 77–82), as the counting principles: “cardinal word,” “order-irrelevance,” and “abstractness”—that the last word in a count is the cardinal number of the collection, that the order in which the objects in a collection are counted doesn’t matter, and that it doesn’t matter what the objects are. These abilities are acquired by children in stages from ages 2 to 4. See Carey (2009, p. 241–244) for a description of the process, and for indications of how arduous acquisition of tacit counting knowledge is.
- 2.
See Butterworth (1999), especially p. 52–62, for an accessible discussion—with some citations—of the different types of number vocabulary to be found in different natural languages.
- 3.
Children can recognize the indefinite nature of numbers without having learned a genuine numeral system. Carey (2009, 252) quotes one five-year old as saying: “suppose you think a gazillion is the highest number—well, you can go a gazillion and one, a gazillion and two …,” and she describes the spontaneous invention of arguments that there is no highest number as common. In my exposition, I assume the subject has acquired a numeral system for counting in order to avoid complications with number-languages that are outstripped—at least in this respect—by the subject’s knowledge of number itself. In any case, children in our culture do, in time, acquire an understanding of numeral systems. That achievement, too, is arduous because of confusions that can arise from the child needing to identify the terms of a numeral system with the number-words in natural languages. Apart from this there is empirical evidence that different brain circuits handle these different number systems. (See, for example, the numerous results on this by Dehaene and his associates. Some of this empirical evidence turns on the loss of certain abilities while others are retained—after strokes and other sorts of specifically-localized brain injuries; but many of them, especially recently, turn on brain imaging during the execution of various arithmetic tasks.)
- 4.
Kripke (1982, 16): “Quounting” a heap of objects is counting in the ordinary sense unless the heap is formed as the union of two heaps, one of which has 57 or more items (assuming the subject has never so-far counted heaps that large), in which case the answer is 5. Also, see Gelman and Gallistel (1986, 51), where an interestingly similar thought experiment is raised.
Kripke’s strategy of undermining what a subject means to do at a present moment by first undermining what the subject could have meant on previous occasions has been widely criticized and (in my view) often misunderstood. Forbes (1983–1984, 225–226) interprets Kripke’s “claim that there is no fact about what the subject meant in the past [as ambiguous] between the claim that he had no determinate understanding in the past and the claim that although he had such an understanding, there is no fact about whether it is the same as his present one. The context dictates that it is the second reading which is intended ….” No, the context indicates that the first reading is intended. The challenge to the subject, by the sceptic, that perhaps she meant quounting and not counting, undercuts possible crosstemporal identity relations between present understanding and past understanding by undercutting the capacity of any past facts about the subject to make determinate whether she meant counting or quounting. The incapacity of present facts about the subject to make determinate whether she means counting or quounting then follows because present facts about the subject are no different in their capacity to fix meaning than past facts are. On the other hand, Boghossian (1989, 515) interprets Kripke’s puzzle as directly about what, if anything, determines content- or meaning-conditions—“the [possession] of a correctness condition” instead of Kripke’s puzzle only being indirectly about this via the antecedent question of what a subject could have meant on previous occasions. This is a straightforward misreading of Kripke’s text. As Forbes (1984, 226) correctly notes, Wright’s (1980) concern (not Kripke’s) is directly about what, if anything, determines content- or meaning-conditions. (Forbes regards Wright’s claim as “more direct and more challenging” than Kripke’s; I think he thinks this because of his misreading of Kripke’s strategy that I attribute to him at the beginning of this paragraph.)
- 5.
Also see Azzouni (2012c).
- 6.
As discussed in, e.g., Correia and Schnieder (2012).
- 7.
Carey (2009), for example, offers an “ontogenetic” description of how the child—already possessing certain innate subpersonal cognitive systems (the parallel individuation of small sets, and the natural-language quantifiers) applicable to particular (restricted) numerical tasks—is enabled over the course of a year and a half by induction and analogy (“Quinean bootstrapping”) to understand certain crucial properties of numerals, e.g., that they continue indefinitely, and to understand how to apply them to count collections of objects. A striking effect of the rule-following paradox, as we’ll see, is that Carey’s empirical hypothesis, and similar others on offer by cognitive scientists, don’t even look relevant to a solution.
- 8.
- 9.
Some of these possible “mistakes” actually occur during a child’s acquisition of counting but most don’t. Children go through specific stages as they acquire counting skills, and cognitive scientists describe children successively, as “one-knowers,” “two-knowers,” “three-knowers,” “four-knowers,” “subset-knowers,” and finally, “cardinal-principle knowers.” They are described—at one stage—as “one-knowers” because they know how to distinguish one object from many, but they can’t distinguish between the cardinal numbers of groups of objects larger than one; “two-knowers,” are analogous, and so on. At one stage in their acquisition of counting skills, children also skip numerals when counting; they don’t do this later. See Carey (2009) and Butterworth (1999) for descriptions, and for references to the literature.
- 10.
“First I group them into collections of five because they’re easier to recognize. I count all the groups, and then multiply by five.” This is a description that a child might give because it strikes her as obvious that her new method gives the same answers as the first method she was taught (strict enumeration).
- 11.
That is, there is an on-going debate about “content normativity” or “the normativity of meaning”—in what sense, if any, normativity is involved in rule following or meaning determination. Some such view about the normativity of meaning is attributed to Kripke on the basis of his being taken to claim that “one is or was disposed to respond in a certain way on a given occasion cannot make it the case that one ought so to respond” (as Ginsborg (2011, 228) puts it, italics hers). See, for example, Boghossian (1989, 2003), Glüer (1999), Wikforss (2001). As I indicate above, and later, “normativity,” “rationality,” “norms,” “correctness” or “appropriateness,” are involved in this topic only in the sense that these presuppose successful rule following, and not in the sense that any of these are presupposed by rule following. Part of this point is that the “ought” currently in play is unremarkably hypothetical, only of the form (put somewhat archly), “If you’d like to live, you ought not to jump off a bridge.” (“If you’d like to live, I’d recommend ….”) Similarly, “if you really want to live, it isn’t appropriate to jump off a bridge, now is it?” (Imagine this is said, in a movie, by an angel to a potential suicide.) Gibbard (2003, 85) writes: “… whether we ought to take a walk can depend on the weather; that doesn’t make the weather normative in any philosophically special sense.” So, to this extent, I’m on the side of those in this debate who reject “meaning normativity.” I say more about this in Sect. 2.3.
- 12.
As I’ll indicate later, although this recognition-requirement seems to apply to counting and to other simple computational concepts, addition, subtraction, etc., it isn’t a requirement on every concept that we take ourselves to understand. On the contrary: It’s often thought that the meaning of a concept is understood by someone (or by all of us) even though he (or all of us) may not be able to recognize when some (or even all) of the uses made of the concept are incorrect. (I won’t think someone doesn’t understand the concept of God even if I disagree with that someone on everything or nearly everything he claims about God.)
- 13.
Kripke (1982, 7–22). Kripke poses the meaning sceptic’s challenge in first-person terms, as whether my previous practice and my occurrent thinking is compatible with “quaddition” instead of (as presumed) addition. I’ve instead described the challenge “third-person,” and—as noted—in terms of counting (instead of “quounting”). Given that we routinely allow ourselves to describe the phenomenology of third parties—pretty much as I’ve done above—this raises no complications. The reader uncomfortable with my use of our ordinary practice of describing the phenomenology of others can easily restructure my discussion in first-person terms. It won’t affect anything essential.
- 14.
- 15.
Wright (1984, 771) is more cautious: “Understanding an expression is, intuitively, more like an ability than a disposition.”
- 16.
E.g., Ginsborg (2011).
- 17.
Boghossian (1989, 527), Fodor (1990, 135–136), McDowell (1984), and others; see Ginsborg (2011, 229), footnote 5 for further examples.
I should add that “rule-following problem” is a misleading misnomer. What’s being challenged is whatever in the subject is a resource to enable the subject to do subsequently what she meant formerly. The resource in question doesn’t have to be a “rule” or a piece of public language with a “meaning.” I mention this because some commentators seem to interpret Kripke as working within one or another restriction of this sort.
- 18.
Indeed, Goldfarb (1985), e.g. 476, reads Kripke as simply engaged in a challenge of “any putative physicalistic reduction of meaning.”
- 19.
See, e.g., Forbes (1983–1984, 234–235).
- 20.
- 21.
Goldfarb (1985, 474) suggests the Fregean “immediate access to the realm of sense,” is a cogent response to Kripke’s meaning sceptic. Tait (1986) may be offering the same rejoinder (I’m not sure). In any case, Russell’s remark about the virtues of theft over honest toil seems pertinent. We’re supposed to be explaining what it is about the subject that enables the subject to grasp counting and not quounting. It’s no answer to say that certain mental faculties in the subject enable this (and that’s the end of it); it’s no more of an answer to say that certain objects (functions) are just the sorts of things that subjects can grasp (and that’s the end of it).
- 22.
This is “subitizing,” and it seems to be restricted to about four items for adults and to three items for children, and for certain animals. See Mandler and Shebo (1982).
- 23.
DIS 2 illegitimately singles out a single method. As noted in Sect. 2.1, a subject may have acquired other appropriate short-cut methods of counting. I set aside attempts to refine DIS 2 to handle this because such refinements won’t affect the overall dialectical trajectory of this chapter.
- 24.
The actual details of our biology—e.g., brain organization—physical laws, and so on, are pertinent to the truth of the counterfactuals involved. See Fox (2011) for a popularized discussion of the issue, with pictures and diagrams.
- 25.
The counterfactual suggestion and the exception-clause suggestion amount to the same strategy executed with grammatical variations. In the former case, diagnoses of the violations of DIS 2 are built into antecedents of counterfactuals; in the latter, a modification of DIS 2 contains the exceptions in antecedents of indicative conditionals.
- 26.
- 27.
- 28.
What follows in this and the next five paragraphs, although a defense of Kripke’s rejection of dispositional approaches, isn’t (as far as I can tell) given by Kripke. (Or, for that matter, by anyone else.)
- 29.
See, e.g., Blackburn (1984, 290) and Forbes (1983–1984). Goldfarb (1985, 477) writes: “A reductionist could claim, for example, that future physiological psychology might reveal two mechanisms, separable on scientific grounds. States of the first amount to a person’s linguistic competence, and would, if untrammeled, always cause correct responses; states of the second are identifiable with interfering features, which explain why on particular occasions the first mechanism does not issue an appropriate response (and the person errs).” Although, in principle, infinitary dispositions that generate the correct functions are neurophysiologically possible—this was ruled out by the early 1990s as far as arithmetical competences are concerned.
- 30.
Some relevant literature: Carey (2009), Dehaene (1997). There is much more, of course. This is a very active research area, primarily because the “dispositions” being studied are so intricate and gerrymandered—not a simple matter of embodied arithmetical rules that people mechanically execute unless something interferes so that they make blunders.
- 31.
Content fixation is the issue Boghossian (1989) takes Kripke to be directly raising; it’s the issue Wright (1980) does directly raise (see foonote 4). But the considerations Boghossian uses to motivate this issue aren’t the ones I’ve just given, for he (2003, 496) writes, describing his earlier views: “I used to underestimate the force of [the infinitude requirement].”
- 32.
This is an extremely important aspect of how we understand rule following, and it’s due to relatively deep facts about our (shared) psychology—in particular to the peculiar ways that we take account of (and fail to take account of) the subpersonal abilities that are relevant to consciously undertaken tasks. This interesting interplay between the personal and the subpersonal has no role in my initial models of private rule following (my first two Robinson Crusoes in Chap. 3); but is part of the realistic psychological aspects of Crusoe 5, to be described in Chap. 5.
- 33.
A routinely-described psychological result is the insecure reaction of a subject in a situation—e.g., in a classroom—where very simple instructions have been given (“raise your hand when the teacher holds up a green card, and lower it when she holds up a red card”), when, after a few episodes of this, everyone else in the class (as previously arranged) conspiratorially violates the instructions.
- 34.
What kind of urge? Imagine an urge like the ones we have (for example) to scratch an itch, or urinate, or eat. It’s resistible, but with (varying) difficulty.
- 35.
I take myself (in this section) to be expounding Kripke’s (1982, 74–93) discussion of assertability conditions. Douglas Patterson (9/24/09—email) has raised the concern that a general replacement of truth conditions (for contents or statements) by assertability conditions leads to the view that it’s sentences “otherwise meaningless, that have assertability conditions.” And so, “what I’m entitled in the community to do is to make a noise, say.” He further suggests that such a view isn’t coherent unless an illicit appeal to truth conditions occurs in the description of what people are licensed to do.
I think this is wrong—despite my eventual denial in this book that truth conditions need be replaced by assertability conditions to provide a sceptical solution to the rule-following paradox. I agree that sentences (without necessary and sufficient conditions of application) are what—on this view—people are entitled to apply or refuse to apply. Two views are possible: (i) The pattern of entitlements of such sentences is their “meaning”; (ii) Because of certain conditions that “meanings” must meet (e.g., that they must result in necessary and sufficient conditions of application) and that such a pattern of entitlements doesn’t meet, these sentences don’t have “meanings.” In neither case, it seems to me, does it follow that an assertion practice (built on “entitlements”) is incoherent, or that it needs supplementation with truth conditions.
The replacement of truth conditions by assertability conditions—Kripke (1982, 86) describes Wittgenstein as claiming—doesn’t rule out ordinary uses of “true” and “false.” For exactly these reasons, as I indicate in Chap. 6, one can still utilize—instead of assertability conditions—truth conditions, provided one understands them in a “deflated” way, rather than as requiring correspondence relations to facts. I’m suggesting, in other words, that Kripke (and perhaps Wittgenstein) has built into the notion of “truth conditions” that they are required to provide correspondence relations to facts (Kripke (1982, 72): “A declarative sentence gets its meaning by virtue of its truth conditions, by virtue of its correspondence to facts that must obtain if it is true” (italics are Kripke’s)). I deny that “truth conditions” need to be so understood. See Chap. 6.
- 36.
A distinction is drawn between “solitary languages” spoken by single individuals and Wittgenstein’s “private languages,” e.g., “sensation languages” (Goldfarb (1985)). Blackburn, Goldfarb and Boghossian think Kripke’s skeptical solution doesn’t rule out “solitary languages” as impossible. I don’t agree.
References
Azzouni, Jody. 2012c. Simple metaphysics and “ontological dependence”. In Metaphysical grounding: Understanding the structure of reality, ed. Fabrice Correia and Benjamin Schnieder, 234–253. Cambridge: Cambridge University Press.
Blackburn, Simon. 1984. The individual strikes back. Synthese 58(3): 281–301.
Boghossian, Paul. A. 1989. The rule-following considerations. Mind 98(392): 507–49.
———. 2003. The normativity of content. In Philosophical Issues, ed. Ernest Sosa and Enrique Villanueva, vol. 13: 31–45. Oxford: Blackwell Publishing.
Butterworth, Brian. 1999. The mathematical brain. London: Papermac.
Carey, Susan. 2009. Where our number concepts come from. Journal of Philosophy 106(4): 220–254.
Correia, Fabrice, and Benjamin Schnieder. 2012. Metaphysical grounding: Understanding the structure of reality. Cambridge: Cambridge University Press.
Dehaene, Stanislas. 1997. The number sense. Oxford: Oxford University Press.
Forbes, Graeme. 1983–1984. Skepticism and semantic knowledge. Proceedings of the Aristotelian Society, New Series, vol. 8, 223–237.
Fox, Douglas. 2011. The limits of intelligence. Scientific American, July, 36–43.
Gelman, R., and C.R. Gallistel. 1986. The child’s understanding of number. Cambridge: Harvard University Press (originally published in 1978).
Gibbard, Alan. 2003. Thoughts and norms. In Philosophical Issues, ed. Ernest Sosa and Enrique Villanueva, Vol. 13, 83–98. Oxford: Blackwell Publishing.
Ginsborg, Hannah. 2011. Primitive normativity and skepticism about rules. The Journal of Philosophy 108(5): 227–254.
Glüer, Kathrin. 1999. Sense and prescriptivity. Acta Analytica 14: 111–128.
Goldfarb, Warren. 1985. Kripke on Wittgenstein on Rules. The Journal of Philosophy 82(9): 472–488.
Kripke, Saul. 1982. Wittgenstein on rules and private language. Cambridge, MA: Harvard University Press.
Mandler, G., and B.J. Shebo. 1982. Subilizing: An analysis of its component processes. Journal of Experimental Psychology: General, 11–22.
Putnam, Hilary. 1981. Reason, truth and history. Cambridge: Cambridge University Press.
Tait, William. 1986. Wittgenstein and the “Skeptical Paradoxes,” reprinted in (William Tait) The provenance of pure reason, 198–211 (2005). Oxford: Oxford University Press.
Wikforss, Åsa Maria. 2001. Semantic normativity. Philosophical Studies 102(2): 203–226.
Wittgenstein, Ludwig. 1956. Remarks on the foundations of mathematics. Oxford: Basil Blackwell.
———. 1958. Philosophical Investigations. Trans. G.E.M. Anscombe, 3rd ed. New York: The Macmillan Company.
Wright, Crispin. 1980. Wittgenstein on the foundations of mathematics. London: Duckworth.
———. 1984. Kripke’s account of the argument against private language. The Journal of Philosophy 81(12): 759–778.
———. 1989. Critical notice of Colin McGinn’s Wittgenstein on meaning. Mind 98: 289–305. Reprinted in (Alexander Miller and Crispin Wright) Rule-following and meaning, 108–128. Montreal/Kingston: McGill-Queen’s University Press. References are to the reprint.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Azzouni, J. (2017). Kripke’s Version of Wittgenstein’s Paradox and His Solution. In: The Rule-Following Paradox and its Implications for Metaphysics. Synthese Library, vol 382. Springer, Cham. https://doi.org/10.1007/978-3-319-49061-8_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-49061-8_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-49060-1
Online ISBN: 978-3-319-49061-8
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)