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Normativity, probability, and meta-vagueness


This paper engages with a specific problem concerning the relationship between descriptive and normative claims. Namely, if we understand that descriptive claims frequently contain normative assertions, and vice versa, how then do we interpret the traditionally rigid distinction that is made between the two, as ’Hume’s law’ or Moore’s ’naturalistic fallacy’ argument offered. In particular, Kripke’s interpretation of Wittgenstein’s ’rule-following paradox’ is specially focused upon in order to re-consider the rigid distinction. As such, the paper argues that if descriptive and normative claims are not mutually exclusive, then we need a new framework with which to understand this relationship. In this regard, the paper borrows from concerns with vagueness, particularly using a degree-theoretic approach in terms of subjective probability, in an attempt to graphically figure out these differences. Consequently, the paper tentatively proposes the hyperbola model in which degrees of normativity and degrees of descriptivity could be expressed and measured. It is hoped, as a result, that this tentative proposal will contribute to deepening the debate on vagueness in general.

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  1. By ‘meta-vagueness’ I mean the vagueness of predicates which are predicated of sentences rather than subjects. This corresponds to a semiotic distinction between object-language and meta-language. Suppose A to be a sentence (that may include a vague predicate like red, child, or good, and so on), and P to be a predicate. Thus, if ‘A is P’ is vague, then the vagueness here could be called ‘meta-vagueness’, as this is the vagueness of the sentence. Please note that meta-vagueness should be differentiated from higher-order vagueness on the boundary between penumbra and definitely true/false cases. Regarding the vagueness of the predicate ‘vague’, Sorensen (1985) strongly suggests that any argument involving a vague predicate, including the predicate ‘vague’ itself, must be ‘a self-refuting thesis’ (Sorensen 1985, p. 136). I do not believe that the same pessimistic conclusion should be drawn in the case of a descriptive/normative distinction, as will be shown by my argument as a whole. In any case, the vagueness of ‘vague’ must be investigated further.

  2. Richard Dietz, one of my colleagues, suggests that ‘child’ is more illuminating example than ‘cruel’ as having an inferential role both in descriptive reasoning and in normative reasoning. I fully agree that ‘child’ is another, maybe more helpful, example of this point. For the time being, however, I follow Putnam’s example, ‘cruel’, in order to situate my argument more easily in the current debate on the relation between descriptivity and normativity. Actually, ‘cruel’ could play an inferential role in descriptive reasoning as well as in normative reasoning, in that, for example, a dog cruelly treated is damaged.

  3. In Japan where the death penalty (by hanging) is still retained under the constitutional restriction of not imposing cruel punishment, there has been the debate on whether killing by hanging is cruel or not. It takes about fourteen minutes to end a convict’s life by hanging, but the court judged that it is not cruel by saying that convicts lose their consciousness instantly after being hanged. Of course, it is still a controversial issue.

  4. Dietz also raised a question about whether components of descriptivity and normativity in my examples only coexist as individual constituents (like different colours in one and the same picture) so that this does not lead to vagueness of the distinction between them. To this question, I will answer that descriptivity and normativity cannot separately coexist in the same sentence without being interrelated (unlike in the case of colours) because of their countervailing asymmetry which I will discuss later in Sect. 7 by investigating how we should react if the relevant sentence conflicted with the matter of fact. Rather what I want to focus on is the relational fact that the more descriptive, the less normative, and vice versa.

  5. Of course, this formulation is made by simplifying the structure of the argument. Precisely speaking, the Imperial ruling is established according to various elements including historical and mythological contexts in addition to temporal length. However, this may not be a flaw in the argument, because such simplification is usually made in constructing the sorites paradox concerning typically vague predicates like ‘hot’ or ‘child’. What matters is whether the structure of the argument admits the sorites paradox or not.

  6. It was pointed out by an anonymous reviewer that the absence of a sharp distinction is not tantamount to the absence of a distinction—it is just a vague distinction. So why, the reviewer asks, is Kripke’s formulation not compromised given the existence of a vague distinction? This observation is absolutely true. Actually, my aim is not to criticize Kripke’s argument, but to elucidate the relation between two meta-predicates, that is, descriptivity and normativity. Therefore, if Kripke in reality admitted the vague boundary between two meta-predicates, my argument would result in bringing to light a hidden structure of Kripke’s argument. For the time being, however, Kripke so obviously develops his argument at least in this context by appealing to the sharp contrast between descriptivity and normativity that I provisionally presuppose he uses the sharp contrast between two meta-predicates as a necessary condition for his argument.

  7. The linkage between my argument and Waismann’s open texture argument was pointed out to me by an anonymous reviewer, for whose insightful observation I am most grateful.

  8. Philip Pettit has also developed a framework for understanding social norms by appealing to the notion of sanction. He offers the concept of the ‘the intangible hand’, through which people may suffer a sanction as a result of their own actions. This sanction motivates people to observe social norms (Pettit 1999, p. 225).

  9. Regarding the degree of descriptivity, I will discuss it further in Sect. 7 below.

  10. As far as the degree of difference between a normative and descriptive claim can be defined in terms of degrees of belief, the Bayesian approach (see Howson and Urbach 1993) may allow for dynamic interactions or updates, such as by interfacing with the social environment. This would require further exploration.

  11. I gave a talk on one part of my idea about DN in connection with the problem of causation by absence on 3 April 2015 at the 89th Annual Meeting of the American Philosophical Association Pacific Division, which was held in Vancouver of Canada. Ernest Sosa kindly gave me some comments, in which he asked how my idea of DN could treat the case of people lacking an ability to obey a norm. That is a quite significant question. For the time being, as long as I formulate that both SS and PS are measured from a mainly interpersonal point of view, SS should be evaluated with a consideration about people lacking an ability to obey a norm. But, this requires further clarification in detail, which is one of my future tasks.

  12. Sober (1994) provides an actual example of analysing the grue paradox in terms of probability.

  13. That descriptive facts must involve some kind of normative claim can also be seen in other cases. For instance, if we connect the notion of normativity with that of rationality, as Raz discussed, we have to consider normativity as seriously concerning the rationality of every epistemic belief. As Raz notes, ‘When studying reasons we study normative aspects of the world ... Our ability to reason is central to our rationality in all its manifestations, that is regarding reasons for belief, action, emotion, or anything else’ (Raz 2000, p. 43). If we understand the notion of normativity as a kind of guide or map drawn from our active deliberations on theoretical questions, as well as practical ones (such as Hookway developed), then normativity would conceptually permeate any description: ‘We can examine the normative standards that guide us when we try to carry out theoretical deliberations and conduct inquiries. These norms will guide us in formulating cognitive goals and selecting methods of deliberation or inquiry to employ in pursuit of them’ (Hookway 2000, pp. 60–61).

  14. According to Field, paraconsistent dialetheism takes \(\hbox {P(A)} + \hbox {P}(\lnot \hbox {A})\) to be greater than or equal to 1, while paracomplete theory takes \(\hbox {P(A)} + \hbox {P}(\lnot \hbox {A})\) to be less than or equal to 1 (Field 2008, p. 364). I regard P(A) and \(\hbox {P}(\lnot \hbox {A})\) as corresponding to DD and DN on the ground that there must be asymmetry between DD and DN. Of course, this is just a possibility, requiring more meticulous research. That is another subject for another paper.

  15. This crucial drawback was correctly pointed out by my colleague, Richard Dietz. I appreciate his acute observation. Actually, an anonymous reviewer also raised a question concerning the value of k. Should there be any constraints on the possible values k may take? I have no firm view on this question. As for the diagram, I presume that the maximum value for k in the diagram 3 should be 0.04. At the end of the day, I regard this question as empirical in nature.


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Correspondence to Masaki Ichinose.

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Ichinose, M. Normativity, probability, and meta-vagueness. Synthese 194, 3879–3900 (2017).

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  • Normativity
  • Descriptivity
  • Probability
  • Vagueness
  • Rule-following
  • Kripke