I present a puzzle which seems simple, but is found to have interesting implications for confirmation. Its dissolution also helps us to throw light on the relationship between first- and second-order probabilities construed as rational degrees of belief.

Barker and Achinstein misread Goodman's definitions of 'grue' and 'bleen'. If we stick to Goodman's definition of 'grue' as applying "to all things examined before t just in case they are green but to other things just in case they are blue" (my italics), and his parallel definition of 'bleen', then Barker and Achinstein's arguments are seen to be irrelevant. The result is to by-pass the question whether Mr. Grue sees things as grue rather than as green while showing that (...) it is possible for human conceptual schemes to employ different sensory terms. These two issues are separate. (shrink)

Extant literature on Goodman’s ‘New Riddle of Induction’ deals mainly with two versions. I consider both of them, starting from the (‘epistemic’) version of Goodman’s classic of 1954. It turns out that it belongs to the realm of applications of inductive logic, and that it can be resolved by admitting only significant evidence (as I call it) for confirmations of hypotheses. Sect. 1 prepares some ground for the argument. As much of it depends on the notion of evidential significance, this (...) concept is defined and its introduction motivated. Further, I introduce and explain the distinction between support and confirmation: put in a slogan, ‘confirmation is support by significant evidence’. The second section deals with the Riddle itself. I demonstrate that, given the provisions of the first section, not ‘anything confirms anything’: significant green-evidence confirms only green-hypotheses (and no grue-hypotheses), and significant grue-evidence confirms only grue-hypotheses (and no green-hypotheses), whichever terms we use for expressing these evidences or hypotheses. The third section rounds off my treatment. First I show that Frank Jackson’s use of his counterfactual condition is unsuccessful. Further, I argue that no unwanted consequences result, if one starts from the other, ‘objective’, definition of ‘grue’, as it constitutes no more than a mere fact of logic that cannot do any harm. Finally, I present a grue-case involving both kinds of definition, where the exclusive confirmation of either the green- or the grue-hypothesis is shown. (shrink)

Goodman'snew riddle of induction can be characterized by the following questions: What is the difference between grue and green?; Why is the hypothesis that all emeralds are grue not lawlike?; Why is this hypothesis not confirmed by its positive instances?; and, Why is the predicate grue not projectible? I argue in favor of epistemological answers to Goodman's questions. The notions of lawlikeness, confirmation, and projectibility have to be relativized to (actual and counterfactual) epistemic situations that are determined by the available (...) background information. In order to defend this thesis, I discuss an example that is less strange than the grue example. From the general conclusions of this discussion, it follows that grue is not projectible in the actual epistemic situation, but it is projectible in certain counterfactual epistemic situations. (shrink)

The problem of justifying induction has vexed philosophers for centuries. It has been entangled with issues concerning logic and probability and the philosophy of science. This article proposes a new approach to untangling these issues and resolving the overall problem. This new approach is by way of the perspective of realist ontology. Induction should not be seen as a debased form of logic and the search for inflexible rules for justifying particular sorts of inductive inference should be abandoned. Rather, induction (...) can be justified pragmatically as a general practice because reality possesses ordered patterns. Induction, being dependent on repetition, can sometimes identify aspects of those patterns simply because the instances of repetition really do occur. Thus, confidence in conclusions based on inductive inferences is not properly to be derived from unchanging logical rules governing when and where inductive inferences may be successfully applied, but from the involvement of such inferences with a wider, deeper process in which not only deductive inferences and other inductive inferences are involved, but also a set of relations to broader bodies of relevant knowledge. That is, unlike particular deductive inferences, inductive inferences may not be engaged in isolation from all other information about the world and completely in the abstract. Induction is always connected with a process of retroduction. (shrink)

The purpose of this article is to examine the applicability of the theory of projection for Anthropological hypotheses. The claim is made that Goodman's classic statement of the problem does not apply in its entirety to actual Anthropological hypotheses. The recent Freeman-Mead debate is employed as a framework for the discussion, illustrating that the issue of projectibility, while central for the social sciences, is best used as a backdrop to illustrate several important methodological problems. For Anthropology, and other related social (...) sciences, the central methodological problem, which is directly related to the projectibility one, is the development and justification of evidence-rules that can be used for a theory of confirmation. A preliminary attempt is then made to articulate the nature of these rules within the general Hempelian framework of qualitative confirmation. (shrink)

The extensions of Goodman’s ‘grue’ predicate and Kripke’s ‘quus’ are constructed from the extensions of more familiar terms via a reinterpretation that permutes assignments of reference. Since this manoeuvre is at the heart of Putnam’s model-theoretic and permutation arguments against metaphysical realism (‘Putnam’s Paradox’), both Goodman’s New Riddle of Induction and the paradox about meaning that Kripke attributes to Wittgenstein are instances of Putnam’s. Evidence cannot selectively confirm the green-hypothesis and disconfirm the grue-hypothesis, because the theory of which the green-hypothesis (...) is a part has an unintended model in which the grue-hypothesis is equally confirmed; and there are no meaning-facts that determine reference, because the objects referred to by the referring terms of any language or set of intentional mental states are permutable in a way that is consistent with the truth-values of all other sentences in that language or beliefs in that set. The upshot is that the three paradoxes need to be solved in a unified way. (shrink)

Israel 2004 claims that numerous philosophers have misinterpreted Goodman’s original ‘New Riddle of Induction’, and weakened it in the process, because they do not define ‘grue’ as referring to past observations. Both claims are false: Goodman clearly took the riddle to concern the maximally general problem of “projecting” any type of characteristic from a given realm of objects into another, and since this problem subsumes Israel’s, Goodman formulated a stronger philosophical challenge than the latter surmises.

One of the fundamental problems in the fields of inductive logic and the philosophy of science is the one concerning inferences or projections containing so-called “grue-like” or “pathological” predicates. This problem was first put into sharp focus by Nelson Goodman, who called it the “new riddle of induction.”Goodman has shown that the few attempts by others to solve this problem are not adequate. However, very little has been written concerning Goodman's own attempt to solve the problem, namely his theory of (...) entrenchment. The purpose of this article is to show that Goodman's entrenchment theory also is inadequate as a solution to the new riddle of induction. I shall try to do this by presenting two kinds of counterexamples to the entrenchment theory: one kind illustrating a general objection to the theory as a whole ; the other kind, specific objections to particular parts of the theory. (shrink)

Yehezkel presents an interesting argument to resolve Goodman’s New Riddle of Induction, and in effect to claim that a purely syntactical theory of confirmation is possible. In this paper I suggest that Yehezkel’s argument relies on two premises not proven in his paper. The first premise is that if “a New Riddle of Deduction” can be formulated then there is no New Riddle of Induction. This premise seems to be wrong as I claim in part 2 of this paper. The (...) second premise is that a New Riddle of Deduction can be formulated. Despite Yehezkel’s efforts to prove this claim, it seems to me false, or at least, unproven. I therefore conclude that there is no New Riddle of Deduction, and therefore, no actual resolution to the New Riddle of Induction. (shrink)

Goodman published his "riddle" in the middle of the 20th century and many philosophers have attempted to solve it. These attempts almost all shared an assumption that, I shall argue, might be wrong, namely, the assumption that when we project from cases we have examined to cases we have not, what we project are predicates. I shall argue that this assumption, shared by almost all attempts at a solution, looks wrong, because, in the first place, what we project are generalizations (...) and not predicates, and a generalization is projectible relative to a given context. In this paper I develop the idea of explainable-projectible generalizations versus unexplainable-unprojectible generalizations, relative to a specific context. My main claim is that we rationally project a generalization if and only if we rationally believe that there is something that explains the general phenomenon that the generalized statement in question asserts to obtain, and that a generalization is projectible, if and only if its putative truth can be explained, whether we know that it can be or not. (shrink)

This paper begins with a rigorous critique of David Stove''s recent bookThe Rationality of Induction. In it, Stove produced four different proofs to refute Hume''s sceptical thesis about induction. I show that Stove''s attempts to vindicate induction are unsuccessful. Three of his proofs refute theses that are not the sceptical thesis about induction at all. Stove''s fourth proof, which uses the sampling principle to justify one particular inductive inference, makes crucial use of an unstated assumption regarding randomness. Once this assumption (...) is made explicit, Hume''s thesis once more survives.The refutation of Stove''s fourth proof leads to some observations which relate Goodman''s grue paradox with randomness of a sample. I formulate a generalized version of Goodman''s grue paradox, and argue that whenever a sample, no matter how large, is drawn from a predetermined smaller interval of a population that is distributed over a larger interval, any conclusion drawn about the characteristics of the population based on the observed characteristics of the sample is fatally vulnerable to the generalized grue paradox. (shrink)

Kripke's lectures, published as Wittgenstein on Rules and Private Language , posed a sceptical problem about following a rule, which he cautiously attributed to Wittgenstein. He briefly noticed an analogy between his new kind of scepticism and Goodman's riddle of induction. ‘Grue’, he said, could be used to formulate a question not about induction but about meaning: the problem would not be Goodman's about induction—‘Why not predict that grass, which has been grue in the past, will be grue in the (...) future?’—but Wittgenstein's about meaning: ‘Who is to say that in the past I did not mean grue by “green”, so that now I should call the sky, not the grass, “green”?’. (shrink)

This paper examines problems of order and periodicity which arise when the attempt is made to define a confirmation function for a language containing elementary number theory as applied to a universe in which the individuals are considered to be arranged in some fixed order. Certain plausible conditions of adequacy are stated for such a confirmation function. By the construction of certain types of predicates, it is proved, however, that these conditions of adequacy are violated by any confirmation function defined (...) for the type of language in question. Various possible solutions to these difficulties are explored and found to be inadequate. In particular, a proposal which stems from the suggestion to restrict a fundamental principle of confirmation to hypotheses containing only non-positional predicates is cited. This proposal, however, is shown to prevent confirmation functions from taking periodicities into account, and so is deemed unsatisfactory. A general theorem is proved to the effect that if non-positional predicates are taken to satisfy the conditions of adequacy which have been formulated, then no periodicity predicates whatsoever (i.e., predicates used in formulating hypotheses which foretell periodicities) can be subject to these conditions, on pain of contradiction. Yet it seems that periodicity predicates must be subject to these conditions of adequacy if a confirmation function is to recognize periodic occurrences. Thus, an impasse seems to be reached. In the final sections we consider the beginnings of one possible solution to these difficulties. Our proposal involves treating sets of individuals, rather than individuals themselves, as instances of an hypothesis which predicts a periodicity. On this basis we formulate new conditions of adequacy which are free from the previous difficulties and which will permit a confirmation function that satisfies them to take periodicities into account. (shrink)

This paper reports (in section 1 “Introduction”) some quotes from Nelson Goodman which clarify that, contrary to a common misunderstanding, Goodman always denied that “grue” requires temporal information and “green” does not require temporal information; and, more in general, that Goodman always denied that grue-like predicates require additional information compared to what green-like predicates require. One of the quotations is the following, taken from the first page of the Foreword to chapter 8 “Induction” of the Goodman’s book “Problems and Projects”: (...) “Nevertheless, we may by now confidently conclude that no general distinction between projectible and non- projectible predicates can be drawn on syntactic or even on semantic grounds. Attempts to distinguish projectible predicates as purely qualitative, or non-projectible ones as time-dependent, for example, have plainly failed”. Barker and Achinstein in their famous paper of 1960 tried to demonstrate that the grue-speaker (named Mr. Grue in their paper) needs temporal information to be able to determine whether an object is grue, but Goodman replied (in “Positionality and Pictures”, contained in his book “Problems and Projects”, chapter 8, section 6b) that they failed to prove that Mr. Grue needs temporal information to determine whether an object is grue. According to Goodman, since the predicates “blue” and “green” are interdefinable with the predicates “grue” and “bleen”, “if we can tell which objects are blue and which objects are green, we can tell which ones are grue and which ones are bleen” [pages 12-13 of “Reconceptions in Philosophy and Other Arts and Sciences”]. But this paper points out that an example of interdefinability is also that one about the predicate “gruet”, which is a predicate that applies to an object if the object either is green and examined before time t, or is non-green and not examined before time t. The three predicates “green”, “gruet”, “examined before time t” are interdefinable: and even though the predicates “green” and “examined before time t” are interdefinable, being able to tell if an object is green does not imply being able to tell if an object is examined before time t (the interdefinability among three elements is a type of interdefinability present, for example, also among the logical connectives). Thus, it is wrong the Goodman’s thesis according to which if it is possible to determine without having temporal information whether the predicate “green” has to be applied to an object, then it is also possible to determine without having temporal information whether a predicate interdefinable with “green” has to be applied to an object. Another example of interdefinability is that one about a decidable predicate PD, which is interdefinable with an undecidable predicate PU: therefore even though we can tell whether an object is PD and whether an object is non-PD, we cannot tell whether an object is PU (since PU is an undecidable predicate) and whether an object is non-PU. Although the predicates PD and PU are interdefinable, the possibility to determine whether an object is PD does not imply the possibility to determine whether an object is PU (since PU is an undecidable predicate). Similarly, although the predicates “green” and “grue” are interdefinable, the possibility to determine whether an object is “green” even in absence of temporal information does not imply the possibility to determine whether an object is “grue” even in absence of temporal information. These and other examples about “grue” and “bleen” point out that even in case two predicates are interdefinable, the possibility to apply a predicate P does not imply the possibility to apply a predicate interdefinable with P. And that the possibility to apply the predicate “green” without having temporal information does not imply the possibility to apply the predicate “grue” without having temporal information. Furthermore, knowing that an object is both green and grue implies temporal information: in fact, we know by definition that a grue object can only be: 1) either green (in case the object is examined before time t); 2) or blue (in case the object is not examined before time t). Thus, knowing that an object is both grue and green, we know that we are faced with case 1, the case of a grue object that is green and examined before time t. Then the paper points out why the Goodman-Kripke paradox is a paradox about meaning that cannot have repercussions on induction. Finally the paper points out why Hume’s problem is a problem different from Goodman’s paradox and requires a specific treatment. -/- . (shrink)