Nelson Goodman’s and Carl Hempel’s paradoxes both relate to Hume’s general problem of induction: they are two major thought experiments in the attempted elaboration of a formal theory for inductive confirmation. Revealing of a time where research of a formal inductive theory was paramount in the mind of many philosophers, these two paradoxes are elements in a dialogue of friendly criticisms and refutation attempts between two contemporaries. In a parallel with the use of evidence in the confirmation/refutation of scientific hypotheses, these thought experiments shed light upon the problems of their predecessing theories of confirmation. In this essay, I will concentrate on the theory devised by Hempel himself as it played a key role in the exchange between the two philosophers: Hempel’s theory was elaborated to accommodate for the “Ravens Paradox” while Goodman later articulated his paradox out of that particular theory. I will start with a chronological presentation of both paradoxes, in an attempt to best reveal how they logically ensue from one another in the wider picture: the overarching problem of induction which in effect asks, “can induction truly be justified?” I will then show how Goodman’s paradox, while famous for its attack on specifically Hempel’s theory, puts the finger on an overarching problem common to any attempt of formal theory of confirmation.
Let us first define induction as an ampliative form of inference – inductive reasoning stems from the observation of particular instances which lead you to make conclusions of greater span. Such conclusions’ span goes beyond the initial particular statements, thus inductive reasoning is not truth-preserving. For example, say I observe a considerable number of swans, all white, I can induce “all swans are white”, but that does not make it true, and indeed, we have discovered black swans who originate from Australia. As such, what we call the ‘old problem of induction’ looks at justifying inductive inferences and our use of such a method; a method which, left unchanged, can lead us to making false conclusions.
In “Studies in the Logic of Confirmation”, Hempel makes an attempt at a formal definition of a valid induction and the conditions under which a generalisation is confirmed or disconfirmed by certain instances, in the hope of devising a theory that mirrors formal theories of deductions. He begins his argument with a critique of a former articulation of inductive confirmation theory called “Jean Nicod’s Criterion”. Nicod provided a definition for hypotheses of the form: "For any object x: if x is a P, then x is a Q”, and any object would confirm such a hypothesis only if both the antecedent and consequent. Hempel builds his falsification in two points. Firstly, “the applicability of this criterion is restricted to hypotheses of universal conditional form”, while what is desired is a criterion which can be applied to hypotheses of any form – existential statements for example. Secondly, Hempel articulates the following hypothesis: “all ravens are black”, from which a paradox consequently arises. By considering Nicod’s Criterion as a necessary and sufficient condition for confirmation, every black raven one sees is confirming evidence of the hypothesis. Equally, the hypothesis is equivalent to “all non-black things are non-ravens”; it follows that these equivalent propositions will be confirmed or disconfirmed by the same instances. “Any red pencil, any green leaf, and yellow cow, etc., becomes confirming evidence for the hypothesis”. The paradox lies in the fact that we risk testing our initial hypothesis about ravens, observing anything but ravens, something he calls “armchair ornithology”.
Hempel builds on the shortcomings of Nicod’s work to devise his own theory of confirmation. In response to the first failing of Nicod’s theory of confirmation, Hempel construes confirmation as a logical relationship between two statements: “observation reports”, which describe the given evidence, and the hypothesis. This, rather than a relation between non-linguistic objects (the evidence) and hypotheses as sentences. In the second part of his work, he attempts to accommodate for the “paradox of ravens” in his own theory. Hempel argues that if one knows a priori that the object of our study is a non-raven, then observing that it is not black should not further support our hypothesis. On the other hand, if we possess no such knowledge, discovering that it is non-black should confirm our hypothesis. Goodman accepted Hempel’s handling of the “Raven’s Paradox” but in an optic of further testing Hempel’s theory, he proceeded with the elaboration of the “grue” paradox.
It is best to introduce Goodman in the context of Hume’s work which proved a great source of inspiration for his work on induction. Hume argued that inductive inferences could not be justified because reasoning from experience presupposes the assumption that past experience is a reliable indicator of what we have yet to observe: this itself is induction. Thus, it would seem induction cannot be justified without circularity. Hume’s work led Goodman to label the "old problem” of induction justification a pseudo-problem, which he replaced by his “new riddle of induction”: “what is induction ?” Having put the finger on our willingness to project certain predicates into the future like “Since all emeralds we have so far observed have been green, we adopt the hypothesis that all future emeralds probably are going to be green also”. There commences his riddle: he adopts a time-indexed predicate “grue”, applied to objects if examined prior to a future time t and discovered to be green, or to objects examined after t and found to be blue. In past experience, the many emeralds we have observed have been green, none were blue. However, because all our evidence predates t, we can also qualify these emeralds as “grue”. Any evidence suggesting emeralds are green, can now be collected as evidence for emeralds being “grue”. Yet one of these hypotheses at least has to be false and we can formulate an infinite number of such predicates, which would give rise to an indefinite number of hypotheses confirmed by the same observations. Thus, he concludes that inductive reasoning cannot simply be an appropriate projection, interpretation, of our evidence since our observations can sometimes fit multiple hypotheses.
Goodman first introduced his predicate “grue” following the publication of Hempel’s theory of confirmation and defined his predicates and hypothesis out of its defined rules. Goodman’s formulation exemplified that although one knew which of the two incompatible predictions (“all emeralds are green” and “all emeralds are grue”) should be confirmed by the observation of green emeralds, they were equally confirmed in the present definition given by Hempel. It would seem that the choice of predicates is paramount to make valid inductive inferences: positional predicate – in time or space – like “grue” have to be excluded, unlike more qualitative predicates like “green”. Only, because grue-like predicates are possible, conditions must be added on how to conduct experiments, something Hempel had overlooked.
One obvious response made to Goodman was precisely to exclude grue-like predicates, but this appears rather ad hoc, and his response was that this was all relative to language. He insisted that were a person raised to speak a language where “grue” was a rooted predicate, they would not understand “green” and “blue” other than in terms of “grue” and a future set time t. For them, "green" would apply to all and only those things which are grue prior to a time t and “blue” posterior to that time. Goodman will maintain that "grue" is not intrinsically a more non-temporal predicate than is "green”, that “green” is inherently not positional in time. However, as shown by Barker in his paper “On the New Riddle of Induction”, we can easily prove that there is a legitimate difference between a “projectable” predicate like “green”, whose application can be extended to the future, and more temporal predicates like “grue”.
However, whatever the difference found between the above predicates, unlike what is classically thought, Goodman’s riddle does not pose formal difficulties because of those “grue”-like predicates, to Hempel’s logic of confirmation in particular. As shown by Hooker in “Goodman, ‘Grue’ and Hempel”, so long as we correspond hypotheses framed with temporal or positional predicates to “observation reports” that include the precise times or places of observations, these will go through Hempel’s system, just as any normal hypothesis would have. Interestingly, some authors have argued for the equivalence of the two paradoxes, asserting that Goodman’s “grue” poses a problem to Hempel’s theory of induction if and only if the “raven paradox” does so in an analogous way (see for instance, Boyce (year)). But the overarching difficulty remains: if a predicate such as “grue” is allowed, Hempel’s system, and any other formal theory of inductive confirmation, will offer no possibility of judging between exclusive and equally confirmed hypotheses; if it is not, potential theories do not account for everything, in an ad hoc manner, which is not desirable for empiricists. Ultimately, Hempel himself recognized that Goodman’s “grue” revealed that no purely syntactical theory of confirmation could be devised.
To conclude, Goodman and Hempel were two contemporary minds, in a wider philosophical context where the search for an infallible formal theory of induction was central to philosophical discussion. The chronological approach adopted in the present essay to relate Hempel’s and Goodman’s paradoxes reveals how scientific induction, dear to its defenders, was directly applied to the search of a confirmation theory. Taking a longer view, Nicod’s Criterion was falsified by the “experiment” of “Hempel’s ravens”, his own theory was falsified by Goodman’s “grue”, in logical positivist fashion. Ultimately, Goodman’s “grue” prodded at a fundamental weakness of inductivism, highlighting the apparent impossibility of a “total” theory of induction.