the source of the problem of induction as we know it in the modern context comes from Hume; Book I, Part III, section VI of the Treatise Hume doesn’t use the word induction. Rather, his argument is centered around inferences concerning causal connections. In it Hume argues that causation is the strongest and most important associative relation, over and above our memory and senses. Therefore causation is the basis of our probabilistic reasoning, and our reasoning connects the present fact to that which we infer from it. Hume investigates the nature of this connection between reasoning and inference (he discerns two categories of reasoning; demonstrative, by which he means deductive, and probabilistic, by which he means inductive).
Hume’s project in The Treatise:
“…the development of the empirical science of human nature. The epistemological sector of this science involves describing the operations of the mind, the interactions of impressions and ideas and the function of the liveliness that constitutes belief.” Stanford Encyclopedia
From the Stanford Encyclopedia:
Hume proceeds first negatively, to show that our causal inferences are not due to reason, or any operation of the understanding. Reasoning concerns either relations of ideas or matters of fact. Hume quickly establishes that, whatever assures us that a causal relation obtains, it is not reasoning concerning relations between ideas. Effects are distinct events from their causes: we can always conceive of one such event occurring and the other not. So causal reasoning can’t be a priori reasoning.
Reasoning includes the imagination and can be considered as a kind of cause.
If inductive conclusions were produced by the understanding, inductive reasoning would be based upon the premise that nature is uniform; “that instances of which we have had no experience, must resemble those of which we have had experience, and that the course of nature continues always uniformly the same.” (Hume THN, 89)
We are left without any link between past and future, once induction is recognized as unsound logic. Hume clears space here for his own investigation and solution: (from Stanford Encyclopedia)
This principle can’t be some “intricate or profound” metaphysical argument Hume overlooked. For all of us — ordinary people, infants, even animals — “improve by experience,” forming causal expectations and refining them in the light of experience. Hume’s “sceptical solution” limits our inquiries to common life, where no sophisticated metaphysical arguments are available and none are required.
Hempel – the Raven paradox
a is a raven and is black
b is a raven and is black
c is a raven and is black
all ravens are black
a is a non-raven and is non-black
all non-black things are not ravens
This logic entails that all ravens are black is equal to all non-black things are non-ravens, therefore a white shoe is a non-raven.
The problem here is that observing a white shoe tells us nothing of the color of ravens.
Goodman – Grue
Suppose that at time t we have observed many emeralds to be green. We thus have evidence statements
Emerald a is green,
Emerald b is green,
and these statements support the generalization:
All emeralds are green.
But now define the predicate “grue” to apply to all things observed before t just in case they are green, and to other things just in case they are blue. Then we have also the evidence statements
Emerald a is grue,
Emerald b is grue,
and these evidence statements support the hypothesis
All emeralds are grue.
Hence the same observations support incompatible hypotheses about emeralds to be observed in the future; that they will be green and that they will be blue.
The problem of induction is central and intrinsically tied to the epistemological problem of justification.