Whereas deductive inference begins from a small set of axioms or premises, and inductive inference from a large group of instances of a given phenomenon, abductive inference is more like a shot in the dark.
When we come across a “surprising fact” we need to explain it. Why is it surprising to us? We then come up with an explanation of the curious fact. But that explanation must serve a purpose. That is to make the curious or surprising fact non-curious or unsurprising. The explanation explains away the fact’s curious nature or surprising nature. By explaining or understanding the fact, we take away its anomalous character. Things are only surprising or curious if we can't explain them.
This explanation of the fact, according to C.S. Peirce, would therefore be an “explanatory hypothesis”. This means that such a hypothesis isn't deductively or inductively inferred from anything (as such). It's not a logical conclusion, entailment or implication. It's an explanation of a given fact. The explanation itself isn't thereby factual. It's a means of making sense of a given phenomenon.
Another important point about Peircian hypotheses is that they come before any tests, calculations, experiments, etc. These things are carried out to determine the truth of the hypothesis in question. It's not the case, as many people think, that the hypothesis is formulated after the tests, calculations and experiments. It's not, as it were, inferred from such things. Instead the hypothesis motivates or brings about the tests, calculations and experiments, rather than vice versa.
Hypotheses are essentially creative acts or even acts of intuition (if in a loose sense). They're neither the logical consequence of things, nor are they derived from empirical experimentation.
For example, if I see that a town has been levelled to the ground, I'll immediately formulate the hypothesis, say, that there's been a nuclear bomb dropped on the town. Though I wouldn't have formulated this hypothesis after carrying out radiation tests, calculating the strength of the bomb or collecting the data of destruction to help me inductively infer that there's been a nuclear attack. No, the hypothesis is (as it were) spontaneous. Or even if it weren't exactly spontaneous, it would still not rely on prior tests, experiments or things that I can deductively or inductively infer from them. The whole point of the hypothesis is to get the ball rolling. It's not what comes after the ball has stopped rolling.
What matters about the “explanatory hypothesis” is what it says will happen if certain experiments or tests are carried out. If the results that it conjectures do in fact happen, then the hypothesis is taken as true (if only for a given amount of time).
Firstly we begin with abduction. However, after the acts of abduction it will indeed be the case that scientist will utilise the principle of induction. That is:
i) First comes the abduction.
ii) Then come the tests, experiments and calculations (which attempt to legitimise the abductive inference).
iii) Then all this is put together via various inductive processes.
We infer from these various experiments, tests, calculations, and abductions, a single inductive inference or conclusion.
Many people seem to think that science is all about induction. In fact it's logically about induction, deduction and abduction (amongst many other non-logical things). In fact, because abductive acts come first, they could be deemed to be the most important type of inference out of the three. The abduction, or “explanatory hypothesis”, is used as a basis for further inductive inferences. It becomes a guide for later inductive processes. This means that the hypothesis says this or that, or explains this or that. Then the scientist investigates various examples or instances of these phenomena which are explained by the hypothesis. The scientist then sees whether or not it's the case that the explanatory hypothesis holds true for the many instances of the phenomenon concerned.
Inductively speaking, the abduction must explain more than a single phenomenon. It must also explain every phenomenon of the type. And the different instances of this type are put together or made uniform through various inductive processes. After all this is achieved, the scientific results will be formalised via various deductive logical systems.
For example, the inductive conclusion (itself dependent on the adductive inference) may itself become an axiom or premise in an otherwise deductive logical system. If the inductively-derived conclusion is true, then what will deductively follow from it? What will follow from it which can be discovered simply by analysing the conclusion-come-premise itself? What can we deduce simply by exploring the premise’s logical grammar (as it were)?