Abductive vs. inductive reasoning To me, abductive reasoning and inductive reasoning are very very similar, in that they both go from the specific to the general and they are distinguished only through the examples which are provided in their descriptions: one may use an inductive reasoning for instance to prove a counting formula in combinatorics, while your doctor may look at the symptoms and from that, abduct the original cause of the symptoms. Also, Darwin theory of evolution (as he came to the conclusion), is said to be an example of abduction. 

What is abduction, in mathematically precise terms, if there can be anything like that. Is abduction a mathematically valid form of reasoning?  

Thanks
 A: This Youtube video distinguished and explained most clearly for me:

Understanding these 3 words' etymologies can help: 

$\color{limegreen}{\text{Retroduction :}}$
  The prefix "retro," occurs in loanwords from Latin having to do with going backward. Yet, the prefix "retro" provides an implication of deliberateness–of deliberately "choosing" to go backward for a purpose. Thus "retroactive" means choosing to go back to an earlier date and make something operative as of that date. "Retrofit" means choosing to go back and modify an earlier model of something with an improvement of some sort. The combination of the prefix "retro" (as deliberately "going backward") with the suffix "ductive" from the Latin ducere (to lead) places the meaning of retroduction as "deliberately leading backward." This implies that retroduction is intended to be a deliberate and recursive process involving more than the making of an abductive inference. Its Latin roots indicate that "retroduction" refers, not only to the apprehension of a "surprising fact," and an ensuing hunch, but also that the hunch, once formed, is deliberately and recursively taken "backward" for analysis and adjustment (requiring deduction and induction), before it is engendered into a hypothesis worthy of extensive testing.
$\color{limegreen}{\text{Abduction :}}$
  The prefix "ab" appears in loanwords from Latin where it
  meant "away from." Thus we have words like "abdicate" and
  "abolition"–going "away from" the throne and from slavery,
  respectively. Thus, when the prefix "ab" (away from) is combined with
  the suffix "ductive" (from the Latin ducere, meaning to lead) we have
  the meaning of abduction as "leading away from." The term "abduction"
  fits well with the concept of abduction as moving "away from" a
  particular course or topic, as one would when responding to an
  anomaly, or a "surprising fact." The Latin root for "abduction" does
  not fit with the idea of going backward to explicate and evaluate an
  idea. Rather, this root indicates that the outward movement of an
  abductive inference allows the result of such an inference to be left
  as a completion, or used as the sole means for further exploration of
  possibilities–as in the arts.
$\color{limegreen}{\text{Deduction :}}$
  The prefix "de" from Latin loanwords refers to separation,
  removal, and negation. When we combine the prefix "de" (to separate)
  with the suffix "ductive" (to lead), we have the meaning of deduction
  as "leading to separation, removal, or negation," which are the goals
  and consequences of deductive reasoning.
$\color{limegreen}{\text{Induction :}}$
   The prefix "in," also from the Latin has to do with
  inclusion. Thus, the prefix "in" (to include) combined with the suffix
  "ductive" means "leading into" (or including), as one would do when
  reaching a conclusion by estimating from a sample, or generalizing
  from a number of instances.
Therefore, based upon their Latin derivations (to which Peirce was partial, as he was for Greek roots) our four terms have the following meanings:
Retroduction = deliberately leading backward.
          Abduction = leading away from
          Deduction = leading to separation, removal, or negation.
          Induction = "leading into" (or including) .   

A: Per Git Gud:
Wikipedia's article on abductive reasoning, especially the section comparing it to deductive and inductive reasoning and the section on logic-based abduction should answer your questions.
A: In addition to the above, I saw http://www.cs.utexas.edu/~ml/papers/abind-chapter.pdf and found the description on page 3 to be very instructive.
A: I highly recommend the Youtube video above, and would like to throw in some intuitive clarity: It's like y=f(x). 
Deduction is the most straight forwards: find y. Given an input and the rule, just plug in to find the output. You will always converge at some answer. 
Abduction is finding x. You're given y, and f, implying that x is f inverse of y. Not all functions have one output when inverted, like in math square root, inverse sine, and etc.. You'll often find a set a possible causes, rather than one in particular, so abduction isn't always valid for determining an exact cause. 
Induction is like finding a function f, given a point on its graph (x,y). You can produce an infinite "number" of functions that might pass through that point, but to make it perfect, you have to find all the points. 
Induction, abduction, and deduction, represent solving for x (a cause), y (the effect), and f (the rule of the pattern that changes x to y). That is the abstract concept of a function (relation between two things). They describe patterns (and even randomness), which encompasses the method by which any system imaginable operates. This means these three methods of reasoning are fundamental in using axioms to develop knowledge. The big idea is finding information about a pattern using its other parts (rule, input, and output). 
