Abductive vs. inductive reasoning

Question

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

Answer 1

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.

Answer 2

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).