I've casually bandied about for your delectation (or annoyance and revulsion if you're averse to recondite lexemes, which aren't ever indispensable or even particularly helpful, except for establishing your bona fides among academicians :)), so I should probably explain them, though a full exposition of "Abduction" could certainly be managed without the distraction.
If you know that the truth of A will guarantee the truth of B, then in logic we generally say, more concisely, "A implies B," which is also written, "A-->B," which latter fact you can now immediately forget. (Take my word. Sherlock Holmes even advises it.) As an example, take "if it rains, the ground gets wet." Raining is A. Wet ground is B. If you know A is true, you can pretty much take B for granted. A implies B. Raining implies wetness. Here, we call A (raining) the "antecedent," and B (wetness of the ground) the "consequent." This is an implication that is almost always true. Sure, someone could have covered the whole landscape with a tarp, but we go with the odds. (There is a saying among physicians oft cited in connection with diagnostic methodology: "if you hear hoofbeats, do not first think of zebras.") We embrace the likeliest or the simplest explanations or implications that seem to pertain. (This principle, by the way, has also a name. It's called, "Occam's Razor," after Sir William of Occam, about whom you want to know, believe me, nothing else.) So if you come along and tell me that it's raining, I'm going to assume the consequent is now true and that the ground is wet. Actually, this is *deductive* (not "abductive") reasoning, of the specific variety known as "modus ponens." Forget that, too.
Now, we could also work this backwards, though our conclusions would lose the imprimatur of logically guaranteed truth. For example, you could tell me you've noticed that the ground is wet, and knowing that one possible (and probably the simplest) explanation (choice of implications) for this is that rain is occurring, I might jump to that conclusion. And it wouldn't be a bad jump. Not logically guaranteed (maybe a pack of wild dogs has decided to anoint the sidewalk), but very, very likely. If, however, I lived in a world of delusion in which I didn't believe in the existence of rain (or of global warming, perhaps), or if I somehow believed that rain always evaporated before it hit the ground, then I'd never hit on the right explanation providing me with the right "antecedent." I'd never guess "rain," and I'd come up with some screwy explanation such as that the sidewalk was crying.
That's what keeps happening to us. As Paul said, "video meliora" (we see what's right, the evidence in front of our noses), but we just can't believe the truth (it would conflict with one of the BIG LIES), so we arrive at the wrong conclusion.
Icebergs are melting everywhere, glaciers are disappearing, Greenland is starting to look like Palm Springs. If we weren't forbidden to reach out for the simple explanation (if there's global warming, the ice sheets will melt, and they're melting, so there *is* global warming because of old William of Occam), we'd get all alarmed and might even have listened to what every climatologist on the planet who wasn't a paid whore had been screaming about for at least a decade. This is abduction, but not, for our purposes, the most important application, though it might be nice to prevent NYC, Los Angeles and Tokyo from submerging under 100 feet of water.
Next: where it really matters.