While the name "abduction" has been chosen to resemble the traditional terms "deduction" and "induction", the name does not make sense.

The name for this kind of reasoning should suggest a reverse direction when compared to "deduction", but "abduction" has the same direction as "deduction".

The preposition "ab" is related to the English "off" and "of" and it can usually be translated by "from". The preposition "de" means the same thing as "ab" (which is why in most Romance languages, like French, "de" corresponds to English "of"), except that it suggests a vertical movement from up towards down, while "ab" suggests a horizontal movement.

I do not know which might have been a better word, maybe they should just have stuck with "reverse deduction", because "reduction" is widely used with other meanings, or maybe they could have used "adduction", which has an opposite direction to "abduction" ("ad" corresponds to English "at").

The names "deduction" and "abduction" make grammatical sense when their direct objects are the premises (i.e. the action is "lead from premises"), while the name "adduction" makes grammatical sense when the direct objects of this action are the conclusions (i.e. the action is "lead at conclusions").

IIRC the original English-language author who coined the term (C. S. Pearce) did use the verb form "adduce" as analogous to "deduce", so perhaps the original offending term, "abduction", is a poor back-formation.

From Latin abductiō (“robbing; abduction”), from abdūcō (“take or lead away”), from ab (“away”) + dūcō (“to lead”)[1]. Equivalent to abduct +‎ -ion.

"Away" and "to lead" indead does imply a reverse direction.

https://en.wiktionary.org/wiki/abduction

It does not imply a reverse direction with respect to "deduction", which was the intention of the name, as used in the book discussed here.

Both "abdūcō" and "dēdūcō" mean "lead away", except that the latter implies that the origin of the movement was somewhere above, i.e. the movement was descendant (the prefix "dē-" in descend is the same as the prefix "dē-" in deduce).

"Abduction" implies a reverse direction only vs. "adduction", which means leading towards something, not away of something, like "abduction".

The words "abduction" and "adduction" are frequently used to describe the movements of the human arms and legs, where the place that is the initial point of the movement for abduction or the final point of the movement for adduction is understood to be the median axis of the body. Thus there are abductor muscles and adductor muscles. (For example the abductor muscles of the thigh raise the thigh laterally, away from the other thigh, while the adductor muscles of the thigh bring the two thighs together.)

"Induction" also implies the same direction of movement as "adduction", i.e. towards something, and opposite of deduction/abduction, but "induction" implies that the final point of the movement is inside the target.

Latin dē, especially in etymollogy of deduction, usually means "of" - meaning: regarding, in respect of, concerning etc. (Look de facto, de jure, de dato, de futuro...);

with dūcō - meaning to lead, to guide forming deducere as present active infinitive;

in contrast with the prefix ab which usually means "from" or "away from" as in "ab ovo" where "de ovo" would mean something regarding the egg instead of originating from or away from the egg (also look ab initio, ab extra, ab hinc...).

Similair to the word absent coming from latin absens, absum where ab means "away from" and sum means "I am", taken together meaning "not present".

Thus in the context of logic where deduce would mean going in one way, abduce would imply going away from that way i.e. reverse direction.

There has never been any usage in which "dē" and "ab" signified opposite directions.

In all usages of both words, they designate the starting point of a movement. Anything that moves, moves away from the starting point of that movement.

The only difference in meaning is that for "ab" there was no information about the direction of movement in space, while "dē" implied that the direction of the movement comes from somewhere above towards somewhere below, e.g. something that falls from a tree.

In later stages of the Latin language there was a tendency to no longer pronounce the final consonants of both "ab" and "ad". So in classic Latin going "from X to Y" would be "ab X ad Y", which in later pronunciation would become the ambiguous "a X a Y". Because of this ambiguity, the formerly almost synonymous "dē" became completely synonymous with "ab" and it replaced "ab" in all usages, so that in the Romance languages going "from X to Y" became "de X a Y".

In classic Latin, the less definite "ab" was usually preferred when referring to real or concrete movements, while "dē" was usually preferred for imaginary or abstract movements, e.g. when saying that a child comes from its parents, a literary work comes from its author or a steel tool comes from a blacksmith.

"ab ovo" meant "from egg" and "de ovo" also meant in older Latin "from egg".

However, in the late Roman Republic, after it became fashionable to educate the children in the traditional Greek ways and typically with Greek professors (normally slaves), the necessity of translating the Greek texts has produced a very large number of new words in Latin, either by creating new composite words following the model of the corresponding Greek words, e.g. "hypothesis" => "supposition", or by giving additional new meanings to existing Latin words, to cover the meaning of the Greek words for which they were substituted in translations.

A very large number of Greek philosophical texts had titles such as "About X", i.e. "Peri X" in Greek. When translating such titles, "dē" has been chosen to translate Greek "peri" (which had a meaning close to English "around" or "about").

After Latin became strongly influenced by the Greek literature, "dē X" became frequently used to also mean "about X", not only "from X". However the sense "from X" has always remained valid, and which sense was meant has to be discovered from the context of "dē X".

One who would tell how someone had thrown an egg to the ceiling and the egg had broken and the yolk had fallen from the egg on his head, would say that the yolk had fallen "de ovo" with the meaning "from the egg", and not with the meaning "about the egg".

In "de facto", "de jure" and many of the similar expressions, "de" does not mean "regarding" or "concerning". It means "from", in the sense "caused by".

"De jure" does not mean "concerning the law", it means "from the law", in the sense that the cause why something has been done or must be done is because the law says so.

"De facto" does not mean "concerning the facts", it means "from the facts", in the sense that some present state has its cause in the facts (i.e. past actions) of some entity, e.g. the x86-64 ISA has become a "de facto" standard ISA for personal computers, because for many years the Intel/AMD CPUs in IBM PC/AT compatible computers have outcompeted everything else.

For others like me that wondered "Wait is proof by induction not induction?"

> Although its name may suggest otherwise, mathematical induction should not be confused with inductive reasoning as used in philosophy (see Problem of induction). The mathematical method examines infinitely many cases to prove a general statement, but does so by a finite chain of deductive reasoning involving the variable n, which can take infinitely many values.

https://en.wikipedia.org/wiki/Mathematical_induction

I think there might be a loose connection. When you want to prove something using mathematical induction, you must come up with an "induction hypothesis". This requires cleverness which could possibly be described as the product of inductive reasoning (in the philosophical sense). Essentially it amounts to finding a general principle that can be specialized to prove each of the inductive steps.

The main difference to me is that, with inductive reasoning (in the philosophical sense), you converge on a general principle but it might be wrong—it is only probable. Mathematical induction is the tool needed to close the gap—to turn a hypothesis which could be wrong into a bulletproof mathematical theorem. Of course, if your induction hypothesis didn't turn out to be right, you won't be able to complete the proof.

I use the Coq theorem prover to write induction proofs all the time, and often you have to try out different induction hypotheses until you get one that finally works. The process my brain goes through to come up with these induction hypotheses feels like "inductive reasoning" in the philosophical sense as I understand it.

Yeah, I think you nailed the reason why they both use the same word: "mathematical induction" is deductive once the proof is finished, but coming up with the proof is absolutely an inductive process.

Thanks, never thought about it this way: It seems like the NP of reasoning. If you know the solution everything seems much easier.

I was looking at that.

It seems like induction and abduction are pretty much that same thing but framed differently. Abduction looks for the "simplest explanation: and induction uses statistical machinery to argue for an explanation, which, if you break down the statistical machinery, is something like the lowest entropy, the most efficient encoding of the data, the ... simplest explanation.

The term "abduction" was coined by CS Peirce to describe the process that puts hypotheses on the docket to be tried by induction and deduction. Sometimes folks use it to mean something like "inference to best explanation", but that's not how Peirce meant it.

> Proposals for hypotheses inundate us in an overwhelming flood, while the process of verification to which each one must be subjected before it can count as at all an item, even of likely knowledge, is so very costly in time, energy, and money -- and consequently in ideas which might have been had for that time, energy, and money, that Economy would override every other consideration even if there were any other serious considerations. In fact there are no others. For abduction commits us to nothing. It merely causes a hypothesis to be set down upon our docket of cases to be tried.

— CS Peirce (Collected Papers §5.602)

> abduction commits us to nothing. It merely causes a hypothesis to be set down upon our docket of cases to be tried.

This is the perfect phrasing. Consider the contrast with deduction: if I have a premise, P, there are many deductive inferential rules I could apply to that premise to produce a new truth.

E.g. Given P, we can infer "not not P", and if we're also given "P implies Q" we can infer Q

Abduction flips this on its head: we have Q. And Q could be just the double negation of a P we weren't given. Or it could be the consequence of "P implies Q". There's really no way to know with certainty.

This is forensics. It's exactly the reasoning used in courtrooms: from the evidence (or "consequences"), we need to use science and logic to figure out how we got there (i.e. what were the antecedents).

This is why the most famous fictional user of "abduction" is Sherlock Holmes. But really it's what every forensic investigator (including a detective) is doing. Abduction is most common in judicial settings (i.e. when we are trying to assign blame).

Ah,

That is actually an illuminating perspective cutting through verbiage.

Abduction is the first "step", that the produces the ideas that can be considered ... by induction or deduction. It is the principle that we have to start somewhere.

Basically, an abductive step is something like "I can think of only ten hypotheses that explain X and only one of them works. I could consider an infinity of possibilities but it's not worth it. I will stick with my working hypothesis".

That's wrong. Induction is the process of empirically producing natural laws.

Our givens are the observed state of the experiment before (i.e. the antecedent, or "premise"), and after (i.e. the consequent, or "conclusion" of) an event. These givens are the only observations.

Induction is the process of creating an inferential rule (e.g. a causal hypothesis) that allows us to deduce the given consequent from the given antecedent (e.g. by applying a formula).

Abduction has different givens. In that case, we have observed consequences, and are given a set of inferential rules (e.g. natural laws) that could have produced the observed consequence. We then try to acquire evidence to support theories about which (unobserved) antecedent led to the (observed) consequent, by using one of our given inferential rules.

Your usage of "premise" and "conclusion" as shorthands for "the state of the world at a time t" is somewhat confusing to me, when I hear those words I think of propositions that have truth value, not events/states. You can sort of encode the state of the world at any instant as a proposition which is the conjunction of all possible true propositions describing the world at that instant ("The sky is blue", "Water is wet", "Temperature is 40C",...), but it's not obvious that this is a useful way to think in.

For example, suppose you're a mathematician with a conclusion (conjecture) C and a premise P (axioms), you search for a proof to get C from P, and you find it. That fits your definition of induction. Now suppose you're a pre-copernicus astronomer who have a "conclusion" and a "premise", which are actually just the state of the world recorded at different times, and you want to find a law that gets you the conclusion from the premise, and you find it in a system of epicycles. This too fits your definition of induction.

But those 2 situations are radically different. The mathematician's proof is much more correct and permanent than the astronomer's "proof". If I find another proof of the mathematician's claim, even a very different proof, that doesn't render the first proof obsolete, and it doesn't mean my proof is wrong either. But if I find another law that explains the states of the world the astronomer explained, then unless one of the laws is a special case of the other, either me or the astronomer is wrong, or the world is crazy and somehow obeys 2 sets of laws that aren't reducible to each other at the same time.

In other words, if logical propositions are imagined as a DAG where the edges represent valid rules of inference, then it doesn't matter which path (proof) got you to a node as long as you got there. But if you imagine the states of the world like this, then it actually matters what specific laws got you to a particular node. Maybe actual reality doesn't care and there are infinite systems of physics that all perfectly well explain the universe, but we very much care about path uniqueness, much more than we do in math, chess or other such systems. That's why I think the encoding of world states as propositions is lossy. A state\event is more than the propositions describing it.

> Your usage of "premise" and "conclusion" as shorthands for "the state of the world at a time t" is somewhat confusing to me

I agree, and the correct terms are "antecedent" and "consequent"; a premise is a type of antecedent, and a conclusion a type of consequent. But using those terms right away makes it harder to relate to formal arguments.

Given this definition, abduction seems to me to be the source of many, many arguments between significant others. "I know you! You must think [some random opinion] because of [some past unrelated event]."

In theory, you know the final opinion, and you know how your partner thinks, so you can logically guess what the source of the opinion is.

So abduction is armchair psychology?

Sometimes I understand abduction, other times, not. I'm a bit of a fan of logic programming so I try to describe the three inference modes using definite clauses.

Deduction: we go from facts (positive literals) and rules (definite clauses) to facts:

  swan(alice).   white(X) ← swan(X).
  ----------------------------------
            white(alice)

(If we know that Alice is a swan, and all swans are white, we can infer that alice is white).

Induction: we go from facts and rules to new rules:

  has_feathers(X) ← bird(X).  bird(X) ← swan(X).
  ----------------------------------------------
             has_feathers(X) ← swan(X).

(If we know that all birds have feathers and all swans are birds, we can infer that all swans have feathers).

Abduction: we go from facts and rules to _goals_ (negative literals):

  white(alice).   white(X) ← swan(X).
  -----------------------------------
            swan(alice)

(If we know that Alice is white and all swans are white, we can infer that Alice is a swan).

The last inference is of course not sound, as such - Alice could be white because she's a yogurt, say. But, if we operate under the assumption that only what we can prove is true (the so-called "closed world assumption") and the only rule we have about the concept of "white" is that "all swans are white", then the reasoning step becomes sound: the only way that we know for Alice to be white is for her to be a swan.

What if we know multiple ways for an "X" to be white? For example, what if we have the rule "white(X) ← yogurt(X)"? Well then, we can make this inference also:

  white(alice).   white(X) ← yogurt(X).
  -------------------------------------
            yogurt(alice)

And now we have two contradictory hypotheses about Alice: either she's a swan, or she's a yogurt^. We can resolve the contradiction when we learn more about yogurts and swans, and Alice. For example, if we have the information that Alice has feathers, we can figure out that Alice is a swan.

The closed world assumption is the simplest way to handle uncertainty (or in any case there are no simpler methods). It is also non-monotonic, in that new facts and rules can change the truth value of facts inferred from a theory. Which is great because that way we 're not stuck with the conclusions of initial premises.

Anyway, that's my logic-programmer's model for abduction. And it may be horribly wrong at places (my model for logic programming was horribly wrong the first ten years so...).

_______________

^Correction: Actually... alice can be a swan _and_ a yogurt. swan(alice) and yogurt(alice) are not contradictory unless we have some proof that they are.

Thanks. This is the closest I've come to understanding abductive reasoning and it's because you've explained it in a way my Prolog brain can understand it. Even though I've never written a Prolog program that abduces goals on its own, now I kind of want to try.

Thanks! Note that the first two inferences I show above are valid resolution steps since they have literals that are complements under unification, like swan(alice) and swam(X)← (the latter being a negative literal, since it's in the body). But the third and fourth inferences can't be made with resolution.

There is a whole bunch of work on abductive reasoning in a first-order framework, but I'm far from an expert on it. Antonis Kakas is a prominent name in the subject. For example, see:

Abductive Logic Programming, A.C. Kakas, R. A. Kowalski, F. Toni

https://academic.oup.com/logcom/article-abstract/2/6/719/942...

Although behind a paywall. And yes, that's that Kowalski. Here's an unpaywalled overview:

Abduction in Logic Programming, Marc Denecker and Antonis Kakas

https://web.stanford.edu/class/cs227/Readings/Abudction%20in...

And here's one on the application to learning:

Abductive concept learning, Antonis C. Kakas & Fabrizio Riguzzi

https://link.springer.com/article/10.1007/BF03037531

Yes, this is fantastic and accurate. It captures exactly the uncertainty that underpins this type of reasoning.

Abductive reasoning is usually an evidence-gathering process, so that we can narrow down the set of inferences that still produce meaningful conclusions.

Thus "once you've eliminated the impossible, whatever remains, however improbable, must be the truth."

Its also a great part of the "success"stories. Somebodys subconcious drags up a great idea or he is plain lucky, and then gets ask about the how and why. And the subconcious puzzles together a semi-plausible story, to explain away the heuristics in the head working and the statistics washing an outliar to shore.

This doesn't make a lot of sense because when you're trying to infer premises from conclusions the conclusions are your current premises. And premises you're searching will be your conclusions.

Any reason not to just call that “speculation”?