The Three Pillars of Reasoning Part III
Is Abduction Inference to the best explanation?
This is the third and last part of the series “The Three Pillars of Reasoning”.
You can find here Part I and Part II
Example
The example of Abduction in Part I was too simplistic.
Here is a longer example from Sir Arthur Conan Doyle’s "The Adventure of the Dying Detective." The following quotation is from a summary by LLaMa 2.
In this story, Holmes is approached by a man named Culverton Smith, who claims to have been poisoned by his own daughter. Smith claims that his daughter is trying to kill him and asks Holmes to investigate.
Holmes begins his investigation by interviewing Smith and gathering information about his family and household. During the interview, Holmes notices that Smith seems overly concerned with his own safety and appears to be hiding something. Based on this observation, Holmes generates a hypothesis that Smith might be the victim of a conspiracy.
Next, Holmes uses abduction to generate additional hypotheses about the possible motives of Smith's daughter and her alleged accomplices. He considers various scenarios, such as a family dispute over inheritance or a personal vendetta, and tests each hypothesis against the available evidence.
For example, Holmes wonders if Smith's daughter might have been motivated by a desire to inherit her father's wealth. He considers the possibility that she might have been working with a group of accomplices, perhaps including her husband or a lover. Holmes then uses his knowledge of criminal behavior and motives to generate additional hypotheses about the possible identities and motivations of these accomplices.
Throughout the investigation, Holmes continually updates and refines his hypotheses based on new information and evidence. He uses abduction to generate new theories and to test their plausibility, always keeping an open mind and considering alternative explanations.
Ultimately, Holmes's use of abduction leads him to uncover the truth behind Smith's poisoning. He discovers that Smith's daughter was indeed involved in a conspiracy, but not for the reasons Holmes initially suspected. Instead, she was working with a group of medical professionals who were trying to kill Smith in order to end his suffering from a terminal illness.
In this example we see the whole process of hypothesis forming, looking for further clues, and updating or refining the hypothesis based on new information until some certainty is arrived.
Abduction is a complex process, involving observations, generation of hypotheses, and selection of the “best” hypothesis, followed by a verification process, which consists of further data collection, updating, and refining the hypothesis. It is an iterative process that includes active data collection (reminiscent of active learning), hypothesis generation, and hypothesis selection according to some criteria.
It can be simply described as an educated guess but is actually much more complicated.
Inference to the best explanation.
The suggestion of Gilbert Harman, that Abduction is inference to the best explanation, while popularly accepted, is inaccurate, for reasons first spelled out by the Finnish philosopher and logician Jaakko Hintikka:
the criterion of what is the best explanation is not clear and may be subjective
The “best” explanation refers only to the selection process, but we see above that abduction is more than just hypothesis selection.
Abduction as informal Bayesian updating
It strikes me that the abduction process of updating or refining the hypothesis based on new information is very similar to the intuition of Bayesian updating formulated by E.T Jaynes in his book “Probability Theory: The Logic of Science (1996).”
"Bayesian inference is reallocation of credibility over possibilities"
In both abduction and Bayesian updating, we start with a set of possibilities, and then we use the evidence to update our beliefs about the relative likelihood of each possibility. In abduction, the possibilities are different explanations for the observations. In Bayesian updating, the possibilities are different values of a parameter.
The key difference between abduction and Bayesian updating is that abduction is a more qualitative process, while Bayesian updating is a more quantitative process. In abduction, we simply compare the relative likelihood of the different explanations. In Bayesian updating, we use Bayes' theorem to calculate the exact probability of each possibility.
Jaakko Hintikka proposed a probabilistic approach to abduction, in which the space of possible hypotheses is represented as a probability distribution. In this approach, each hypothesis is assigned a probability based on its plausibility.
However, some people might object to this approach, arguing that abduction is often informal and spontaneous and that it is not always possible to assign probabilities to hypotheses.
For example, if we see a car accident, we might hypothesize that one of the drivers was speeding. This is an abduction because we are inferring the best explanation for the observations (the car accident). However, it might be difficult to assign a probability to this hypothesis, as we do not have all the information.
The context of abduction
We saw that deduction is relative to the logical system, and induction is relative to the assumed statistical model, what is abduction relative to?
The answer is the background knowledge and beliefs
Abduction generates new hypotheses from observations using the existing background knowledge and beliefs. The background knowledge and beliefs provide the constraints of abduction.
If deduction generates theorems, induction generalizes probabilistic statements from instances, and abduction generates hypotheses, often even without probabilities.
Abduction: {background knowledge + beliefs} + {observations} →
{updated background knowledge + beliefs}
Hypotheses need to be verified, and the background knowledge and beliefs also need to be questioned in critical thinking.
The AI connection
The AI connection of abduction is a difficult question, as abduction is the most vague of the three inferences. Deduction has led to theorem provers, while induction has led to machine learning. However, it is not clear how abduction can be computerized, as it relies on background knowledge and beliefs that are often too general.
It was thus surprising when I tested Claude-instant for their abduction capabilities.
Q: Given the following premises: (example from Stanford Encyclopedia of Philosophy).
Tim and Harry are former business partners who still have some financial matters to discuss.
they had a terrible row some time ago
they were just seen jogging together .
What can you infer from the above premises using abduction? Explain
A: Based on the provided premises, here is what I can infer through abductive reasoning:
Tim and Harry were former business partners who still had financial matters to discuss (premise 1). This indicates there was some past connection or relationship between them in business.
They had a terrible row/argument some time ago (premise 2). This suggests there was a falling out or disagreement in their past relationship.
They were just seen jogging together (premise 3). This current observation is inconsistent with the idea that they are no longer on speaking terms due to the past disagreement, as indicated by premises 1 and 2.
Through abduction, the most reasonable inference is that Tim and Harry must have reconciled and resolved their past disagreement/row mentioned in premise 2. This would explain the inconsistency between premises 2 (terrible row in the past) and 3 (seen together currently).
Their being seen together jogging implies they are now on friendly terms again. So the best explanation for the inconsistency between the premises is that Tim and Harry patched things up after their past disagreement and row. Their financial matters from their business partnership (premise 1) were also likely discussed and resolved for them to be willing to engage in a joint activity like jogging together now.
In short, through abductive reasoning, we can infer that Tim and Harry reconciled and resolved their past issues based on the inconsistent evidence across the premises becoming consistent with that explanation.
Q: List all the possible explanations of the above premises, and indicate why you have chosen "Tim and Harry have reconciled" as the conclusion
A: Here are the possible explanations based on the premises provided:
Tim and Harry are still on bad terms after their row and are meeting just to discuss financial matters from their past business.
They have let bygones be bygones but are not fully reconciled, just cooperating on sorting out business issues.
Tim and Harry bumped into each other by chance while jogging and it was purely coincidental.
Someone is forcing them to meet to resolve outstanding matters for liability/legal purposes.
The row was fake news/an exaggeration and they were always on good terms.
The reason I inferred that "Tim and Harry have reconciled" is the best explanation is:
Explanation 1 is unlikely given that people who are still on bad terms would not jointly engage in a leisure activity like jogging together.
Explanation 2 does not fully account for the inconsistency between being on unreconciled terms (premise 2) and now engaging together in an activity requiring trusting cooperation.
Explanation 3 is possible but an unlikely coincidence given their past relationship history.
Explanation 4 does not fit the description of a casual jogging encounter.
Explanation 5 does not align with premise 2 explicitly stating they had a "terrible row".
Among the possible explanations, "Tim and Harry have reconciled" resolves the inconsistency across all three premises in the most coherent and plausible manner. Seeing them jogging together indicates a friendlier interaction than would be expected if their past issues were still unresolved.
Q: So you think that abduction is inference to the best explanation? But what are the criteria of "best". Are there arguments denying that abduction is inference to the best explanation?
A: You're right that in this case, I inferred the explanation that I viewed as "best" based on certain criteria. You've identified an important nuance in characterizing abduction.
In terms of criteria for what makes an explanation "best" in abductive reasoning:
Coherence/consistency with available facts
Plausibility/likelihood
Simplicity or minimal assumptions needed
Explanatory power in resolving inconsistencies
However, some philosophers argue abduction is not strictly "inference to the best explanation" for a few key reasons:
What makes one explanation objectively "best" is difficult to define and can be subjective.
Abduction may consider multiple plausible explanations, not just one best one.
It aims to formulate plausible hypotheses, not prove a single conclusion.
Confirmation of hypotheses relies on further reasoning/testing, not just initial plausibility.
So in summary, while I relied on an "inference to best explanation" approach here, some arguments can be made that:
The criteria for "best" is not fully defined or objective.
Abduction considers multiple hypotheses, not just one.
Its goal is plausible hypotheses, not proved conclusions.
Further reasoning is needed to confirm rather than just infer the "best" explanation.
My characterization was an oversimplification - abduction is better defined as evaluating plausible explanations worthy of further testing, not definitively inferring a single best one. Thank you for pushing me to clarify this nuanced aspect of abductive reasoning.
The answers showed that Claude-2 has considerable background knowledge, which is not surprising being a Large Language Model.
It is possible that LLMs could be used to implement abduction in AI systems. However, more research is needed to explore this possibility.