Abductive Reasoning: Fundamentals and Applications

By Fouad Sabry

What Is Abductive Reasoning

In abductive reasoning, one makes a series of observations and then draws the conclusion that is both the simplest and the one that is most likely to follow from those facts. Beginning in the latter third of the 19th century, the American philosopher Charles Sanders Peirce was the one who initially conceived of and advocated for this idea.

How You Will Benefit

(I) Insights, and validations about the following topics:

Chapter 1: Abductive reasoning

Chapter 2: Charles Sanders Peirce

Chapter 3: Scientific method

Chapter 4: Propositional calculus

Chapter 5: Modus ponens

Chapter 6: Modus tollens

Chapter 7: Statistical inference

Chapter 8: Inference

Chapter 9: Abductive logic programming

Chapter 10: Working hypothesis

(II) Answering the public top questions about abductive reasoning.

(III) Real world examples for the usage of abductive reasoning in many fields.

(IV) 17 appendices to explain, briefly, 266 emerging technologies in each industry to have 360-degree full understanding of abductive reasoning' technologies.

Who This Book Is For

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of abductive reasoning.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: Jun 26, 2023

Book preview

Chapter 1: Abductive reasoning

Abduction, another name for abductive reasoning, is a type of logical inference that looks for the most straightforward and likely conclusion from a series of observations. Beginning in the final part of the 19th century, American philosopher Charles Sanders Peirce developed it.

In contrast to deductive reasoning, abductive reasoning produces a possible conclusion but does not necessarily prove it. Uncertainty or doubt, which are represented in phrases like best available or most likely, are not eliminated by abductive conclusions. The optimal explanation can be inferred using abductive reasoning, which is widely used by diagnostic expert systems.

Deductive reasoning allows deriving b from a only where b is a formal logical consequence of a .

Alternatively put, deduction results in the predicted consequences.

assuming that the predictions are accurate, A sound deduction ensures that the conclusion is accurate.

For example, given that Wikis can be edited by anyone ( a_{1} ) and Wikipedia is a wiki ( a_{2} ), it follows that Wikipedia can be edited by anyone ( b ).

Inductive reasoning is the process of inferring some general principle b from a body of knowledge a , where b does not necessarily follow from a .

a might give us very good reason to accept b , but does not ensure b .

For example, if the swans we have seen thus far are all white, We can infer that it's plausible for all swans to be white.

We have solid grounds to believe the premise's conclusion, However, the conclusion's veracity is not assured.

(Indeed, Apparently, some swans are black.)

Abductive reasoning allows inferring a as an explanation of b .

Because of this conclusion, abduction allows the precondition a to be abducted from the consequence b .

Thus, the two types of reasoning—deductive and abductive—differ in their goals, right or left, of the proposition a entails b serves as conclusion.

As such, abduction is formally equivalent to the logical fallacy of affirming the consequent because of multiple possible explanations for b .

For example, in a game of pool, after looking and observing the eight ball approaching us, We might assume that the eight ball was struck by the cue ball.

The movement of the eight ball would be explained by the cue ball's strike.

It acts as a theory to account for our observation.

In light of all the scenarios where the eight ball could move,, We cannot be confident that the cue ball actually impacted the eight ball because of our abduction, but our kidnapping, still useful, can help us become more aware of our environment.

Despite the fact that any physical activity we observe could have many different causes, In an effort to better understand our environment and rule out some alternatives, we frequently abduce a single explanation (or a few explanations) for this process.

Properly used, In Bayesian statistics, abductive reasoning can be a helpful source of priors.

In logic, explanation is accomplished through the use of a logical theory T representing a domain and a set of observations O .

Abduction is the process of deriving a set of explanations of O according to T and picking out one of those explanations.

For E to be an explanation of O according to T , It should meet two requirements:

O follows from E and T ; E is consistent with T .

formal logical, O and E are assumed to be sets of literals.

The two conditions for E being an explanation of O according to theory T are formalized as:

{\displaystyle T\cup E\models O;}

T \cup E is consistent.

Among the possible explanations E satisfying these two conditions, some other condition of minimality is usually imposed to avoid irrelevant facts (not contributing to the entailment of O ) being included in the explanations.

Abduction is then the process that picks out some member of E .

The simplicity is one factor to consider when choosing a member to represent the best explanation, the earlier likelihood, or the explanation's capacity to explain.

First order classical logic has been given a proof-theoretical abduction method based on the sequent calculus, as well as a dual one based on semantic tableaux (analytic tableaux).

A computational system called abductive logic programming adds abduction to regular logic programming.

It separates the theory T into two components, Among these, one uses a standard logic program, used to generate E by means of backward reasoning, the second of which is a group of integrity restrictions, used to narrow the pool of explanation candidates.

Another way to formalize abduction is by flipping the function that determines how the hypotheses will manifest their effects.

Formally, we are given a set of hypotheses H and a set of manifestations M ; Due to the domain expertise, they are connected, represented by a function e that takes as an argument a set of hypotheses and gives as a result the corresponding set of manifestations.

Alternatively put, for every subset of the hypotheses H' \subseteq H , their effects are known to be e(H') .

Abduction is performed by finding a set H' \subseteq H such that M \subseteq e(H') .

Alternatively put, abduction is performed by finding a set of hypotheses H' such that their effects e(H') include all observations M .

The notion that the outcomes of the hypotheses are independent, or, for every H' \subseteq H , it holds that e(H') = \bigcup_{h \in H'} e(\{h\}) .

If this condition is met, Set covering may take the form of kidnapping.

The process of validating a given hypothesis via abductive reasoning is known as abductive validation. Alternatively known as successive approximation reasoning. According to this principle, an explanation is legitimate if it provides the best explanation for a given collection of available data. See Occam's razor for more information on how to define the best explanation in terms of simplicity and beauty. Science frequently uses abductive validation to support its hypotheses, and Peirce argues that it is a fundamental element of human mind:

On this beautiful spring morning, I observe an azalea in full bloom as I look out of my window. No, no! Although it is the only way I can define what I see, I don't see that. That is a proposition, a phrase, and a fact. However, what I see is merely an image, which I partially make understandable using a statement of fact. Although this assertion is abstract, what I observe is concrete. Every time I even mention anything I observe in a sentence, I commit an abduction. The truth is that every thread of our knowledge is made up entirely of pure hypotheses that have been strengthened via induction. Without making an abduction at every step, hardly the slightest increase in knowledge beyond the point of vacuous staring can be made.

Facts cannot be explained by a hypothesis more exceptional than these facts themselves, was Peirce's own dictum. The least extraordinary hypothesis must be accepted. Abductive validation is a technique for determining the most likely hypothesis that should be accepted after gathering potential hypotheses that could explain the facts.

By incorporating varying degrees of epistemic uncertainty in the input arguments, subjective logic generalizes probabilistic logic, i.e.

rather than probability, The analyst has the option to provide arguments as personal thoughts.

Thus, subjective logic abduction is a generalization of the probabilistic abduction already discussed.

Subjective opinions are the input arguments in subjective logic, and they can be binomial when they apply to a binary variable or multinomial when they apply to an n-ary variable.

A subjective opinion thus applies to a state variable X which takes its values from a domain \mathbf {X} (i.e.

a state space of exhaustive and mutually disjoint state values x ), and is denoted by the tuple {\displaystyle \omega _{X}=(b_{X},u_{X},a_{X})\,\!} , where {\displaystyle b_{X}\,\!} is the belief mass distribution over \mathbf {X} , {\displaystyle u_{X}\,\!} is the epistemic uncertainty mass, and {\displaystyle a_{X}\,\!} is the base rate distribution over \mathbf {X} .

These parameters satisfy {\displaystyle u_{X}+\sum b_{X}(x)=1\,\!} and {\displaystyle \sum a_{X}(x)=1\,\!} as well as {\displaystyle b_{X}(x),u_{X},a_{X}(x)\in [0,1]\,\!} .

Assume the domains \mathbf {X} and \mathbf {Y} with respective variables X and Y , the set of conditional opinions {\displaystyle \omega _{X\mid Y}} (i.e.

one conditional opinion for each value y ), and the base rate distribution a_{Y} .

Considering these factors, the subjective Bayes' theorem denoted with the operator {\displaystyle \;{\widetilde {\phi }}} produces the set of inverted conditionals {\displaystyle \omega _{Y{\tilde {\mid }}X}} (i.e.

one inverted conditional for each value x ) expressed by:

{\displaystyle \omega _{Y{\tilde {|}}X}=\omega _{X|Y}\;{\widetilde {\phi \,}}\;a_{Y}} .

Using these inverted conditionals together with the opinion \omega_{X} subjective deduction denoted by the operator {\displaystyle \circledcirc } can be used to abduce the marginal opinion {\displaystyle \omega _{Y\,{\overline {\|}}\,X}} .

The following list demonstrates how the many expressions for subjective abduction are equal:

{\displaystyle {\begin{aligned}\omega _{Y\,{\widetilde {\|}}\,X}&=\omega _{X\mid Y}\;{\widetilde {\circledcirc }}\;\omega _{X}\\&=(\omega _{X\mid Y}\;{\widetilde {\phi \,}}\;a_{Y})\;\circledcirc \;\omega _{X}\\&=\omega _{Y{\widetilde {|}}X}\;\circledcirc \;\omega _{X}\;.\end{aligned}}}

The symbolic notation for subjective abduction is {\displaystyle {\widetilde {\|}}} , and the operator itself is denoted as {\displaystyle {\widetilde {\circledcirc }}} .

The operator for the subjective Bayes' theorem is denoted {\displaystyle {\widetilde {\phi \,}}} , and subjective deduction is denoted {\displaystyle \circledcirc } .

When comparing subjective logic abduction to probabilistic abduction, the former has the advantage of allowing for the explicit expression and consideration of both aleatoric and epistemic uncertainty regarding the input argument probabilities. Thus, it is conceivable to conduct abductive analysis in the face of ambiguous arguments, which inevitably leads to varying degrees of ambiguity in the conclusions drawn as a result.

Abduction was a modern logical concept first articulated by the American philosopher Charles Sanders Peirce. He has referred to this type of inference as abduction, assumption, and retroduction over the years. He saw it as an issue in philosophy's normative branch of logic, not just in formal or mathematical logic, and subsequently as a topic in research economics as well.

Abduction and induction are two steps of the creation, expansion, etc. of a hypothesis in scientific investigation, and they are sometimes combined into one general term - the hypothesis. Because of this, the abductive phase of hypothesis development is conceived as induction in the scientific method that was developed by Galileo and Bacon. Thus, Karl Popper's explanation of the hypothetico-deductive paradigm, in which the hypothesis is believed to be only a guess, strengthened this collapse in the 20th century (in the spirit of Peirce). However, it becomes evident that this guess has already been tested and strengthened in thought as a required stage of its attaining the status of hypothesis when the construction of a hypothesis is considered the product of a process. Before they ever get to this point, many abductions are actually rejected or significantly modified by later abductions.

Peirce treated abduction as the application of a well-known rule to account for an observation prior to 1900. For instance, it is a well-known truth that when it rains, grass gets wet. Therefore, one can infer that it has rained in order to explain why the grass on this lawn is wet. Abduction might result in incorrect conclusions if other theories that could account for the observation are not considered; for instance, the grass might be damp from dew. The term abduction is still often used in the social sciences and in artificial intelligence in this way.

Peirce consistently described it as the type of inference that results in a hypothesis by providing an explanation, albeit an uncertain one, for some extremely puzzling or unexpected (anomalous) observation made in a given claim. He stated in writings dating back to 1865 that all conceptions of cause and force are obtained by hypothetical inference, and in writings from the 1900s that all theoretical explanatory content is reached through abduction. In other ways, Peirce changed his perspective on kidnapping over time.

Later, his viewpoint developed:

A kidnapping is a guess.).

Abduction conjures up a novel or unusual theory to rationally, instinctively, and cost-effectively explain a puzzling or intricate phenomena. That is its primary objective.

Longer term, it seeks to streamline inquiry itself. Its justification is