The Three Pillars of Reasoning, Part I
Deduction, Induction, and Abduction and their Context
There are three main types of logical reasoning: deduction, induction, and abduction. While deduction and induction are relatively well-known, abduction is more obscure yet just as integral to cognition and science.
Here we will first examine the three forms of reasoning at a superficial level, which is how we understand them in everyday language. Then, we will explore them in a more general context.
Deduction
Deductive reasoning works from the general to the specific, deriving conclusions that must necessarily follow from given premises according to the rules of logic.
For example:
Premise 1: All men are mortal
Premise 2: Socrates is a man
Conclusion: Therefore, Socrates is mortal
The conclusion above is guaranteed if the premises are true because it logically follows based on the form of the argument. Deduction allows absolutely certain inferences but only gives us new conclusions contained implicitly within existing knowledge. It reveals nothing fundamentally novel.
Induction
In contrast to deduction, inductive reasoning works from the specific to the general. It takes observed instances and makes probabilistic inferences about unobserved cases.
For example:
Observation 1: That raven is black
Observation 2: That raven is black
Observation 3: That raven is black
Conclusion: Therefore, all ravens are black
Unlike deduction, the conclusion of inductive arguments is not guaranteed to be true, even if the premises are true. No matter how many black ravens we observe, a white raven could appear tomorrow. However, inductive reasoning allows us to make probabilistic guesses about future cases based on past patterns. This is useful for creating workable models of the world from limited evidence.
Science relies heavily on induction to infer general laws from limited empirical observations. However, induction can only extend existing knowledge - never create fundamentally new knowledge.
Abduction.
Abductive reasoning, pioneered by Charles Peirce, works unlike deduction or induction. With abduction, we start with a set of facts and then infer to the simplest, most likely explanation for those facts. Abduction is the process of puzzling over observations, creatively dreaming up hypotheses, and forming explanatory conjectures. It relies not just on logical reasoning but intuitive leaps as well.
Consider this scenario:
Fact 1: The grass is wet
Fact 2: It wasn't wet last night
Fact 3: There are no sprinklers around
Hypothetical Explanation: It rained earlier today
Unlike deduction, the abductive conclusion does not necessarily follow from the premises. And unlike induction, it does not summarize observations - it creatively explains them. Abductive inference is inherently speculative, producing hypothetical solutions rather than definite conclusions. It involves intuitive processes such as analogy, metaphor, and recognizing patterns that allow us to imaginatively link facts in new ways to form hypotheses even when we cannot logically deduce the explanation. Our experiences and mental models help guide these intuitive leaps.
Abduction powers diagnostic reasoning and allows us to make sense of weird facts. Doctors abduce diseases from symptoms, detectives infer motives from clues, and scientists create theories for unexplained data. Without abduction, we would be stuck in an endless loop of observing facts without grasping their significance. Abduction relies on both logical reasoning and intuition to generate potential explanations worthy of further investigation.
Zooming out
Let's take a step back and consider the above forms of reasoning in a more general context. This will help us to better understand their strengths and weaknesses, and to promote critical thinking.
Deduction is always relative to a logical system
To be able to deduce from premisses requires rules of inference within a defined logical system. A logical system has axioms (assumptions), rules of inference, and a generator of well-formed statements. A well-formed statement is called a theorem if it can be derived from the axioms using the rules of inference.
Think of the Euclidean geometry grounded in Euclid’s five axioms. Set theory is another such system. Or propositional calculus, first-order predicate calculus, second-order predicate calculus, or modal logic. There are dozens of logical systems.
Propositional and first-order predicate calculus have weak expressive power (for example they cannot be used to axiomatize the natural numbers), but they are consistent and complete.
However, Gödel showed second-order logic is either inconsistent or incomplete. Consistency means freedom from contradiction, while completeness relates to semantics - a system is complete if every theorem is semantically valid, and vice versa.
Inferencing from First Principles
Early artificial intelligence systems focused on deductive, symbolic reasoning to make inferences from first principles about a domain of knowledge. A major breakthrough was John Allen Robinson's discovery of the Resolution rule, which provided a single inference procedure that was sound and complete for predicate calculus. The Resolution rule was well-suited for efficient computer implementation.
However, these systems faced challenges of scale and combinatorial explosion as the length of inference chains grew. This limitation echoes bounded rationality in human reasoning, even when augmented by computing power.
This by the way, also shows limitations of Thinking from First Principles in general.
Applications to Critical Thinking and Creativity.
Understanding the context of deductive systems helps us with critical thinking and creativity.
Logical truths depend on the axioms and rules of inference defining a system. Two contradictory conclusions can each be “logical” if supported within their respective systems. Resolving such disagreements requires examining underlying assumptions and first principles.
Understanding how logical systems are constructed also spurs creative thinking by exposing opportunities to question and reframe basic axioms. Denying Euclid's fifth axiom gave rise to non-Euclidean geometries. Rejecting the law of the excluded middle produced intuitionistic logic. Challenging accepted first principles can yield radical new perspectives.
This article will be continued in Part II