The following is excerpted from Analogical Reasoning by John Sowa and Arun Majumdar.

Analogy and Perception

Before discussing the use of analogy in reasoning, it is important to analyze the concept of analogy and its relationship to other cognitive processes. General-purpose dictionaries are usually a good starting point for conceptual analysis, but they seldom go into sufficient depth to resolve subtle distinctions. A typical dictionary lists synonyms for the word analogy, such as similarity, resemblance, and correspondence. Then it adds more specialized word senses, such as a similarity in some respects of things that are otherwise dissimilar, a comparison that determines the degree of similarity, or an inference based on resemblance or correspondence.

Logical and Analogical Reasoning

In developing formal logic, Aristotle took Greek mathematics as his model. Like his predecessors Socrates and Plato, Aristotle was impressed with the rigor and precision of geometrical proofs. His goal was to formalize and generalize those proof procedures and apply them to philosophy, science, and all other branches of knowledge. Yet not all subjects are equally amenable to formalization. Greek mathematics achieved its greatest successes in astronomy, where Ptolemy’s calculations remained the standard of precision for centuries. But other subjects, such as medicine and law, depend more on deep experience than on brilliant mathematical calculations. Significantly, two of the most penetrating criticisms of logic were written by the physician Sextus Empiricus in the second century AD and by the legal scholar Ibn Taymiyya in the fourteenth century.

Sextus Empiricus, as his nickname suggests, was an empiricist. By profession, he was a physician; philosophically, he was an adherent of the school known as the Skeptics. Sextus maintained that all knowledge must come from experience. As an example, he cited the following syllogism:

Every human is an animal.
Socrates is human.
Therefore, Socrates is an animal.

Sextus admitted that this syllogism represents a valid inference pattern, but he questioned the source of evidence for the major premise Every human is an animal. A universal proposition that purports to cover every instance of some category must be derived by induction from particulars. If the induction is incomplete, then the universal proposition is not certain, and there might be some human who is not an animal. But if the induction is complete, then the particular instance Socrates must have been examimed already, and the syllogism is redundant or circular. Since every one of Aristotle’s valid forms of syllogisms contains at least one universal affirmative or universal negative premise, the same criticisms apply to all of them:  the conclusion must be either uncertain or circular.

The Aristotelians answered Sextus by claiming that universal propositions may be true by definition: since the type Human is defined as rational animal, the essence of human includes animal; therefore, no instance of human that was not an animal could exist. This line of defense was attacked by the Islamic jurist and legal scholar Taqi al-Din Ibn Taymiyya. Like Sextus, Ibn Taymiyya agreed that the form of a syllogism is valid, but he did not accept Aristotle’s distinction between essence and accident (Hallaq 1993). According to Aristotle, the essence of human includes both rational and animal. Other attributes, such as laughing or being a featherless biped, might be unique to humans, but they are accidental attributes that could be different without changing the essence. Ibn Taymiyya, however, maintained that the distinction between essence and accident was arbitrary. Human might just as well be defined as laughing animal, with rational as an accidental attribute.

Denouncing logic would be pointless if no other method of reasoning were possible. But Ibn Taymiyya had an alternative: the legal practice of reasoning by cases and analogy. In Islamic law, a new case is assimilated to one or more previous cases that serve as precedents. The mechanism of assimilation is analogy, but the analogy must be guided by a cause that is common to the new case as well as the earlier cases. If the same cause is present in all the cases, then the earlier judgment can be transferred to the new case. As an example, it is written in the Koran that grape wine is prohibited, but nothing is said about date wine. The judgment for date wine would be derived in four steps:

1. Given case:  Grape wine is prohibited.
2. New case:  Is date wine prohibited?
3. Cause:  Grape wine is prohibited because it is intoxicating; date wine is also intoxicating.
4. Judgment:  Date wine is also prohibited.

In practice, the reasoning may be more complex. Several previous cases may have a common cause but different judgments. Then the analysis must determine whether there are mitigating circumstances that affect the operation of the cause. But the principles remain the same: analogy guided by rules of evidence and relevance determines the common cause, the effect of the mitigating circumstances, and the judgment.

Besides arguing in favor of analogy, Ibn Taymiyya also replied to the logicians who claimed that syllogistic reasoning is certain, but analogy is merely probable. He admitted that logical deduction is certain when applied to purely mental constructions in mathematics. But in any reasoning about the real world, universal propositions can only be derived by induction, and induction must be guided by the same principles of evidence and relevance used in analogy. Figure 1 illustrates Ibn Taymiyya’s argument:  Deduction proceeds from a theory containing universal propositions. But those propositions must have earlier been derived by induction using the methods of analogy. The only difference is that induction produces a theory as intermediate result, which is then used in a subsequent process of deduction. By using analogy directly, legal reasoning dispenses with the intermediate theory and goes straight from cases to conclusion. If the theory and the analogy are based on the same evidence, they must lead to the same conclusions.

Figure 1:  Comparison of logical and analogical reasoning

The question in Figure 1 represents some known aspects of a new case, which has unknown aspects to be determined. In deduction, the known aspects are compared (by a version of structure mapping called unification) with the premises of some implication. Then the unknown aspects, which answer the question, are derived from the conclusion of the implication. In analogy, the known aspects of the new case are compared with the corresponding aspects of the older cases. The case that gives the best match may be assumed as the best source of evidence for estimating the unknown aspects of the new case. The other cases show alternative possibilities for those unknown aspects; the closer the agreement among the alternatives, the stronger the evidence for the conclusion.

Both Sextus Empiricus and Ibn Taymiyya admitted that logical reasoning is valid, but they doubted the source of evidence for universal propositions about the real world. What they overlooked was the pragmatic value of a good theory:  a small group of scientists can derive a theory by induction, and anyone else can apply it without redoing the exhaustive analysis of cases. The two-step process of induction followed by deduction has proved to be most successful in the physical sciences, which include physics, chemistry, molecular biology, and the engineering practices they support. The one-step process of case-based reasoning, however, is more successful in fields outside the so-called “hard” sciences, such as business, law, medicine, and psychology. Even in the “soft” sciences, which are rife with exceptions, a theory that is successful most of the time can still be useful. Many cases in law or medicine can be settled by the direct application of some general principle, and only the exceptions require an appeal to a long history of cases. And even in physics, the hardest of the hard sciences, the theories may be well established, but the question of which theory to apply to a given problem usually requires an application of analogy. In both science and daily life, there is no sharp dichotomy between subjects amenable to strict logic and those that require analogical reasoning.

The informal arguments illustrated in Figure 1 are supported by an analysis of the algorithms used for logical reasoning. Following is Peirce’s classification of the three kinds of logical reasoning and the way that the structure-mapping operations of analogy are used in each of them:

• Deduction.  A typical rule used in deduction is modus ponens:  given an assertion p and an axiom of the form p implies q, deduce the conclusion q. In most applications, the assertion p is not identical to the p in the axiom, and structure mapping is necessary to unify the two ps before the rule can be applied. The most time-consuming task is not the application of a single rule, but the repeated use of analogies for finding patterns that may lead to successful rule applications.
• Induction.  When every instance of p is followed by an instance of q, induction is peformed by assuming that p implies q. Since the ps and qs are rarely identical in every occurrence, a form of analogy called generalization is used to derive the most general implication that subsumes all the instances.
• Abduction.  The operation of guessing or forming an initial hypothesis is what Peirce called abduction. Given an assertion q and an axiom of the form p implies q, the guess that p is a likely cause or explanation for q is an act of abduction. The operation of guessing p uses the least constrained version of analogy, in which some parts of the matching graphs may be more generalized while other parts are more specialized.

As this discussion indicates, analogy is a prerequisite for logical reasoning, which is a highly disciplined method of using repeated analogies. In both human reasoning and computer implementations, the same underlying operations can be used to support both.