An analogy is a way to compare two things (or groups of things) because they share a common feature.

In logic, analogy is a type of reasoning that compares two specific examples by showing how they are similar in some way. It is different from deduction, induction, and abduction. It is also used when at least one part of the argument or the conclusion is a general idea instead of a specific one. The general form of an analogy is "A is to B as C is to D."

In a broader sense, analogical reasoning is a way of moving information from one subject (the source) to another (the target). It is also the way this process is expressed in language. The word "analogy" can also describe the relationship between the source and the target, which is often a similarity, like in the biological idea of analogy.

Analogy is very important in how humans think. Some people say analogy is at the center of how people understand and learn new ideas.

Etymology

The English word "analogy" comes from the Latin word "analogia." This Latin word is based on the Greek word "ἀναλογία," which means "proportion." The Greek word is made up of two parts: "ana-" meaning "upon, according to" (also "again" or "anew") and "logos," which means "ratio" (also "word, speech, or reckoning").

Models and theories

Analogy is important in many areas, such as solving problems, making decisions, explaining ideas, understanding things, remembering, being creative, predicting, feeling emotions, and communicating. It helps with basic tasks like recognizing people, places, and objects, such as identifying faces in pictures or using facial recognition systems. Hofstadter said that analogy is the main part of thinking.

An analogy is not just a figure of speech, but a type of thinking. People use analogies in language through examples, comparisons, metaphors, similes, stories, and parables, but not through metonymy. Phrases like "and so on," "as if," and "like" depend on people understanding analogies. Analogy is used in everyday language, science, philosophy, law, and other subjects. Examples can be found in proverbs and idioms.

Many ideas are related to analogy, such as comparison, similarity, mathematical patterns, and how things are alike. In cognitive linguistics, the idea of conceptual metaphors is similar to analogy. Analogy is also used in arguments that compare things and in experiments where results from one group are used to understand another group, like testing rats and applying the results to humans.

People have studied analogy for a long time, starting in ancient times with philosophers, scientists, and lawyers. Recently, interest in analogy has grown, especially in cognitive science.

• Archytas, a friend of Plato, described three types of analogy: mathematical, harmonic, and geometric, with geometric being the true analogy.
• Aristotle talked about analogy in his writings, like Metaphysics and Nicomachean Ethics.
• Roman lawyers used analogical reasoning and the Greek word analogia.
• In Islamic law, analogical reasoning was used in qiyas to help make legal decisions.
• Medieval lawyers talked about analogia legis and analogia iuris.
• During the Middle Ages, people used and studied analogy more.
• In Christian theology, analogies were used to explain God’s attributes. Aquinas said some words, like "healthy," have different but related meanings. Thomas Cajetan wrote a famous book about analogy. All these examples kept the old Greek and Roman ideas about analogy.

Cajetan described different types of analogy, including:

• Analogy of attribution: When something is described as having a quality, like "This food is healthy."
• Analogy of proportionality: When two pairs of things have the same relationship, like "2 is to 1 as 4 is to 2."
• Metaphor: Using words in a non-literal way, like "steely determination."

In ancient Greek, the word analogia meant proportionality, like in math. It was sometimes translated to Latin as proportio. Analogy was seen as a way to show the same relationship between two pairs of things, whether they are numbers or objects.

Analogy and abstraction are different ways of thinking. Analogy is often easier to understand. For example, comparing a hand to its palm and a foot to its sole focuses on their shared feature of having an inner surface, not all their differences.

The SAT college entrance test used analogy questions, like "Hand is to palm as foot is to ___?" The answer is "sole." While people often know the answer quickly, explaining the exact relationship between the pairs is harder. This relationship isn’t always clear from dictionary definitions.

Kant said that analogy can show the same relationship between completely different things.

Greek philosophers like Plato and Aristotle saw analogy as a shared idea or pattern between things. They used comparisons, metaphors, and stories as arguments. These methods helped people understand abstract ideas better.

James Francis Ross wrote a book called Portraying Analogy in 1982, which showed that analogy is a common and predictable part of language.

Some thinkers, like Ibn Taymiyya, Francis Bacon, and John Stuart Mill, said analogy is a type of inductive reasoning. They believed analogy uses known traits to predict other traits, like saying if something is true for one thing, it might be true for another.

Today, cognitive scientists use a broad view of analogy, similar to how Plato and Aristotle did, but they explain it using a theory called structure-mapping. This theory says analogy works by matching parts of one thing (the source) to parts of another (the target). This matching happens not only with objects but also with relationships between objects. This idea is used in studying metaphors and similarity. Structure-mapping has been tested in psychology and computer science, where it helps explain how people think and how computers can mimic human reasoning.

Applications and types

Logicians study how analogies are used in arguments that compare things. An analogy can be shown using phrases like "is to" and "as," such as "Smile is to mouth, as wink is to eye." In math and logic, this relationship can be written using colons. A single colon shows a ratio, and two colons show equality. For example, "Smile : mouth :: wink : eye" is how this might be written.

In the study of language, analogy helps change words that seem to break rules into forms that follow common patterns. For example, the old past tense of "help" was "holp," but now it is "helped," following the pattern used by many other verbs like "jumped" or "carried." This process is called morphological leveling. Sometimes, analogies can also cause words to break rules, like the American English past tense of "dive," which is "dove," similar to "drive" becoming "drove."

New words can also be created through analogy. For example, "software" was made by comparing it to "hardware." Other examples include "firmware" and "vapourware." Another example is the word "underwhelm," which was made by comparing it to "overwhelm."

Some people think analogy is a way to explain how new words or structures are created, while others believe it is the same as rules that have become standard in language. This idea matches how scientists study thinking today.

In the Neogrammarian school of thought, analogy is a term used to describe changes in language that are not caused by sound changes or borrowing from other languages.

Analogies are often used to create new ideas or test them, which is called the heuristic function of analogical reasoning. In science, analogies can also help support theories, especially in areas like theology, philosophy, or cosmology, where direct proof is not always possible.

In science, analogies can be used in models or simulations to help understand complex systems. For example, electrical circuits are sometimes compared to hydraulic circuits. Another example is the "analogue ear," which uses electrical or mechanical devices to mimic how ears work.

Some analogies can be described mathematically through isomorphism, which means two structures are exactly the same in certain ways. For example, the space of real numbers (R²) and complex numbers (C) are isomorphic as vector spaces, but complex numbers have more structure.

In mathematics, category theory uses the idea of functors, which are like analogies between different categories. Functors preserve the structure of objects and relationships between them, similar to how analogies work in the mind.

Large Language Models (LLMs) can use math to find analogies by comparing the differences between vectors that represent concepts. For example, they might compare "dog" and "puppy" to find a similar relationship for "cat" and "kitten." However, not all vector math is used for analogies, like the famous example "KING – MAN + WOMAN = QUEEN."

Scientists are still studying how LLMs compare to human thinking when it comes to analogies.

By 2006, a computer program using the Vector Space Model (VSM) could answer analogy questions from the SAT test as well as humans. The program compares how words relate by looking at large amounts of text. It chooses the answer that has the most similar relationships.

Human analogical reasoning avoids mistakes that happen in some AI models, called systematicity. Scientists use math to explain how the mind might naturally use analogies by focusing on relationships between structures, not just objects.

In 1997, Keith Holyoak and Paul Thagard created a theory that explains how analogies work by considering structure, meaning, and purpose. They say the best analogies are those that match exactly, but even partial matches can help. An analogy is useful if it helps solve a problem. This theory has challenges when multiple sources are involved, but these can be fixed. Later, the theory was adapted to work with neural networks, which are computer models inspired by the brain.

Mark Keane and Brayshaw created a model called the Incremental Analogy Machine (IAM) to study how analogies work in the mind. Their model includes rules for how memory and structure affect the use of analogies.