===
Logic ===
Logicians analyze how analogical reasoning is used in
arguments from analogy. An analogy can be stated using
is to and
as when representing the analogous relationship between two pairs of expressions, for example, "Smile is to mouth, as wink is to eye." In the field of mathematics and logic, this can be formalized with
colon notation to represent the relationships, using single colon for ratio, and double colon for equality. In the field of testing, the colon notation of ratios and equality is often borrowed, so that the example above might be rendered, "Smile : mouth :: wink : eye" and pronounced the same way.
Linguistics In
historical linguistics and
word formation, analogy is the process that alters words-forms perceived as breaking rules or ignoring general patterns to more typical forms that follow them. For example, the
English verb help once had the simple past-tense form
holp and the
past participle form
holpen. These older forms have now been discarded and replaced by
helped, which came about through the analogy that many other past-tense forms use the
-ed ending (
jumped,
carried,
defeated, etc.). This is called
morphological leveling. Analogies do not always lead to words shifting to fit rules; sometimes, they can also leading to the breaking of rules; one example is the
American English past tense form of
dive:
dove, formed on analogy with words such as
drive to
drove or
strive to
strove. •
Neologisms (new words) can also be formed by analogy with existing words. A good example is
software, formed by analogy with
hardware; other analogous neologisms such as
firmware and
vapourware have followed. Another example is the humorous term
underwhelm, formed by analogy with
overwhelm. • Some people present analogy as an alternative to
generative rules for explaining the
productive formation of structures such as words. Others argue that they are in fact the same and that rules are analogies that have essentially become standard parts of the linguistic system, whereas clearer cases of analogy have simply not (yet) done so (e.g. Langacker 1987.445–447). This view agrees with the current views of analogy in cognitive science which are discussed above. Analogy is also a term used in the
Neogrammarian school of thought as a
catch-all to describe any morphological change in a language that cannot be explained merely sound change or borrowing.
Science Analogies are mainly used as a means of creating new ideas and hypotheses, or testing them, which is called a heuristic function of analogical reasoning. Analogical arguments can also be probative, meaning that they serve as a means of proving the rightness of particular theses and theories. This application of analogical reasoning in science is debatable. Analogy can help prove important theories, especially in those kinds of science in which
logical or
empirical proof is not possible such as
theology,
philosophy or
cosmology when it relates to those areas of the cosmos (the universe) that are beyond any data-based observation and knowledge about them stems from the human insight and thinking outside the senses. Analogy can be used in theoretical and applied sciences in the form of models or simulations which can be considered as strong indications of probable correctness. Other, much weaker, analogies may also assist in understanding and describing nuanced or key functional behaviours of systems that are otherwise difficult to grasp or prove. For instance, an analogy used in physics textbooks
compares electrical circuits to hydraulic circuits. Another example is the
analogue ear based on electrical, electronic or mechanical devices.
Mathematics Some types of analogies can have a precise
mathematical formulation through the concept of
isomorphism. In detail, this means that if two mathematical structures are of the same type, an analogy between them can be thought of as a
bijection which preserves some or all of the relevant structure. For example, \mathbb{R}^2 and \mathbb{C} are isomorphic as vector spaces, but the
complex numbers, \mathbb{C} , have more structure than \mathbb{R}^2 does: \mathbb{C} is a
field as well as a
vector space.
Category theory takes the idea of mathematical analogy much further with the concept of
functors. Given two categories C and D, a functor
f from C to D can be thought of as an analogy between C and D, because
f has to map objects of C to objects of D and arrows of C to arrows of D in such a way that the structure of their respective parts is preserved. This is similar to the
structure mapping theory of analogy of Dedre Gentner, because it formalises the idea of analogy as a function which makes certain conditions true.
Artificial intelligence Large Language Model (LLM) may perform analogy by applying the mathematical difference between two vectors in multi-dimensional space, representing two concepts (e.g dog and puppy), to another vector (e.g. for cat) to find an analogous vector (e.g. ideally, something close to the vector for kitten.) (Not all vector operations are analogical: the commonly cited example from
Natural Language Processing (NLP) KING - MAN + WOMAN = QUEEN is not.) The issue of how
Large Language Models relate to human analogical thinking is a matter of current research (as of 2025).
Pre-LLM By 2006, a computer algorithm using the
Vector Space Model (VSM) had achieved human-level performance on multiple-choice analogy questions from the
SAT test. The algorithm measures the similarity of relations between pairs of words (e.g., the similarity between the pairs HAND:PALM and FOOT:SOLE) by statistically analysing a large collection of text. It answers SAT questions by selecting the choice with the highest relational similarity. The analogical reasoning in the human mind is free of the false inferences plaguing conventional
artificial intelligence models, (called
systematicity). Steven Phillips and
William H. Wilson use
category theory to mathematically demonstrate how such reasoning could arise naturally by using relationships between the internal arrows that keep the internal structures of the categories rather than the mere relationships between the objects (called "representational states"). Thus, the mind, and more intelligent AIs, may use analogies between domains whose internal structures
transform naturally and reject those that do not.
Keith Holyoak and
Paul Thagard (1997) developed their multiconstraint theory within structure mapping theory. They defend that the "
coherence" of an analogy depends on structural consistency,
semantic similarity and purpose. Structural consistency is the highest when the analogy is an
isomorphism, although lower levels can be used as well. Similarity demands that the mapping connects similar elements and relationships between source and target, at any level of abstraction. It is the highest when there are identical relations and when connected elements have many identical attributes. An analogy achieves its purpose if it helps solve the problem at hand. The multiconstraint theory faces some difficulties when there are multiple sources, but these can be overcome.
Mark Keane and Brayshaw (1988) developed their
Incremental Analogy Machine (IAM) to include working memory constraints as well as structural, semantic and pragmatic constraints, so that a subset of the base analogue is selected and mapping from base to target occurs in series.
Empirical evidence shows that humans are better at using and creating analogies when the information is presented in an order where an item and its analogue are placed together. Eqaan Doug and his team challenged the shared structure theory and mostly its applications in computer science. They argue that there is no clear line between
perception, including high-level perception, and analogical thinking. In fact, analogy occurs not only after, but also before and at the same time as high-level perception. In high-level perception, humans make
representations by selecting relevant information from low-level
stimuli. Perception is necessary for analogy, but analogy is also necessary for high-level perception. Chalmers et al. concludes that analogy actually is high-level perception. Forbus et al. (1998) claim that this is only a metaphor. It has been argued (Morrison and Dietrich 1995) that Hofstadter's and Gentner's groups do not defend opposite views, but are instead dealing with different aspects of analogy. Based on Hofstadter's ideas, Douglas Blank (1997) demonstrated that abstract analogy-making tasks (using both perceptual and symbolic inputs) could be learned by a connectionist network using a two-step, feed-forward network.
Anatomy In
anatomy, two anatomical structures are considered to be
analogous when they serve similar
functions but are not
evolutionarily related, such as the
legs of
vertebrates and the legs of
insects. Analogous structures are the result of
independent evolution and should be contrasted with structures which
shared an evolutionary line. Engineering Often a physical
prototype is built to model and represent some other physical object. For example,
wind tunnels are used to test scale models of wings and aircraft which are analogous to (correspond to) full-size wings and aircraft. For example, the
MONIAC (an
analogue computer) used the flow of water in its pipes as an analogue to the flow of money in an economy.
Cybernetics Where two or more biological or physical participants meet, they communicate and the stresses produced describe internal models of the participants.
Pask in his
conversation theory asserts an
analogy that describes both similarities and differences between any pair of the participants' internal models or concepts exists.
History In historical science, comparative historical analysis often uses the concept of analogy and analogical reasoning. Recent methods involving calculation operate on large document archives, allowing for analogical or corresponding terms from the past to be found as a response to random questions by users (e.g., Myanmar - Burma) and explained.
Morality Analogical reasoning plays a very important part in
morality. This may be because morality is supposed to be
impartial and fair. If it is wrong to do something in a situation A, and situation B corresponds to A in all related features, then it is also wrong to perform that action in situation B.
Moral particularism accepts such reasoning, instead of deduction and induction, since only the first can be used regardless of any moral principles.
Psychology Structure mapping theory Structure mapping, originally proposed by
Dedre Gentner, is a theory in psychology that describes the psychological processes involved in reasoning through, and learning from, analogies. More specifically, this theory aims to describe how familiar knowledge, or knowledge about a base domain, can be used to inform an individual's understanding of a less familiar idea, or a target domain. According to this theory, individuals view their knowledge of ideas, or domains, as interconnected structures. In other words, a domain is viewed as consisting of objects, their properties, and the relationships that characterise their interactions. The process of analogy then involves: • Recognising similar structures between the base and target domains. • Finding deeper similarities by mapping other relationships of a base domain to the target domain. • Cross-checking those findings against existing knowledge of the target domain. In a system of flowing water, the water is carried through pipes and the rate of water flow is determined by the pressure of the water towers or hills. This relationship
corresponds to that of electricity flowing through a circuit. In a circuit, the electricity is carried through wires and the current, or rate of flow of electricity, is determined by the voltage, or electrical pressure. Given the similarity in structure, or structural alignment, between these domains, structure mapping theory would predict that relationships from one of these domains, would be inferred in the other using analogy. This is critical in their cognitive development as continuing to focus on specific objects would reduce children's ability to learn abstract patterns and reason analogically. Comparison is more likely when the objects to be compared are close together in space and/or time, In the
common law tradition, it is most typically used for extending the scope of
precedent.
Religion Catholicism The Fourth Lateran Council of 1215 taught:
For between creator and creature there can be noted no similarity so great that a greater dissimilarity cannot be seen between them. The theological exploration of this subject is called the
analogia entis. The consequence of this theory is that all true statements concerning God (excluding the concrete details of Jesus' earthly life) are rough analogies, without implying any falsehood. Such analogical and true statements would include
God is,
God is Love,
God is a consuming fire,
God is near to all who call him, or God as Trinity, where
being,
love,
fire,
distance,
number must be classed as analogies that allow human cognition of what is infinitely beyond positive or
negative language. The use of theological statements in
syllogisms must take into account their analogical essence, in that every analogy breaks down when stretched beyond its intended meaning.
Doctrine of the Trinity In traditional Christian doctrine, the
Trinity is a
Mystery of Faith that has been revealed, not something obvious or derivable from first principles or found in any thing in the created world. Because of this, the use of analogies to understand the Trinity is common and perhaps necessary. The Trinity is a combination of the words "tri," meaning "three," and "unity," meaning "one." The "Threeness" refers to the persons of the Trinity, while the "Oneness" refers to substance or being. Medieval
Cistercian monk
Bernard of Clairveaux used the analogy of a kiss: Many analogies have been used to explain the Trinity, however, all analogies fail when taken too far. Examples of these are the analogies that state that the Trinity is like water and its different states (solid, liquid, gas) or like an egg with its different parts (shell, yolk, and egg white). However, these analogies, if taken too far, could teach the heresies of modalism (water states) and partialism (parts of egg), which are contrary to the Christian understanding of the Trinity. One more analogy used is one that uses the mythological dog, Cerberus, that guards the gates of Hades. While the dog itself is a single organism—speaking to its substance—Cerberus has different centers of awareness due to its three heads, each of which has the same dog nature.
Protestantism In some Protestant theology, "analogy" may itself be used analogously in terms, more in a sense of "rule" or "exemplar": for example the concept "
analogia fidei" has been proposed as an alternative to the concept
analogia entis but named analogously.
Islam Islamic jurisprudence makes ample use of analogy as a means of making conclusions from outside sources of law. The bounds and rules employed to make analogical deduction vary greatly between
madhhabs and to a lesser extent individual scholars. It is nonetheless a generally accepted source of law within
jurisprudential epistemology, with the chief opposition to it forming the
dhahiri (ostensiblist) school. ==See also==