Contribution
A theory of how the meaning of an analogy is derived from the meanings of its parts. It replaces feature-overlap accounts of similarity, which it grants for literal comparisons, with a typed account: predicates are sorted by arity (attribute or relation) and by order (first-order, or higher-order taking propositions as arguments), and each kind of comparison is defined by which kind of predicate is shared. Its one selection principle, systematicity, says which relations are carried over. “The central idea is that an analogy is an assertion that a relational structure that normally applies in one domain can be applied in another domain” (p. 156).
Key insight
What an analogy keeps is relational structure, and the relations it keeps are the ones held together by higher-order relations. In “The atom is like the solar system”, MORE MASSIVE THAN (sun, planet) goes across and HOTTER THAN (sun, planet) does not, though both are binary relations between the same objects. “Only MORE MASSIVE THAN participates in the central-force system of predicates” (p. 164): the relations DISTANCE, ATTRACTIVE FORCE, REVOLVES AROUND and MORE MASSIVE THAN are bound by the equations of central-force motion, so that “changing one of these relations affects the others” (p. 163), and HOTTER THAN is bound by nothing.
Assumptions
- Domains are “systems of objects, object-attributes and relations between objects” (p. 156), represented “as propositional networks of nodes and predicates” (p. 157).
- Two syntactic distinctions. “Attributes are predicates taking one argument, and relations are predicates taking two or more arguments” (p. 157). First-order predicates take objects; second- and higher-order ones take propositions, as in CAUSE [COLLIDE (x,y), STRIKE (y,z)] (p. 157). “in general an nth order relation has at least one (n-1)th order argument” (n. 7, p. 162).
- The representation is psychological, and chosen. “These representations, including the distinctions between different kinds of predicates, are intended to reflect the way people construe a situation, rather than what is logically possible” (p. 157). Footnote 4 makes the dependence explicit: “Logically, a relation R(a,b,c) can perfectly well be represented as Q(x), where Q(x) is true just in case R(a,b,c) is true. Psychologically, the representation must be chosen to model the way people think about the domain.” Footnote 3 adds that many attributes are implicit relations: LARGE (x) means “X is large for its class”, and only relations that apply within the domain are “psychologically stored and processed as true relations”.
- Identical predicates. “The assumption that predicates are brought across as identical matches is crucial to the clarity of this discussion” (n. 6, p. 158); similarity of predicates is to be restated as identity of component predicates.
- One-to-one object maps in most explanatory analogies (n. 5, p. 158), with exceptions allowed.
- Object correspondences are fixed first, in the exposition. Footnote 10 (p. 164) weakens this: the object correspondences “are decided before the relational mappings are finished”, and in complex analogies relation matching may decide them.
Key results
The mapping rules (p. 158)
With object correspondences M: bᵢ → tᵢ,
- “Discard attributes of objects”: A(bᵢ) ↛ A(tᵢ).
- “Try to preserve relations between objects”: R(bᵢ, bⱼ) → R(tᵢ, tⱼ).
- “(The Systematicity Principle) To decide which relations are preserved, choose systems of relations”: R′(R₁(bᵢ, bⱼ), R₂(bₖ, bₗ)) → R′(R₁(tᵢ, tⱼ), R₂(tₖ, tₗ)).
“Notice that this discussion has been purely structural; the distinctions invoked rely only on the syntax of the knowledge representation, not on the content” (p. 158).
Kinds of comparison (pp. 159–161, Table 1)
| kind | attributes mapped | relations mapped | example (Table 1) |
|---|---|---|---|
| literal similarity | many | many | ”The K5 solar system is like our solar system” |
| analogy | few | many | ”The atom is like our solar system” |
| abstraction | few, and few in the base too | many | ”The atom is a central force system” |
| anomaly | few | few | ”Coffee is like the solar system” |
Mere-appearance matches, with attributes shared and relations not (“A sunflower looks like the sun”), are set aside: “their explanatory power is sharply limited” (p. 161). “the contrast between analogy and literal similarity is a continuum, not a dichotomy” (p. 161), and so is the one between analogy and general law, as the base’s objects become “more abstract and variable-like” (p. 161). Most metaphors are analogies; some are attribute matches (“She’s a giraffe”), and some mix the two, with less regular mapping rules (p. 162).
Systematicity (pp. 162–165)
- Statement: “A predicate that belongs to a mappable system of mutually interconnecting relationships is more likely to be imported into the target than is an isolated predicate” (p. 163).
- The Rutherford case (p. 163): relations (1)–(4) of the solar system and (5)–(8) of the atom are tied by F = Gmₛmₚ/R² = mₚaₚ = mₛaₛ and the Coulomb counterpart, which “embody higher-order relations that connect the lower-order relations (1) through (4) into a mutually constraining structure” (p. 164).
- Heat and water (pp. 164–165): of a static description (a lidded vessel) and a causal one (puncture the vessel and water flows out), the causal one should be mapped. Systematicity achieves this without appeal to content, because “Dynamic causal information … will usually be represented in a more deeply embedded structure than simple stative information” (p. 165): “It is a purely syntactic mechanism that guarantees that the set of candidate mappings will be as interesting—in the sense that a mutually interconnected system of predicates is interesting—as the knowledge base allows” (p. 165).
- Plausibility is a separate matter. “our ultimate acceptance of the analogy will depend on whether its candidate set of predicates is plausible; but this is a separate matter” (n. 11, p. 165).
Empirical support (pp. 165–166)
One set of studies, summarized in a paragraph and described more fully in Gentner 1980b (a conference paper): subjects wrote interpretations of comparisons such as “A cigarette is like a time bomb”, and naive judges rated each assertion as attributional or relational. “Relational information predominates over attributional information in analogy interpretations, but not in object descriptions generated by the same subjects” (pp. 165–166), and “subjects rated as most apt those analogies for which they wrote the greatest amount of relational information” (p. 166). No numbers are given. Developmental and science-learning studies are cited (p. 166).
The analogical shift conjecture (pp. 167–168)
“An informal observation is that the earliest comparisons are chiefly literal-similarity matches, followed by analogies, followed by general laws” (p. 167), from one subject reasoning about water in a constricted pipe. The proposed reason: literal matches are accessible and uninformative, analogies the reverse.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | In analogy, relations between objects are mapped from base to target and object attributes are not | moderate | worked examples (battery–reservoir, 3:6::2:4, Rutherford); one study summarized without numbers (pp. 165–166) |
| C2 | Which relations are mapped is decided by systematicity: a relation in a system governed by higher-order relations is more likely to be imported than an isolated one | weak to moderate | the Rutherford and heat–water examples (pp. 163–165); no direct test reported |
| C3 | The interpretation rules depend only on the syntax of the representation (arity and order), not on content | strong, as a property of the theory | by construction (pp. 158, 165); but the syntax is chosen to fit psychology (n. 4), so the content-independence is relative to that choice |
| C4 | Literal similarity, analogy, abstraction and anomaly are distinguished by how many attributes and how many relations are shared, and the contrasts are continua | moderate | definitional (Table 1, p. 161) |
| C5 | Learners’ spontaneous comparisons move from literal similarity to analogy to general law | weak | ”An informal observation” of one subject (p. 167) |
Concepts
- Base / target — the domain that “serves as a source of knowledge” and the domain “being explicated” (p. 157).
- Attribute — a one-place predicate. Relation — a predicate of two or more places (p. 157).
- Order — set by the arguments: first-order relations take objects; “A second-order relation has at least one first-order relation among its arguments” (n. 7, p. 162).
- Systematicity — membership in “a mappable system of mutually interconnecting relationships” governed by higher-order relations (p. 163).
- Mere appearance match — attributes shared, relations not (p. 161).
- Abstraction — a comparison whose base is “an abstract relational structure”, with no nonmapped predicates (pp. 159–160).
Connections
- Tversky (1977), the contrast model, is the foil: good for literal similarity, wrong for analogy (pp. 155–156).
- Rumelhart and Abrahamson (1973), multidimensional-space models of analogy, are among those said to deal well with attributes and to be “extremely limited in their ability to express relations between objects, and especially higher-order relations” (p. 166). The same paper is LIT-1801’s precursor for the parallelogram rule.
- Winston (1980, 1981) matches only predicates in causal chains, “somewhat more restrictive than the structure-mapping principle that participation in any higher-order chain results in preferential mapping” (pp. 166–167). Gick and Holyoak are cited for schema induction.
- Forbus and Gentner (1983) and the later Structure-Mapping Engine are the computational descendants; Chalmers, French and Hofstadter’s critique of SME’s hand-built representations (LIT-414, read in NOTE-359) is aimed at the dependence that footnote 4 here concedes.
Bearing on the record
- LIT-1818 (Gromov–Wasserstein alignment): A6’s reading holds for the first two rules, and the paper shows exactly what GW leaves out. GW matches two spaces through the pairwise distances within each, with no shared coordinates. That is rules 1 and 2 for one relation: object attributes (coordinates) are discarded and a binary relation is preserved. Three differences follow from the text. (a) Structure-mapping carries relations of many types, each kept as an “identical match” of predicate (n. 6); GW carries one relation, a distance, and preserves its values approximately. A distance matrix cannot say whether two objects stand in ATTRACTS or in REVOLVES AROUND, so A6’s point that “two systems can share pairwise distances and realize different kinds of relation” is right in Gentner’s own terms. (b) GW has no higher-order relations, so it has no counterpart of systematicity, the principle that decides which relations an analogy keeps. (c) Structure-mapping lets most relations fail to map and still counts the analogy good (HOTTER THAN is dropped), while GW’s objective charges every pair’s mismatch. The exchange’s description in LIT-1810 is accurate. One qualification: the paper states systematicity as a likelihood (“more likely to be imported”, p. 163), while the abstract and summary say the relations mapped “are determined by” or “are those that are dominated by” higher-order relations (pp. 155, 168).
- CLAIM-113: a case of transformation and invariant. An analogy is a map between domains that preserves relational structure and discards attributes; the kinds of comparison in Table 1 are classes of map sorted by what they preserve. That fits CLAIM-113’s distinction between exact preservation and structures that are changed. What the paper adds is a selection principle: of the relations a map could preserve, those bound by higher-order relations are the ones that matter. CLAIM-113 has nothing corresponding.
- CLAIM-135: structure-mapping lives inside a chosen representation. Footnote 4 concedes that any relation can be recoded as an attribute and that the choice is made “to model the way people think about the domain”. So the theory’s rules recover relational correspondence relative to a representation fixed in advance, and they do not themselves say which representation is right. That is CLAIM-135’s gap between identifying structure under stated assumptions and establishing that the structure is the right kind. The paper also says acceptance of an analogy depends on plausibility, “a separate matter” (n. 11).
- Should produce: THEORY-1154 (relations, not attributes, are mapped, and systematicity selects among them; Proposed).
- No instruction for machine-learning practice.
Limitations
- Thin evidence, by the author’s own placement. The Empirical Support section is one paragraph on one set of studies with no statistics, plus citations (pp. 165–166). The paper is a framework, and it says so.
- The representation is supplied. The theory takes the predicate structure of each domain as given, and footnote 4 says that structure is a modelling choice. Whether a predicate is an attribute or a relation, and of which order, is decided before the rules apply.
- Systematicity is stated two ways: as a likelihood (p. 163) and as determination (abstract, p. 155; summary, p. 168). The examples show preference, not determination.
- Identity of predicates is assumed for clarity (n. 6), and graded similarity of relations is left to a restatement as identity of components that is not carried out.
- Small errors. Table 1’s literal-similarity example is “The K5 solar system”, where the text’s Example 1 is “The X12 star system” (pp. 159, 161). The atom’s equation calls “−1 … the electromagnetic constant” (p. 163); the sign marks attraction, and the constant is not −1 in any common system of units. Neither affects the argument.
- No processing model. The order of processing “is clearly variable” (n. 10, p. 164), and the paper gives no algorithm.
Open questions
- Does membership in a higher-order system raise the chance that a predicate is mapped, with content held fixed? A test needs base representations elicited independently of the analogy, then varies connectedness. Clement and Gentner’s later “Systematicity as a selection constraint in analogical mapping” (Cognitive Science, 1991) is reported to run such a test; it is not held here.
- Can systematicity be given a numerical form that a GW-type alignment could use, for example by aligning typed relations and relations among relations (fused or hypergraph variants of GW)?
- How does the theory decide the attribute/relation split when footnote 4 allows either? An account of representation-building is needed, which is the critique of LIT-414.