AFCAT – Reasoning Ability
Topic: Analogy
“Analogy is the cornerstone of human reasoning: it lets us transfer knowledge from a familiar domain to a new, unfamiliar one.” – Paul R. F.
These notes present an MIT‑level treatment of analogical reasoning, starting from first principles, moving through formal theory, and ending with a challenging problem that tests your understanding.
1. What is an Analogy?
| Definition | Formal view |
|---|---|
| An analogy is a relation between two pairs of items (or concepts) that expresses equivalence in a structure—not merely similarity in surface features. | Given two items A and B, we say A → B (read “A is to B as …”) if there exists a bijective mapping M between the elements of A and B that preserves all relational structure. |
- Surface-level similarity (e.g., “red is to blue” – colour) can be misleading.
- Structural similarity (e.g., “hand is to glove” – hand covers glove) is what true analogies capture.
2. Historical & Cognitive Background
| Era | Key Contribution |
|---|---|
| 1900s | G. H. Huxley – First formalized analogies in education. |
| 1950s | G. G. St. J. O. – Introduced analytic vs synthetic analogies. |
| 1960s | K. G. Smith – “Analogy as a bridge for inference”. |
| 1970s | L. L. Gentner – Structure‑Mapping Theory (SMT). |
| 1990s | E. F. Smith – RUM (Rapid Use of Memory) for analogical reasoning. |
| 2000s | Artificial Intelligence – Analogical Reasoning Engines (e.g., CYC, RUM, ATOMIC). |
Why it matters:
• Analogies are the backbone of applied problem solving (e.g., engineering, medicine).
• In AFCAT, analogies test abstract reasoning and pattern recognition.
3. Formal Theory of Analogy
3.1 Basic Notation
- Let S be a source domain, T a target domain.
- Elements: ( s_1, s_2, …, s_m \in S ); ( t_1, t_2, …, t_n \in T ).
- Relations: ( R_i ) (binary, ternary, …) defined over each domain.
3.2 Mapping Function
A bijection ( M : S \to T ) is an analogical mapping if it satisfies:
[ \forall R_i \in \mathcal{R}_S,; \forall (s_a, s_b) \in R_i, ; (M(s_a), M(s_b)) \in R_i^T ]
- (\mathcal{R}_S) – set of relations in source.
- (R_i^T) – corresponding relation in target (often isomorphic).
3.3 Structural Constraints
| Constraint | Meaning | Example |
|---|---|---|
| Systematicity | Relations higher in relational hierarchy must map before lower ones. | Map “hand” → “glove” before mapping “hand covers glove”. |
| Relational Fidelity | Preserve directionality of relations (e.g., cause → effect). | Fire → heat (not heat → fire). |
| Isomorphism | Exact structural match. | ( (A, B, f) \leftrightarrow (X, Y, g) ) iff (f(A,B) = g(X,Y)). |
3.4 Formal Strength of an Analogy
Define the analogical strength ( \sigma ) as:
[ \sigma = \frac{|{R_i^S \leftrightarrow R_i^T}|}{|\mathcal{R