Advanced Study Notes on Analogy
(Reasoning Ability – Andaman & Nicobar Islands Forest Guard, MIT‑Level)
Analogy is the cognitive process of recognizing structural similarity between two (or more) domains and inferring relationships, properties, or behaviors that hold in one domain from knowledge of the other.
In reasoning‑ability tests, analogies are the cornerstone of many problems. To master them you must understand why analogies work, how they are formally represented, and what pitfalls can arise.
1. Overview
| Term | Definition | Example |
|---|---|---|
| Base | The source domain (known) | Fire |
| Target | The novel domain to be understood | Fireworks |
| Mapping | Correspondence between elements of base & target | Flame ↔ Spark |
| Analogy problem | Task: find the missing element of a series | Fire : Flame :: Fireworks : ___ |
| Analogy type | Lexical, conceptual, relational, structural | “Hot” → “Warm” (lexical), “Heart” → “Central Hub” (conceptual) |
2. Formal Foundations (From First Principles)
2.1 Sets, Relations, Functions
- Set (S) = ({x_1, x_2, \dots, x_n}).
- Relation (R \subseteq S_1 \times S_2).
- Function (f: S_1 \rightarrow S_2) is a relation where each (x \in S_1) maps to exactly one (y \in S_2).
2.2 Structural Mapping
Let
- Base domain (B = (P_B, R_B)): a set of elements (P_B) and a set of relations (R_B).
- Target domain (T = (P_T, R_T)).
A mapping (M: P_B \rightarrow P_T) is a partial function such that for each relation (r \in R_B) we have a corresponding relation (r' \in R_T) satisfying:
[ \forall (x, y) \in r, \quad (M(x), M(y)) \in r'. ]
Theorem (Relational Preservation)
If (M) preserves all relations of (B), then any property (P) expressible as a first‑order formula over (R_B) holds for (M(P)) in (T).
Proof Sketch: Express (P) as a formula (\phi(x_1, \dots, x_k)). Since each atomic relation in (\phi) is preserved by (M), the truth value of (\phi) is maintained under (M).
3. Structural Mapping Theory (SMT)
3.1 Core Principles
- Similarity – The mapped elements must be similar in function or role, not just in appearance.
- Transitivity – If (A) maps to (B) and (B) to (C), then (A) maps to (C).
- Consistency – The mapping must be one‑to‑one (injective) to avoid contradictions.
- Exclusiveness – Each element in the target is mapped to at most one element in the base.
3.2 Example
| Base | Target | Mapping |
|---|---|---|
| Heart | Central Hub | (M(\text{Heart}) = \text{Hub}) |
| Pump | Relay | (M(\text{Pump}) = \text{Relay}) |
| Blood | Data | (M(\text{Blood}) = \text{Data}) |
| Circulate | Transmit | (M(\text{Circulate}) = \text{Transmit}) |
4. Types of Analogy
| Type | Description | Example |
|---|---|---|
| Lexical | Similar word forms or meanings | Cold : Heat :: Light : Dark |
| Conceptual | Shared abstract concept | Justice : Fairness |
| Relational | Similar relational structure | Teacher teaches student :: Coach trains athlete |
| Structural | Same underlying structure | Bridge connects two islands :: Antenna connects two antennas |
5. Cognitive Foundations
- Mental Set – Prior knowledge biases the mapping.
- Analogical Transfer – Transfer of solution from base to target (Gick & Holyoak, 1989).