Quiz: Vagueness & Uncertainty
Methods of AI — SoSe 2026
Hub: Fuzzy Logic (full atomic note) · Lernzettel: lernzettel_vagueness-uncertainty_30-04-26
Q1 — Vagueness & Uncertainty
Question: What is the fundamental difference between vagueness and uncertainty? Which does fuzzy logic address, which does probabilistic logic address?
Answer
- Vagueness: the concept itself has imprecise, gradual boundaries — “tall”, “heap”, “reddish”. The property applies to a degree. Handled by fuzzy logic.
- Uncertainty: the fact is either true or false, but we don’t know which. We have incomplete information. Handled by probabilistic logic.
Key distinction: in fuzzy logic, “the ball is reddish” is true to degree 0.6 — the value represents membership. In probabilistic logic, “it will rain” has probability 0.6 — the value represents likelihood of a crisp fact.
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Q2 — Vagueness & Uncertainty
Question: Calculate: given μ_red(ball) = 0.6 and μ_round(ball) = 1.0, compute (a) μ_{red ∩ round}(ball), (b) μ_{red ∪ round}(ball), (c) μ_{red^C}(ball).
Answer
Using standard Zadeh operations (min/max):
(a)μ_{red ∩ round}(ball) = min(0.6, 1.0) = **0.6**
(b)μ_{red ∪ round}(ball) = max(0.6, 1.0) = **1.0**
(c)μ_{red^C}(ball) = 1 − 0.6 = **0.4**
Bonus:μ_{red ∩ red^C}(ball) = min(0.6, 0.4) = 0.4≠ 0 (classical law A ∩ ¬A = ∅ breaks for fuzzy sets!)
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Q3 — Vagueness & Uncertainty
Question: What conditions must a t-norm satisfy? Name two examples.
Answer
A t-norm t: [0,1]² → [0,1] must satisfy:
- Neutral element 1: t(x, 1) = x for all x
- Commutativity: t(x, y) = t(y, x)
- Associativity: t(t(x,y), z) = t(x, t(y,z))
- Monotone increasing: x ≤ x’ and y ≤ y’ → t(x,y) ≤ t(x’,y’)
Examples:
- Standard (Zadeh): t(x,y) = min(x,y)
- Algebraic product: t(x,y) = x·y
S-norms (for disjunction) have neutral element 0, same other properties.
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Q4 — Vagueness & Uncertainty
Question: State Kolmogorov’s 3 axioms of probability. Derive P(A^C) = 1 − P(A) from them.
Answer
- Non-negativity: P(E) ≥ 0 for all events E
- Normalization: P(Ω) = 1
- Finite additivity: if E₁, …, Eₙ are disjoint: P(E₁ ∪ … ∪ Eₙ) = P(E₁) + … + P(Eₙ)
Derivation of P(A^C) = 1 − P(A):
- A and A^C are disjoint, and A ∪ A^C = Ω
- By additivity: P(A ∪ A^C) = P(A) + P(A^C)
- By normalization: P(Ω) = 1
- Therefore: P(A) + P(A^C) = 1 → P(A^C) = 1 − P(A) ✓
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Q5 — Vagueness & Uncertainty
Question: Why does probabilistic logic maintain the property that tautologies have P=1 and contradictions have P=0, while fuzzy logic does NOT?
Answer
In probabilistic logic, formulas are associated with events in a probability space. A tautology T is satisfied by ALL interpretations → event E_T = Ω → P(T) = P(Ω) = 1. A contradiction C is satisfied by NO interpretation → E_C = ∅ → P(C) = 0.
In fuzzy logic, truth values are directly computed, not via probability spaces. For vague predicates: I(F ∨ ¬F) = max(I(F), 1−I(F)). If I(red) = 0.6: max(0.6, 0.4) = 0.6 ≠ 1. This is intentional — “this is red or not red” is only 60% true for a reddish object.
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Q6 — Vagueness & Uncertainty
Question: Define conditional probability. If P(Bird) = 0.8 and P(Penguin|Bird) = 0.2, what is P(Penguin)? (Assume P(Penguin|¬Bird) ≥ 0)
Answer
Definition: P(B|A) = P(B ∩ A) / P(A)
Using total probability:
P(Penguin) = P(Penguin|Bird)·P(Bird) + P(Penguin|¬Bird)·P(¬Bird)
= 0.2 · 0.8 + P(Penguin|¬Bird) · 0.2
≥ 0.2 · 0.8 = 0.16So P(Penguin) ≥ 0.16. The lower bound comes from ignoring the non-Bird penguins.
This illustrates reasoning in probabilistic knowledge bases.
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Q7 — Vagueness & Uncertainty
Question: The Heap Paradox: what is it and why does it motivate fuzzy logic?
Answer
Sorites paradox (heap paradox):
P1: 1,000,000 grains = heap. P2: heap − 1 grain = still heap. By induction → 1 grain = heap. Contradiction.
The problem: “heap” is a vague predicate — there’s no sharp threshold. Classical logic forces binary (heap or not heap), making the gradual transition paradoxical.
Fuzzy logic solution: assign a graded truth value. 1,000,000 grains: heap to degree 1. 500,000: heap to degree ~0.8. 100: heap to degree ~0.1. 1: heap to degree ~0. No sharp cutoff → no paradox. The degree decreases gradually as grains decrease.
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Tags: methods-of-ai quiz ai-generated
Hub (atomic note): Fuzzy Logic
Lernzettel: lernzettel_vagueness-uncertainty_30-04-26
Superlink: Methods of AI Lecture