Lernzettel: Vagueness & Uncertainty
Methods of AI — SoSe 2026 · 1-page exam sheet
For more depth: Fuzzy Logic (full atomic note — all techniques, Kolmogorov derivations, worked examples) · quiz_vagueness-uncertainty_30-04-26 (7 practice questions)
Core Ideas
- Fuzzy logic addresses vagueness — partial truth (“reddish”). μ ∈ [0,1] = degree of membership.
- Probabilistic logic addresses uncertainty — degree of belief in a crisp fact. P ∈ [0,1] = probability.
- Same numerical range, different meaning.
- Probabilistic logic preserves tautologies/contradictions (P(T)=1, P(C)=0); fuzzy logic does not (intentional).
Mini-glossary
| Term | Meaning |
|---|---|
| Vagueness | predicate boundary is fuzzy (“tall”, “heap”) — degree of truth |
| Uncertainty | predicate is sharp, we lack info — degree of belief |
| μ_A(x) | membership function U → [0,1]; μ_A(x) = degree x belongs to A |
| Fuzzy set | A = (U, μ_A); crisp sets are special case μ_A : U → {0,1} |
| t-norm t(x,y) | generalises AND. Axioms: neutral 1, commutative, associative, monotone |
| s-norm s(x,y) | generalises OR. Axioms: neutral 0, commutative, associative, monotone |
| P(E) | probability of event E; obeys Kolmogorov axioms |
| P(B|A) | = P(B ∩ A) / P(A), requires P(A) > 0 |
| Tautology | T with P(T) = 1 (probabilistic); fuzzy does NOT guarantee this |
Standard fuzzy operations (Zadeh) ⭐
| Operation | Formula |
|---|---|
| Intersection A ∩ B | μ = min(μ_A, μ_B) |
| Union A ∪ B | μ = max(μ_A, μ_B) |
| Complement A^C | μ = 1 − μ_A |
⚠️ A ∩ A^C ≠ ∅ and A ∪ A^C ≠ U in general (e.g. μ_A = 0.6 → min = 0.4, max = 0.6).
Other valid families: algebraic (x·y, x+y−xy), quotient (xy/(x+y−xy), (x+y−2xy)/(1−xy)). All are valid t/s-norms.
⭐ Full t-norm / s-norm table & axioms → OR)
Kolmogorov axioms (compact)
- Non-negativity: P(E) ≥ 0
- Normalisation: P(Ω) = 1
- Finite additivity: A ∩ B = ∅ ⇒ P(A ∪ B) = P(A) + P(B)
Derived: P(∅)=0, P(A^C)=1−P(A), A⊆B ⇒ P(A)≤P(B), P(A∪B)=P(A)+P(B)−P(A∩B), tautology → P=1, contradiction → P=0.
Conditional: P(B|A) = P(B∩A) / P(A). Probabilistic KB formulas: F : p and (G|F)[p].
Common Exam Traps ⚠️
- Vagueness ≠ uncertainty. Partial truth vs. unknown crisp truth.
- Fuzzy: A ∨ ¬A is NOT necessarily 1 — max(0.6, 0.4) = 0.6. Intentional for vague predicates.
- Probabilistic: tautologies always P=1, contradictions always P=0 — fuzzy does NOT guarantee this.
- min/max are one Zadeh choice for t/s-norm — not the only one. Algebraic (x·y, x+y−xy) is equally valid.
- Algebraic t-norm ≠ min: x·y ≤ min(x,y), strict when both < 1.
- t-norm neutral element is 1, s-norm is 0 — don’t mix up.
- Heap paradox motivates fuzzy logic (classical induction yields contradiction).
- Conditional KB satisfaction: P satisfies
(G|F)[p]iff P(G|F) = p, requires P(F) > 0. - Centroid vs. max-membership defuzzification can disagree on the same aggregated fuzzy set.
- ⚠️ Dempster-Shafer is NOT in Session 08 slides — treat as supplementary external material only.
Quick Comparison Table
| Classical Logic | Fuzzy Logic | Probabilistic Logic | |
|---|---|---|---|
| Truth values | {0, 1} | [0, 1] | [0, 1] |
| Propositions | True or false | True to a degree | True or false |
| Truth values represent | True or False | Degree of membership | Probability of truth |
| A ∨ ¬A | Always 1 | Can be < 1 | Always 1 |
| A ∧ ¬A | Always 0 | Can be > 0 | Always 0 |
| Addresses | — | Vagueness | Uncertainty |
(“Truth values represent” row matches Session 08 slide 75.)
Full techniques (fuzzy inference, defuzzification, Bayesian update) → TECHNIQUES (full reference) ⭐
Related Q&A & Notes
Practice quiz
Targeted exam questions in Questions for Methods of AI
- Q56–63 (basic: fuzzy sets, membership, t-norm/s-norm; Dempster-Shafer items are not in Session 08 — supplementary only) · Q91 (T-Norms / S-Norms with examples) · Q123–125 (deep / exam-trap: fuzzy μ vs. probability, t-norm axioms, defuzzification methods that disagree)
Atomic notes
- Fuzzy Logic · Dempster-Shafer Theory (external — not in Session 08)
See also
Tags: methods-of-ai lernzettel ai-generated
Full reference: Fuzzy Logic
Quiz: quiz_vagueness-uncertainty_30-04-26
Superlink: Methods of AI Lecture
Questions hub: Questions for Methods of AI
Created: 30/04/26 · Updated: 23/05/26