CLASSICAL (CRISP) SET THEORY: an element either belongs to a set or it does not. Membership is binary. μ_A(x) ∈ {0, 1} THE PROBLEM: MOST HUMAN CONCEPTS HAVE NO SHARP BOUNDARY. "tall" — is 175 cm tall? 176? Where exactly does tall begin? "warm", "fast", "expensive", "nearly finished", "quite likely" Forcing a crisp boundary produces absurd behaviour at the edge: if "hot" means above 30 °C, then 29.9 °C is not hot at all and 30.0 °C is fully hot. A CONTROLLER USING THAT DEFINITION WILL SWITCH VIOLENTLY AROUND THE THRESHOLD, which is exactly the practical failure fuzzy control was invented to fix. THIS IS ALSO A KNOWLEDGE REPRESENTATION PROBLEM, in the terms of ACtE0903: the encoder must state a fact more precisely than it is true. "It is quite warm" has no crisp encoding, so the qualification is dropped — which was named there as the commonest KR error. FUZZY SET THEORY (Zadeh, 1965): membership is a MATTER OF DEGREE. μ_A(x) ∈ [0, 1] μ_A(x) = 0 definitely not in A μ_A(x) = 1 definitely in A μ_A(x) = 0.7 substantially in A, partially THE CRUCIAL DISTINCTION, and it is examined constantly: FUZZINESS IS NOT PROBABILITY. PROBABILITY expresses UNCERTAINTY ABOUT A CRISP FACT. "There is a 0.7 chance this bottle contains water" — it either does or does not, and I do not know which. Once I look, the probability becomes 0 or 1. FUZZINESS expresses a GRADED PROPERTY that is certainly true to a degree. "This tea is warm to degree 0.7" — I know exactly what the temperature is. Looking again changes nothing. The vagueness is in the CONCEPT "warm", not in my knowledge. A useful test: DOES MORE INFORMATION RESOLVE IT? If yes, it is probability. If no, it is fuzziness. MEMBERSHIP FUNCTIONS — how the degree is defined. The common shapes: TRIANGULAR, defined by three points a < b < c: μ(x) = 0 for x ≤ a or x ≥ c = (x − a)/(b − a) for a < x ≤ b = (c − x)/(c − b) for b < x < c TRAPEZOIDAL, four points — a flat top, so a range of values is fully in the set GAUSSIAN, a smooth bell — differentiable, which matters for tuning SIGMOIDAL, an S-curve for "at least" concepts A WORKED SET OF TEMPERATURE MEMBERSHIPS, using triangles: COLD = triangle(0, 10, 20) WARM = triangle(15, 25, 35) HOT = triangle(25, 35, 45) T = 18 °C: cold 0.20 warm 0.30 hot 0.00 T = 22 °C: cold 0.00 warm 0.70 hot 0.00 T = 25 °C: cold 0.00 warm 1.00 hot 0.00 T = 28 °C: cold 0.00 warm 0.70 hot 0.30 T = 32 °C: cold 0.00 warm 0.30 hot 0.70 CHECK T = 18: cold = (20 − 18)/(20 − 10) = 2/10 = 0.20, and warm = (18 − 15)/(25 − 15) = 3/10 = 0.30. NOTE THAT IT IS BOTH SOMEWHAT COLD AND SOMEWHAT WARM, and that the degrees need NOT sum to 1 — that is a probability constraint, not a fuzzy one. THE OVERLAP IS DELIBERATE: it is what gives fuzzy control its smooth transitions, because as the temperature rises the influence of one rule fades in while another fades out. A LINGUISTIC VARIABLE is a variable whose values are words: the linguistic variable Temperature takes the LINGUISTIC VALUES (or fuzzy terms) cold, warm, hot. HEDGES modify them: "very warm" μ² — squaring reduces middling memberships, sharpening the set "somewhat warm" √μ — increasing them, broadening it "not warm" 1 − μ

THE STANDARD OPERATIONS, which generalise the crisp ones: COMPLEMENT μ_¬A(x) = 1 − μ_A(x) UNION (OR) μ_{A∪B}(x) = max( μ_A(x), μ_B(x) ) INTERSECTION μ_{A∩B}(x) = min( μ_A(x), μ_B(x) ) (AND) WHY MIN AND MAX: they are the operations that reduce to the crisp ones when memberships are 0 or 1, and they satisfy most of the usual set laws. check: min(1, 0) = 0 = crisp AND; max(1, 0) = 1 = crisp OR. WORKED, with A = WARM and B = HOT at T = 28 °C: μ_A = 0.70, μ_B = 0.30 NOT warm = 1 − 0.70 = 0.30 warm AND hot = min(0.70, 0.30) = 0.30 warm OR hot = max(0.70, 0.30) = 0.70 very warm = 0.70² = 0.49 somewhat warm = √0.70 = 0.837 ONE LAW THAT FAILS, and it is a favourite exam point: THE LAW OF EXCLUDED MIDDLE does NOT hold. A ∪ ¬A = max(μ, 1 − μ), which is 1 only when μ is 0 or 1. At μ = 0.5 it is 0.5, not 1. SIMILARLY THE LAW OF CONTRADICTION FAILS: A ∩ ¬A = min(μ, 1 − μ) = 0.5 at μ = 0.5, not 0. SO SOMETHING CAN BE BOTH WARM AND NOT-WARM TO DEGREE 0.5. THAT IS NOT A BUG — it is the point. In classical logic a proposition and its negation cannot both hold; in fuzzy logic a graded property and its complement genuinely overlap in the middle. DE MORGAN'S LAWS DO hold with min/max: 1 − max(a, b) = min(1 − a, 1 − b) ✓ 1 − min(a, b) = max(1 − a, 1 − b) ✓ ALTERNATIVE OPERATORS exist — the general families are called T-NORMS (for AND) and T-CONORMS or S-NORMS (for OR). Besides min/max, the common pair is the ALGEBRAIC PRODUCT and SUM: AND: a · b OR: a + b − a·b At a = 0.7, b = 0.3: product AND = 0.21 against min's 0.30, so the product is a STRICTER conjunction. The choice affects how sharply rules interact, and min/max remain the default because they are cheap and behave predictably. OTHER FUZZY SET CONCEPTS worth naming: SUPPORT the set of x with μ(x) > 0 CORE the set of x with μ(x) = 1 HEIGHT the maximum membership attained NORMAL SET a set whose height is 1 CROSSOVER the x where μ(x) = 0.5 POINT α-CUT the crisp set of x with μ(x) ≥ α — the bridge back to crisp reasoning, since every fuzzy set is determined by its family of α-cuts

FUZZY LEARNING means acquiring the fuzzy model — the membership functions and the rules — rather than hand-crafting it. Three sources, and the progression mirrors the knowledge-acquisition story of ACtE0904: 1. FROM AN EXPERT (not learning, but the baseline) Interview the operator, who says "if the load is large and the dirt is heavy, wash for a long time". The rules come out in fuzzy terms naturally, WHICH IS THE POINT — the expert's knowledge was already vague, and fuzzy logic is the first formalism that does not force them to invent precision they do not have. THIS IS THE STRONGEST PRACTICAL ARGUMENT FOR FUZZY SYSTEMS. 2. TUNING THE MEMBERSHIP FUNCTIONS FROM DATA Keep the expert's rule structure and adjust the parameters of the membership functions (the triangle corners) to minimise error on recorded input–output examples. Gradient descent works if the functions are differentiable, which is why Gaussian membership functions are preferred for tuning over triangular ones. 3. LEARNING THE RULES THEMSELVES · CLUSTERING the input–output data and reading one rule off each cluster · GENETIC ALGORITHMS searching over rule sets — the subject of a later topic in this section · NEURO-FUZZY SYSTEMS, of which ANFIS (adaptive-network- based fuzzy inference system) is the standard: a fuzzy system expressed as a layered network so that backpropagation can train it. THE APPEAL OF NEURO-FUZZY IS EXACTLY THE HYBRID ARGUMENT from the AI concepts topic: a neural network learns but cannot explain; a fuzzy system explains but does not learn; combining them gives a system that is trained from data AND whose rules can be read afterwards. FUZZY CLUSTERING deserves separate mention because it is genuinely useful: FUZZY C-MEANS is k-means with graded membership — each point belongs to every cluster to a degree, summing to 1 across clusters, rather than being assigned to exactly one. WHY IT IS BETTER FOR OVERLAPPING DATA: a point midway between two clusters gets 0.5 and 0.5 rather than being forced into one, so the information that it is ambiguous survives. k-means discards that, which is a real loss when the boundary matters. WHERE FUZZY LOGIC IS THE RIGHT TOOL: · CONTROL problems where the expert can describe the policy in words but nobody can write the differential equations · systems needing SMOOTH transitions rather than switching · domains where the concepts are INHERENTLY GRADED — comfort, quality, risk, severity · where the rules must remain READABLE by a domain expert WHERE IT IS NOT: · where the uncertainty is genuinely PROBABILISTIC — use probability, which has axiomatic foundations that fuzzy logic lacks · where precise optimisation is required · where there is plenty of labelled data and no need for interpretability, in which case a statistical model will usually do better THE HONEST ASSESSMENT: fuzzy logic's foundations are weaker than probability's — the choice of membership functions and operators is a modelling decision with no principled derivation, and Zadeh's framework attracted sustained criticism on exactly that ground. What it has is a genuine practical niche: TURNING VAGUE EXPERT DESCRIPTIONS INTO WORKING CONTROLLERS, which it does better than any alternative, and thousands of deployed appliances are the evidence.