Unit 3: Logical reasoning and uncertainty

• Propositional logic uses statements (P, Q) and connectives (¬, ∧, ∨, →, ↔). It cannot express "all" or "some".
• First-order (predicate) logic adds objects, predicates, functions and quantifiers: ∀ (for all) and ∃ (there exists).

• Unification finds a substitution that makes two expressions identical: Knows(John, x) and Knows(John, Mary) unify with {x/Mary}. Lifting applies inference rules to quantified sentences using unification.

• Resolution proves a statement by contradiction: convert sentences to conjunctive normal form (CNF), add the negated goal, and resolve clauses until the empty clause appears.
• A truth maintenance system (TMS) tracks why each belief is held and retracts conclusions when their supporting facts change.

Planning finds a sequence of actions to reach a goal. STRIPS represents each action with preconditions, an add list and a delete list.

The Blocks World (stacking blocks on a table with a robot hand) is the classic planning domain.

• Monotonic logic never withdraws conclusions; non-monotonic reasoning allows conclusions to be withdrawn when new information arrives ("Birds fly" — but not penguins). Default reasoning and circumscription are examples.
• Probabilistic reasoning handles uncertainty with probabilities. Bayes' theorem: P(H given E) = P(E given H) × P(H) / P(E). Bayesian networks represent dependencies among variables as a directed graph.

In a classical (crisp) set, membership is 0 or 1. In a fuzzy set, membership is a degree between 0 and 1 — useful for vague ideas like "tall" or "hot".