INTRODUCTION TO ARTIFICIAL INTELLIGENCE Massimo PoesioLECTURE 3: Logic: predicate calculus, psychological evidence
PREDICATE CALCULUS • The propositional calculus is only concerned with connectives – statements not containing connectives are left unanalyzed • Massimo is happy: p • In predicate calculus, or predicate logic, atomic statements are decomposed into TERMS and PREDICATES • Massimo is happy: HAPPY(m) • Students like AI: LIKE(students,AI) • In this way it is possible to state general properties about predicates: for instance, every professor at the University of Trento is happy, etc.
FIRST-ORDER LOGIC • Predicate calculus becomes FIRST ORDER LOGIC when we add QUANTIFIERS – logical symbols that make it possible to make universal and existential statements (i.e., to translate statements A, E, I and O of syllogisms)
THE EXISTENTIAL QUANTIFIER • Used to traduce statements like • Some birds are swallows • Notation: • ∃(backwards E, for Exist – Peano, 1890) • ‘Some birds are swallows’ • There exists an x, such that x is a bird, and x is a swallow • (∃ x) (BIRD(x) & SWALLOW(x))
THE UNIVERSAL QUANTIFIER • To represent • All men are mortal • But also: Swallows are birds • Notation: • ∀for inverted A (alle) • Conversion of universal statements requires conditional: • For every x, is x is a man, then x is mortal • (∀ x) (MAN(x) → MORTAL(x))
THE SYNTAX OF FOL: VOCABULARY • TERMS • Constants • Variables • PREDICATES: 1 argument ( HAPPY), two arguments (LIKES), etc • CONNECTIVES (from the propositional calculus): ~, &, ∨, →, ↔ • QUANTIFIERS: ∀ ∃
THE SYNTAX OF FOL: PHRASES • If P is an n-ary predicate and t1, … tn are terms, then P(t1,…,tn) is a formula. • If φ and ϕ are formulas, then ~φ, φ & ϕ , φ ∨ϕ , φ →ϕ and φ ↔ ϕ are formulas • If ϕ is a formula and x is a variable, then (∀ x) ϕ and (∃ x) ϕ are formulas.
SCOPE AND BINDING • Let x be a variable and ϕ a formula, and let (∀ x) ϕ and (∃ x) ϕ be formulas. then ϕ is the SCOPE of x in these formulas. • An occurrence of x is BOUND if it occurs in the scope of (∀ x) or (∃ x) • Examples (PMW p. 141)
THE SEMANTICS OF FOL • As in the case of propositional calculus, statements (formulas) can be either true or false • But the other phrases of the language have set-theoretic meanings: • Terms denote set elements • Unary predicates denote sets • N-ary predicates denote n-ary relations • Quantifiers denote relations between sets
SET THEORY RECAP Fred HAPPY PEOPLE John Matilda Massimo Lucy HAPPY(m) = T HAPPY(f) = F
SET THEORY RECAP: RELATIONS PEOPLE SUBJECTS John AI Matilda Logic Fred Maths Massimo LIKES(j,AI) = T LIKES(m,Maths) = F
SET THEORY RECAP: QUANTIFIERS AIRPLANES BIRDS SWALLOWS Tweety Lou Airplane1 Roger Loreto FLYING THINGS Swallows are birds Birds fly
THE SEMANTICS OF FOL • If t is a term and P a unary predicate, then [P(t)] = TRUE iff [t] ∈[P] • If φ and ϕ are formulas, then • [~φ] = TRUE iff [φ] = FALSE • [φ & ϕ] = TRUE iff [φ] = TRUE and [ϕ] = TRUE • [(∀ x) ϕ] = TRUE iff for every value a for x in model M, [ϕ(a/x)] = TRUE • [(∃ x) ϕ] = TRUE iff there is at least one object a in model M such that [ϕ(a/x)] = TRUE
SOME TAUTOLOGIES OF FOL • Laws of Quantifier Distribution: • (∀x) (φ(x) & ϕ(x)) ≡ (∀x) φ(x) & (∀x) ϕ(x) • “Every object is formed of elementary particles and has a spin” iff “Every object is formed of elementary particles” and “Every object has a spin” • Law of Quantifier Negation: • ~ (∀x) (φ(x)) ≡ (∃y) (~ φ(y)) • “It is not the case that every object is made of cheese” iff “there is an object which is not made of cheese”
FROM SYLLOGISMS TO FOL • Four types of syllogism: • Universal affirmative: All Ps are Qs • Universal negative: All Ps are not Qs (No P is a Q) • Particular affirmative: Some P is a Q • Particular negative: Some P is not a Q
THE SQUARE OF OPPOSITION
THE SQUARE OF OPPOSITION
FROM SYLLOGISMS TO FOL • Syllogism in FOL: • Universal affirmative: (∀ x) (P(x) → Q(x)) • Universal negative: (∀y) (P(y) → ~ Q(y)) • Particular affirmative: (∃z) (P(z) & Q(z)) • Particular negative: (∃ w) (P(w) & ~ Q(w))
FROM SYLLOGYSM TO FOL An example of BARBARA: A Birds fly A Swallows are birds A Swallows fly
BARBARA IN PREDICATE CALCULUS (∀x) (BIRD(x) → FLY(x)) (∀y) ( SWALLOW(y) → BIRD(y)) (∀z) ( SWALLOW(z) → FLY(z))
SET THEORETIC DEMONSTRATIONS OF VALIDITY OF SYLLOGISMS R Q Q P A: All Ps are Qs R A: All Qs are Rs P A: All Ps are Rs (A more general method exists)
REPRESENTING KNOWLEDGE IN LOGIC, 2 • Modern logics make it possibile to represent every type of knowledge • Different types of knowledge have different EXPRESSIVE POWER
REPRESENTING KNOWLEDGE IN LOGIC, 2 • “Tutte le biciclette hanno due ruote” • Propositional calculus: p • Predicate logic + quantifiers: • (∀ x) (BICYCLE(x) → HAS_TWO_WHEELS(x)) • Can be used to represent DARII • Explicit representation of the number 2: • (∀ x) (BICYCLE(x) → HAS_WHEELS(x,2)) • Set of wheels:
DEDUCTION IN FOL • The system of inference rules for FOL includes all the inference rules from the propositional calculus, together with four new rules for quantifier introduction and elimination • The tableaus system has also been extended
NATURAL DEDUCTION FOR FOL, 1 (∀y) P(y) UNIVERSAL INSTANTIATION ∴P(c) (for any constant c) P(c) (for any constant c) UNIVERSAL GENERALIZATION ∴ (∀y) P(y)
UI AND UG EXAMPLES (∀y) MADE-OF-ATOMS(y) UNIVERSAL INSTANTIATION ∴ MADE-OF-ATOMS(c) (for any c)
NATURAL DEDUCTION FOR FOL, 2 (∃y) P(y) EXISTENTIAL INSTANTIATION ∴ P(k) (for a new k) P(c) (for a constant c) EXISTENTIAL GENERALIZATION ∴ (∃ y) P(y)
BEYOND FIRST ORDER LOGIC • Artificial Intelligence research moved beyond first order logic in several directions: • Beyond using logic as a formalization of valid inference only, developing logics for non-valid (or NONMONOTONIC / UNCERTAIN) reasoning • Developing simpler logics in which inference can be done more efficiently (description logics, discussed in later lectures)
PSYCHOLOGICAL EVIDENCE ON REASONING • First order logic and the propositional calculus are good formalizations of ‘sound’ reasoning, and are therefore the basis for work on proving mathematical truths • But are they a good formalization of the way people reason? • Evidence suggests that this is not the case • The WASON SELECTION TASK perhaps the best known example of this evidence
THE WASON SELECTION TASK • Subjects are asked to verify the truth of a statement (typically, a conditional statement) by turning over cards
WASON TEST: EXAMPLE If A CARD SHOWS AN EVEN NUMBER ON ONE SIDE, then THE OPPOSITE FACE IS RED Answer: the second and fourth card
READINGS • Basics: • B. Partee, A. ter Meulen, R. Wall, Mathematical Methods in Linguistics, Springer, ch. 5, 6, 7 • (in Italian): D. Palladino, Corso di Logica, Carocci • To know more: • History of logic: P. Odifreddi, Le menzogne di Ulisse, Tea, ch. 1-7 • Inference: P. Blackburn, J. Bos, Representation and Inference for Natural Language, CSLI • K. Stenning and M. van Lambalgen, Human Reasoning and Cognitive Science, MIT Press • Logic on the Web: • http://www.thelogiccourse.com/ • Do the Wason selection task: http://coglab.wadsworth.com/experiments/WasonSelection.shtml
