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### 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: 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

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))

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)

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

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