Download Quick recap of logic: Predicate Calculus - clic

INTRODUCTION TO ARTIFICIAL
INTELLIGENCE
Massimo Poesio
LECTURE 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
John
Fred
Matilda
Massimo
LIKES(j,AI) = T
LIKES(m,Maths) = F
SUBJECTS
AI
Maths
Logic
SET THEORY RECAP: QUANTIFIERS
AIRPLANES
BIRDS
Tweety
SWALLOWS
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
AFFIRMATIVE
NEGATIVE
UNIVERSAL
A (Adfirmo)
E (nEgo)
PARTICULAR
I (adfIrmo)
O (negO)
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
Q
A: All Ps are Qs
Q
R
P
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
UNIVERSAL
INSTANTIATION
UNIVERSAL
GENERALIZATION
(∀y) P(y)
∴P(c) (for any constant c)
P(c) (for any constant c)
∴ (∀y) P(y)
UI AND UG EXAMPLES
UNIVERSAL
INSTANTIATION
(∀y) MADE-OF-ATOMS(y)
∴ MADE-OF-ATOMS(c)
(for any c)
NATURAL DEDUCTION FOR FOL, 2
EXISTENTIAL
INSTANTIATION
EXISTENTIAL
GENERALIZATION
(∃y) P(y)
∴ P(k) (for a new k)
P(c) (for a constant c)
∴ (∃ 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