1. Predicate logic
29 Oct. 2007
Jens Nilsson

2. Predicate logic Propositional logic compared to predicate logic
Predicate logic: Syntax
Predikate logic: Semantic
Aho & Ullman: 14.1 – 14.5
Motivation for predicate logic:
One way for a computer to convey meaning
Form foundation for some programming languages

3. Introduction
Ex 1: If ”Ice cream is cold” → ”There is something that is cold”
Ex 2: If ”Everything consists of matter” → ”The car consists of matter”
In propositional logic: A := ” Ice cream is cold” och B: ” There is something that is cold” A := ” Everything consists of matter” och B : ”The car consists of matter”
A → B is not derivable in propositional logic.
In predikate logic:
isCold(ice_cream) consistsOfMatter(the_car) isCold(the_sun) consistsOfMatter(the_air)

4. Propostitional logic vs. Predicate Logic (1)
Propostitional logic is a special case of Predicate Logic (everything that can be described in Propostitional logic can be descipbed in predicate logic), which entails these similarities:
atomic statements have exactly one of two possible truth values, True or False
consistsOfMatter(the_car) = 1 p = 1
isCold(the_sun) = 0 q = 0
expressions can be composed into larger expressions using operators
consistsOfMatter(the_car) AND consistsOfMatter(the_air), (compare with p AND q)

5. Propostitional logic vs. Predicate Logic (2)
Similarities (cont.)
The same algebraic laws hold:
Associativity: (A OR B) OR C  A OR (B OR C) (A AND B) AND C  A AND (B AND C)
Distributivitity: A OR (B AND C)  (A OR B) AND (A OR C) A AND (B OR C)  (A AND B) OR (A AND C)
Commutativity: A OR B  B OR A A AND B  B AND A
DeMorgans law: NOT (A AND B)  (NOT A) OR (NOT B) NOT (A OR B)  (NOT A) AND (NOT B)
Difference: The truth value of composed expressions in propositional logic is unambigouosly determined by its parts.
This does not hold for predicate logic, since a interpretation is need (see comming slides about semantic).

6. Predicate logic: Syntax (1)
Symbols:
Predicate symbols: p, q, r, ...
Symbols of individuals:
Individual constants: a, b, c, ...
Individual variables: X, Y, Z, ...
Domain (set of individuals):
D = {a, b, c, ...} (I = {a, b, c, ...})
Inductive definition of logic formulas:
Atomic formulas: a predicate symbol followed by zero or more individual symbols, e.g. p, q(a),r(a, b), q(X)

7. Predicate logic: Syntax (2)
Formlulas with operators: If A och B är logical formulas, then the following are also logical formulas:
Negation: A (~A, NOT A, )
Conjunction: A  B (A & B, A AND B, AB)
Disjunction: A  B (A OR B, A + B)
Implication: A → B (A  B)
Equivalence: A  B (A  B)
Formulas with quantifications: If A is a logic formula and  is an individual variable, then thees are also logical formulas:
 A (universal quantifier)
 A (existential quantifier)

8. Predicate logic: Syntax (3)
Example:
course(focs)
teaching(jens, focs)
studying(jens, mathematics)
studying(X, computer_science)  studying(X, mathematics)
 X student(X)
 Y (student(Y) → happy(Y))
 X  Y (student(Y)  knows(X, Y))
Möjlig domän:
D = {focs, ... , jens, ... , mathematics, computer_science, ...}

9. Predicate logic: Syntax (4)
Priority of operators
Quantifiers (, )
Negation
Conjunction
Disjunction
Implication
Equivalence
Examples:
Is  p(X)  q(X) the same as:  (p(X)  q(X)) or ( p(X))  q(X)?
( p(X))  q(X) ?
Is X p(X)  q(X) the same as : X (p(X)  q(X)) or X (p(X))  q(X)?
X (p(X))  q(X) ?
Is p(X)  q(X) → r(X) the same as : p(X)  ( q(X) → r(X)) or (p(X)  q(X)) → r(X)?
(p(X)  q(X)) → r(X) ?

10. Predicate logic: Syntax (5)
Scope, bounded and free variables
Let Q be an arbitrary quantifier where Q is  or , and  is an indiviual variable.
In formulas with the form Q A, the quantifier has scope over A.
The quantifier Q1 binds every occurance of the variable  in A that is bounded by another quantifier Q2 within A.
A variable that is not bounded is free.
A formula without free variables is called a ground formula
Can be compared with local and global declaration of variables in C (or Java)

11. Predicate logic: Syntax (6)
Example (which variables are bounded?):
 X student(X)
[ X bounded by  X ]
 X student(X)  happy(X)
[ X1 bounded, X2 free]
 X (student(X)  happy(X))
[ X1 , X2 bounded]
 X (student(X)   X happy(X))
[ X1 bounded by  X, X2 bounded of  X ]
 X happy(Y)
[ Y free]
happy(Y)
Possible domain: D = {The set of all people}

12. Predicate logic: Semantic (1)
Predicate logic in natural language:
Predicate: mushroom(), purple() och poisonous()
All purple mushrooms are poisonius:  X((mushroom(X)  purple(X)) → poisonous(X))
No purple mushrooms are poisonius: X((mushroom(X)  purple(X)) →poisonous(X))
All mushrooms are either purple or poisonous:
X(mushroom(X) → (purple(X)  poisonous(X)))

13. Predicate logic: Semantic (2)
Interpretation
A non-empty domain
One interpretation for each constant:
One individual constant is assigned one individual from the domain.
One n-ary predicate is assigned a function  : Dn → {True, False} (i.e. a function from individuals to truth values)

14. Predicate logic: Semantic (3)
Evaluation rules
Atomic formulas:
If A is an atomic formula with the predicate p and the arguments a1, ... , ak, and if the current interpretation assigns p the function , then A is true if and only if (a1, ... , ak) = True.
Complex formulas:
Negation: A is true if and only of A is false.
Conjunction: A  B is true if and only both A och B are true.
Disjunction: A  B is true if and only at least one of A och B are true.
Implication: A → B is true if and only A is false or B is true.
Equivalence: A  B is true if and only A and B have the same truth value.

15. Predicate logic: Semantic (4)
Evaluation rules, cont.
Formulas with quantifiers
 A is true if and only if A is true in all interpretations that are exactly the same as in  A, except that the variable  can be assigned any individual constant in the domain.
For all X, it is the case that ...
 A is true if and only if A is true in any interpretations that is exactly the same as in  A, except that the variable  can be assigned any individual constant in the domain.
Of all X, there is at least one X such that ...

16. Predicate logic: Semantic (5)
Interpretation
D = {jens, adam, eve, mathematics, computer_science, da1021, focs}
Interpretation of predicates:
course = {da1021, focs}
student = {adam, eve}
happy = {jens, eve}
teaching = {(jens, focs)}
studying = {(adam, computer_science), (eve, computer_science), (eve, mathematics)}
knows = {(adam, eve), (eve, adam), (jens, adam), (adam, jens)}
Variable assignment: X = jens, Y = eve

17. Predikatlogik: Semantik (5)
Interpretation(example), cont.
Evaluation of formulas:
course(focs) (true)
course(computer_science) (false)
teaching(jens, focs) (true)
studying(jens, mathematics) (false)
 studying(jens, mathematics) (true)
studying(X, computer_science)  studying(X, mathematics) (false?)
X student(X) (true)
Y(student(Y) → happy(Y)) (false)
XY(student(Y)  knows(X, Y)) (false)

18. Predicate logic in language (1)
Translate English to predicate logic:
Mary doesn’t jog:
jog(mary) j(m)
Fred smokes and Mary drinks:
smoke(fred)  drink(mary) s(f)  d (m)
Carl, who is a millonaire, is a socialist:
millonaire(carl)  socialist(carl) m(c)  s(c)
Jean admires Robert, who is a ganster:
admires(jean, robert)  ganster(robert) a(j, r)  g(r)

19. Predicate logic in language (2)
Predicate logic as meta language:
Natural language contains a number of quantifiers:
Some students wrote a paper
Many students wrote a paper
Most students wrote a paper
All students wrote a paper
Every student wrote a paper
No student wrote a paper
A lot of students wrote a paper
Some students:
EX(student(X) AND paper(X, w))
Every student:
AX (student(X) -> paper(X, w))
No student:
NOT EX(student(X) AND paper(X, w))
AX(student(X) -> NOT paper(X, w))

20. Predicate logic in language (3)
Some advantages with formal semantic in form of predicate logic
Amiguous scope for quvantifiers in natural language
Example: Everybody loves somone
Does it mean:
Everyone has someone they love ? or
There is some person who is loved by everyone ?
In predicate logic:
XY loves(X, Y)
YX loves(X, Y)
Räckvidden för kvantifierare i predikatlogik är otvetydigt
Rita liten bild här: fina gubbar i cirkel med pilar
bred och smal räckvidd

21. Predicate logic in language (4)
Some advantages with formal semantic in form of predicate logic (cont.)
Scope for negation:
Everyone didn’t visit London
Does it mean:
For every person x, it is not the case that x visited London ? or
It’s not the case that every person x visited London ?
In predicate logic:
X visit(X, london) or X visit(X, london)
Y visit(X, london) or Y visit(X, london)
predikatlogik ger oss möjlighet att disambiguera meningar som innehåller både negation och kvantifierare
handar igen om vilken som har bred räckvidd: negationen eller kvantifieraren
de två senaste sliden visar att predikatlogik har en fördel vad det gäller klarhet i uttrycket

22. Predicate logic: derivation (1)
Derivation in predicate logic
Eliminating universal quantifiers:
”All objects consists of matter” och ”The car is an object”
”All object consists of matter”: X(isObject(X) → consistsOfMatter(X))
”The car is an object”: isObject(the_car)
(1) X(isObject(X) → consistsOfMatter(X)) Premise
(2) isObject (the_car) Premise
(3) isObject (the_car) → consistsOfMatter(the_car) by -elimination from (1)
(4) consistsOfMatter(the_car) from (2) & (3)

23. Predicate logic: derivation (2)
Introducing existential quantifier:
”All object consists of matter”: X(isObject(X) → consistsOfMatter (X))
” The car is an object with wheels”: isObject (bilen)  hasWheels(bilen)
(1) X(är_objekt(X) → consistsOfMatter(X)) Premise
(2) isObject (the_car)  hasWheels (the_car) Premise
(3) isObject (the_car) → consistsOfMatter(the_car) by -elimination from (1)
(4) isObject (the_car) from (2)
(5) consistsOfMatter(the_car) from(3) & (4)
(6) hasWheels (the_car) from(2)
(7) consistsOfMatter(the_car)  hasWheels (the_car) from(5) & (6)
(8) X(consistsOfMatter(X)  hasWheels (X)) by -introduction from (7)

24. Predicate logic: summary
Predicate logic is an extension of propositional logic
Predicates och quantifiers  more expressive power
Predicate logic can use the same tools for doing derivations as propositional logic