1. Course Instructor: kinza ch

2.  Propositional logic assumes the world contains facts
 First-order logic (like natural language) assumes the world contains
Objects: people, houses, numbers, colors, baseball games, wars, …
Relations: red, round, prime, brother of, part of, comes between, …
Functions: father of, best friend, one more than, plus, …

3. This means that there is only fruit on the table
 Key element of FOL are predicates, which are used to describe objects, properties, and relationships between objects --- e.g. On(x,y)
 A quantified statement is a statement that applies to a class of objects --- e.g.
xOn(x,Table)  Fruit(x)
This means that there is only fruit on the table
The first element is called a quantifier, x is a
Table is a constant
On is a predicate
variable and

4. E.g.,Brother(Richard, John), greaterthan(3,2)
 Constant Symbols:
Stand for objects
e.g., John, 2, Ball,...
 Predicate Symbols
Stand for relations
E.g.,Brother(Richard, John), greaterthan(3,2)
 Function Symbols
Stand for functions
◦ E.g., Sqrt(3), Sum(2,3)…

5. John, 2,... Brother, >,... Sqrt, Sum,... x, y, a, b,...
 Constants
 Predicates
 Functions
 Variables
 Connectives
 Equality
 Quantifiers
John, 2,...
Brother, >,...
Sqrt, Sum,...
x, y, a, b,...
, , , ,  =
, 

6. about a single object: Round(ball), Prime(7). some fact
 Some relations are properties: they state
about a single object: Round(ball), Prime(7).
some fact
 n-ary relations state facts about two or more objects: Married(John,Mary), LargerThan(3,2).
 Some relations are functions: their value is another object: Plus(2,3), Father(Dan).

7.  Atomic sentences state facts using terms and predicate symbols
P(x,y) interpreted as “x is P of y”
 Examples:
LargerThan(2,3) is false.
 Note: Functions do not state facts and form no sentence:
Brother(Pete) refers to John for example (his brother ) and is
neither true nor false.
 Brother_of(Pete,Brother(Pete)) is True.
Binary relation
Function

8. Brother  of (Father (John), Jim)  Short(Jane)
 We make complex sentences with connectives (just like
in propositional logic).
Brother  of (Father (John), Jim)  Short(Jane)
 Binary reletion?
 Function?
 Connectives?
 Objects?
 Property?

9.  Brother(Richard, John)  Brother(John, Richard)
 King(Richard)  King(John)
 King(John)  King(Richard)
 LessThan(Plus(1,2) ,4)  GreaterThan(1,2)

10. argument „John‟ e.g., we can state more general rules like
 Person(John) is true or false because we give it a single
argument „John‟
 We can be much more flexible if we allow variables which can take on values in a domain. e.g., all persons x, all integers i, etc.
e.g., we can state more general rules like
Person(x) => HasHead(x)
or Integer(i) => Integer(plus(i,1))

11. relations functional relations
 Sentences are true
interpretation
with
respect
to a model
and an
(domain
elements)
and relations
 Model contains objects among them
 Interpretation specifies referents for
objects
relations functional relations
constant symbols → predicate symbols → function symbols →
 An atomic sentence predicate(term1,...,termn) is true iff the
objects referred to by term1,...,termn are in the relation referred
to by predicate

12.  <variables> <sentence>
 Everyone at CIIT is smart:
 x At(x,CIIT)  Smart(x)

13.  Typically,  is the main connective with 
 Common mistake: using  as the main connective with :
x At(x,CIIT)  Smart(x)
means “Everyone is at CIIT and everyone is smart”

14.  <variables> <sentence>
 Someone at CIIT is smart:
 x At(x,CIIT)  Smart(x)
 Typically,  is the main connective with , not .

15. “There is a person who loves everyone in the world”
 x y is the same as y x
 x y is the same as y x
 x y is not the same as y x
 x y loves(x, y)
“There is a person who loves everyone in the world”
 y x Loves(x, y)
“Everyone in the world is loved by at least one person”

16. only if term1 and term2 refer to the same object
 term1 = term2 is true under a given interpretation if and
only if term1 and term2 refer to the same object
Father(John) = Henry
 e.g., definition of Sibling in terms of Parent

17.  All persons are either loyal to King or hate him.
 Every one is loyal to someone.
 Every gardener likes the sun
 You can fool some of the people all of the time.

18.  ∀x AnimalLove r(x) → (∀y Animal(y) → ¬Kills(x, y))
 ∀ x ∀ y(Parent(x,y) ^ Female(y) )-> Daughter(y,x))