1. Predicate Calculus

2. First-order logic
Whereas propositional logic assumes the world contains facts,
Relations: red, round, prime, brother of, bigger than, part of, comes between, … 2

3. And
Functions: father of, best friend, one more than, plus, … 3

4. Syntax of FOL: Basic elements
Constants TaoiseachJohn, 2, DIT,...
Predicates Brother, >,...
Functions Sqrt, LeftLegOf,...
Variables x, y, a, b,...
Connectives , , , , 
Equality =
Quantifiers ,  4

5. Atomic sentences
Atomic sentence = predicate (term1,...,termn) or term1 = term2
Term = function (term1,...,termn) or constant or variable
E.g., Brother(TaoiseachJohn,RichardTheLionheart) > (Length(LeftLegOf(Richard)), Length(LeftLegOf(TaoiseachJohn))) 5

6. Complex sentences
Complex sentences are made from atomic sentences using connectives
S, S1  S2, S1  S2, S1  S2, S1  S2,
E.g. Sibling(TaoiseachJohn,Richard)  Sibling(Richard,TaoiseachJohn)
>(1,2)  ≤ (1,2)
>(1,2)   >(1,2) 6

7. Truth in first-order logic
Sentences are true with respect to a model and an interpretation
Model contains objects (domain elements) and relations among them
Interpretation specifies referents for
constant symbols → objects
predicate symbols → relations
function symbols → functional relations
An atomic sentence predicate(term1,...,termn) is true
iff the objects referred to by term1,...,termn
are in the relation referred to by predicate 7

8. Universal quantification
<variables> <sentence>
Everyone at DIT is smart:
x At(x,DIT)  Smart(x)
x P is true in a model m iff P is true with x being each possible object in the model 8

9. Roughly speaTaoiseach, equivalent to the conjunction of instantiations of P
At(TaoiseachJohn,DIT)  Smart(TaoiseachJohn)
 At(Richard,DIT)  Smart(Richard)
 At(DIT,DIT)  Smart(DIT)
 ... 9

10. A common mistake to avoid
Typically,  is the main connective with 
Common mistake: using  as the main connective with :
x At(x,DIT)  Smart(x)
means “Everyone is at DIT and everyone is smart” 10

11. Existential quantification
<variables> <sentence>
Someone at DIT is smart:
x At(x,DIT)  Smart(x)$
x P is true in a model m iff P is true with x being some possible object in the model 11

12. Roughly speaTaoiseach, equivalent to the disjunction of instantiations of P
At(TaoiseachJohn,DIT)  Smart(TaoiseachJohn)
 At(Richard,DIT)  Smart(Richard)
 At(DIT,DIT)  Smart(DIT)
 ... 12

13. Another common mistake to avoid
Typically,  is the main connective with 
Common mistake: using  as the main connective with :
x At(x,DIT)  Smart(x)
is true if there is anyone who is not at DIT! 13

14. Properties of quantifiers
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”
Quantifier duality: each can be expressed using the other
x Likes(x,IceCream) x Likes(x,IceCream)
x Likes(x,Broccoli) x Likes(x,Broccoli) 14

15. Equality
term1 = term2 is true under a given interpretation if and only if term1 and term2 refer to the same object
E.g., definition of Sibling in terms of Parent:
x,y Sibling(x,y)  [(x = y)  m,f  (m = f)  Parent(m,x)  Parent(f,x)  Parent(m,y)  Parent(f,y)] 15

16. Using FOL The kinship domain: Brothers are siblings
x,y Brother(x,y)  Sibling(x,y) 16

17. One's mother is one's female parent
m,c Mother(c) = m  (Female(m)  Parent(m,c))
“Sibling” is symmetric
x,y Sibling(x,y)  Sibling(y,x) 17

18. Knowledge engineering in FOL
Identify the task
Assemble the relevant knowledge
Decide on a vocabulary of predicates, functions, and constants
Encode general knowledge about the domain
Encode a description of the specific problem instance
Pose queries to the inference procedure and get answers
Debug the knowledge base 18

19. Summary First-order logic:
objects and relations are semantic primitives
syntax: constants, functions, predicates, equality, quantifiers 19

20. Semantics for Predicate Calculus
An interpretation over D is an assignment of the entities of D to each of the constant, variable, predicate and function symbols of a predicate calculus expression such that:

21. 1: Each constant is assigned an element of D
2: Each variable is assigned a non-empty subset of D;(these are the allowable substitutions for that variable)
3: Each predicate of arity n is defined on n arguments from D and defines a mapping from Dn into {T,F}
4: Each function of arity n is defined on n arguments from D and defines a mapping from Dn into D

22. The meaning of an expression
Given an interpretation, the meaning of an expression is a truth value assignment over the interpretation.

23. Truth Value of Predicate Calculus expressions
Assume an expression E and an interpretation I for E over a non empty domain D. The truth value for E is determined by:
The value of a constant is the element of D assigned to by I
The value of a variable is the set of elements assigned to it by I

24. More truth values
The value of a function expression is that element of D obtained by evaluating the function for the argument values assigned by the interpretation
The value of the truth symbol “true” is T
The value of the symbol “false” is F
The value of an atomic sentence is either T or F as determined by the interpretation I

25. Similarity with Propositional logic truth values
The value of the negation of a sentence is F if the value of the sentence is T and F otherwise
The values for conjunction, disjunction ,implication and equivalence are analogous to their propositional logic counterparts

26. Universal Quantifier A X S The value for
Is T if S is T for all assignments to X under I, and F otherwise A
X S

27. Existential Quantifier
The value for
Is T if S is T for any assignment to X under I, and F otherwise E
X S

28. Some Definitions A predicate calculus expressions S1 is satisfied.
Definition If there exists an Interpretation I and a variable assignment under I which returns a value T for S1 then S1 is said to be satisfied under I.
S is Satisfiable if there exists an interpretation and variable assignment that satisfies it: Otherwise it is unsatisfiable

29. Some Definitions
A set of predicate calculus expressions S is satisfied.
Definition For any interpretation I and variable assignment where a value T is returned for every element in S the the set S is said to be satisfied,

30. A set of expressions is satisfiable if and only if there exist an intrepretation and variable assignment that satisfy every element
If a set of expressions is not satisfiable, it is said to be inconsistent
If S has a value T for all possible interpretations , it is said to be valid

31. Some Definitions A predicate calculus expressions S1 is satisfied.
Definition If there exists an Interpretation I and a variable assignment under I which returns a value T for S1 then S1 is said to be satisfied under I.
A set of predicate calculus expressions S is satisfied.
Definition For any interpretation I and variable assignment where a value T is returned for every element in S the the set S is said to be satisfied,
An inference rule is complete.
Definition If all predicate calculus expressions X that logically follow from a set of expressions, S can be produced using the inference rule , then the inference rule is said to be complete.

32. A predicate calculus expression X logically follows from a set S of predicate calculus expressions .
For any interpretation I and variable assignment where S is satisfied, if X is also satisfied under the same interpretation and variable assignment then X logically follows from S.
Logically follows is sometimes called entailment

33. Soundness An inference rule is sound.
If all predicate calculus expressions X produced using the inference rule from a set of expressions, S logically follow from S then the inference rule is said to be sound.

34. Completeness
An inference Rule is complete if given a set S of predicate calculus expressions, it can infer every expression that logically follows from S

35. Equivalence
Recall that :
See attached word document