The document discusses predicate logic as a fundamental aspect of computer science, detailing its principles through propositional logic, predicates, quantifiers, and logical expressions. It explains concepts like bound and free variables, tautologies, and the construction of proofs in predicate logic, along with their interpretations and the relationship between provability and truth. Key techniques for manipulating logical expressions and building proofs are also outlined.

1. Predicate Logic
DAT911 - Foundations of Computer Science
Dario Garigliotti
March 9, 2016
Dario Garigliotti Predicate Logic

2. Predicate Logic
In Propositional Logic (P.L.) we could have:
r = It is snowing in Stavanger
s = It is snowing in Oslo
t = It is snowing in Trondheim
Dario Garigliotti Predicate Logic

3. Predicate Logic
In Propositional Logic (P.L.) we could have:
r = It is snowing in Stavanger
s = It is snowing in Oslo
t = It is snowing in Trondheim
A predicate is a generalization of a propositional variable:
r(X) = It is snowing in X
Dario Garigliotti Predicate Logic

4. Predicate Logic
In Propositional Logic (P.L.) we could have:
r = It is snowing in Stavanger
s = It is snowing in Oslo
t = It is snowing in Trondheim
A predicate is a generalization of a propositional variable:
r(X) = It is snowing in X
Atomic formula: a predicate with 0+ arguments (variables or
constants)
”variables” vs ”propositional variables”
0 args: m = It is Wednesday
Ground atomic formula: all arguments are constants
r(Stavanger) = It is snowing in Stavanger
Dario Garigliotti Predicate Logic

5. Predicates and Logical Expressions
A predicate is a generalization of a propositional variable:
r(X) = It is snowing in X
Atomic formula: a predicate with 0+ arguments (variables or
constants)
Logical expressions
Literals: an atomic formula or its negation
p(X, a), ¬p(a, b)
Dario Garigliotti Predicate Logic

6. Predicates and Logical Expressions
A predicate is a generalization of a propositional variable:
r(X) = It is snowing in X
Atomic formula: a predicate with 0+ arguments (variables or
constants)
Logical expressions
Literals: an atomic formula or its negation
p(X, a), ¬p(a, b)
Expressions: by using
logical operators ¬, ∨, ∧, =⇒ , ≡
quantifiers ∀, ∃
Dario Garigliotti Predicate Logic

7. Quantifiers
For our predicate
r(X) = It is snowing in X
if we want to express: It is snowing in at least one city
r(Berlin) ∨ r(London) ∨ r(Paris) ∨ r(Rome) ∨ ...
we do it by
the existential quantifier: (∃X)r(X)
In the case of expressing For all the cities..., we use
the universal quantifier: (∀X)r(X)
Dario Garigliotti Predicate Logic

8. Quantifiers
Recursive definition of logical expressions
Basis: every atomic formula is an expression
Induction: E, F logical expressions, then so are
¬E, E ∧ F, E ∨ F, E =⇒ F, E ≡ F
(∀X)E, (∃X)E, for any variable X
Dario Garigliotti Predicate Logic

9. Quantifiers
Recursive definition of logical expressions
Basis: every atomic formula is an expression
Induction: E, F logical expressions, then so are
¬E, E ∧ F, E ∨ F, E =⇒ F, E ≡ F
(∀X)E, (∃X)E, for any variable X
A little more about quantifiers
Precedence of operators: ∀, ∃ have highest precedence, then
as in P.L.: ¬, ∧, ∨, =⇒ , ≡
Importance of order: (∀X)(∃Y ) vs (∃X)(∀Y )
Bound and free variables
Dario Garigliotti Predicate Logic

10. Quantifiers: Bound and Free Variables
A quantifier ”declares” a variable within a ”scope”
E.g. (∀X)r(X) ∨ (∀X)s(X)
Let’s think on its expression tree
∨ − − − (∀X) − − − r(X)
− − − (∀X) − − − s(X)
An occurrence of a variable X in an expression E is bound by a
quantifier (QX) if in its expression tree, (QX) is the lowest
ancestor involving X
Otherwise, that occurrence is free
Dario Garigliotti Predicate Logic

11. Quantifiers: Bound and Free Occurrences of Variables
An occurrence of a variable X in an expression E is bound by a
quantifier (QX) if in its expression tree, (QX) is the lowest
ancestor involving X
Otherwise, that occurrence is free
Another example: r(X) ∨ (∃X)s(X)
∨ − − − r(X)
− − − (∃X) − − − s(X)
Then we mean: bound and free occurrences of variables
Dario Garigliotti Predicate Logic

12. Interpretations
An example: E ≡ p(X, Y ) =⇒ (∃Z)(p(X, Z) ∧ p(Z, Y ))
An interpretation for E is defined by:
Dario Garigliotti Predicate Logic

13. Interpretations
An example: E ≡ p(X, Y ) =⇒ (∃Z)(p(X, Z) ∧ p(Z, Y ))
An interpretation for E is defined by:
an interpretation I for the predicate based on the assignment
of values in D to its arguments, e.g. I1 : p(U, V ) ≡ U < V
Dario Garigliotti Predicate Logic

14. Interpretations
An example: E ≡ p(X, Y ) =⇒ (∃Z)(p(X, Z) ∧ p(Z, Y ))
An interpretation for E is defined by:
an interpretation I for the predicate based on the assignment
of values in D to its arguments, e.g. I1 : p(U, V ) ≡ U < V
a domain D from which to select values for the variables, e.g.
D = R
Dario Garigliotti Predicate Logic

15. Interpretations
An example: E ≡ p(X, Y ) =⇒ (∃Z)(p(X, Z) ∧ p(Z, Y ))
An interpretation for E is defined by:
an interpretation I for the predicate based on the assignment
of values in D to its arguments, e.g. I1 : p(U, V ) ≡ U < V
a domain D from which to select values for the variables, e.g.
D = R
a value in D for each free occurrences of variables
Dario Garigliotti Predicate Logic

16. Interpretations
An example: E ≡ p(X, Y ) =⇒ (∃Z)(p(X, Z) ∧ p(Z, Y ))
An interpretation for E is defined by:
an interpretation I for the predicate based on the assignment
of values in D to its arguments, e.g. I1 : p(U, V ) ≡ U < V
a domain D from which to select values for the variables, e.g.
D = R
a value in D for each free occurrences of variables
given I1 with D, we can define infinite interpretations such
that E is true
Dario Garigliotti Predicate Logic

17. Interpretations
An example: E ≡ p(X, Y ) =⇒ (∃Z)(p(X, Z) ∧ p(Z, Y ))
An interpretation for E is defined by:
an interpretation I for the predicate based on the assignment
of values in D to its arguments, e.g. I1 : p(U, V ) ≡ U < V
a domain D from which to select values for the variables, e.g.
D = R
a value in D for each free occurrences of variables
given I1 with D, we can define infinite interpretations such
that E is true
what if we use I2 by defining D = Z?
Dario Garigliotti Predicate Logic

18. Interpretations
An example: E ≡ p(X, Y ) =⇒ (∃Z)(p(X, Z) ∧ p(Z, Y ))
An interpretation for E is defined by:
an interpretation I for the predicate based on the assignment
of values in D to its arguments, e.g. I1 : p(U, V ) ≡ U < V
a domain D from which to select values for the variables, e.g.
D = R
a value in D for each free occurrences of variables
given I1 with D, we can define infinite interpretations such
that E is true
what if we use I2 by defining D = Z?
E always true except for those in which Y = X + 1
Dario Garigliotti Predicate Logic

19. Interpretations
We can give a definition of the interpretation of an expression E,
by induction in its expression tree
E ≡ p(X1, ..., Xn)
E ≡ E1 ∧ E2
E ≡ E1 ∨ E2
E ≡ ¬E1
E ≡ (∃X)E1
E ≡ (∀X)E1
Dario Garigliotti Predicate Logic

20. Tautologies - Substitutions and Equivalences
E is a tautology if its value is true for every interpretation of E
The previous example is not:
E ≡ p(X, Y ) =⇒ (∃Z)(p(X, Z) ∧ p(Z, Y ))
E.g. of a tautology: q(X) ∨ ¬q(X)
Dario Garigliotti Predicate Logic

21. Tautologies - Substitutions and Equivalences
E is a tautology if its value is true for every interpretation of E
The previous example is not:
E ≡ p(X, Y ) =⇒ (∃Z)(p(X, Z) ∧ p(Z, Y ))
E.g. of a tautology: q(X) ∨ ¬q(X)
The substitution principle allows to obtain tautologies in
predicate logic from propositional tautologies
E.g. the latter, after replacing in p ∨ ¬p
Dario Garigliotti Predicate Logic

22. Tautologies - Substitutions and Equivalences
E is a tautology if its value is true for every interpretation of E
The previous example is not:
E ≡ p(X, Y ) =⇒ (∃Z)(p(X, Z) ∧ p(Z, Y ))
E.g. of a tautology: q(X) ∨ ¬q(X)
The substitution principle allows to obtain tautologies in
predicate logic from propositional tautologies
E.g. the latter, after replacing in p ∨ ¬p
We can say that E and F are equivalent if E ≡ F is a tautology
Then, by substitution, if E1 ≡ E2, and we replace in F1 one by
another, we obtain an expression F2 such that F1 ≡ F2
Dario Garigliotti Predicate Logic

23. Tautologies Involving Quantifiers
Let’s describe two techniques for tautologies with quantifiers
1 Rectifying expressions
Variable renaming: (QX)E ≡ (QX)E , where E is obtained
from E by replacing all occurrences of bound X for a fresh
(not free in E) Y
Rectified expression by renaming it without repeating variables
Dario Garigliotti Predicate Logic

24. Tautologies Involving Quantifiers
Let’s describe two techniques for tautologies with quantifiers
1 Rectifying expressions
Variable renaming: (QX)E ≡ (QX)E , where E is obtained
from E by replacing all occurrences of bound X for a fresh
(not free in E) Y
Rectified expression by renaming it without repeating variables
2 Moving quantifiers outside
¬((∀X)E) ≡ (∃X)(¬E)
¬((∃X)E) ≡ (∀X)(¬E)
(E ∧ (QX)F) ≡ (QX)(E ∧ F) (X not free in E)
(E ∨ (QX)F) ≡ (QX)(E ∨ F) (X not free in E)
Dario Garigliotti Predicate Logic

25. Tautologies Involving Quantifiers
With those two steps, we can get an equivalent quantifier-free
expression (or prenex-form expression) of the shape
(Q1X1)(Q2X2)...(QkXk)E
E.g. from E ≡ (∀X)p(X) ∨ ¬(∀X)p(X)
Dario Garigliotti Predicate Logic

26. Tautologies Involving Quantifiers
With those two steps, we can get an equivalent quantifier-free
expression (or prenex-form expression) of the shape
(Q1X1)(Q2X2)...(QkXk)E
E.g. from E ≡ (∀X)p(X) ∨ ¬(∀X)p(X)
Rectifying expression: (∀X)p(X) ∨ ¬(∀Y )p(Y )
Dario Garigliotti Predicate Logic

27. Tautologies Involving Quantifiers
With those two steps, we can get an equivalent quantifier-free
expression (or prenex-form expression) of the shape
(Q1X1)(Q2X2)...(QkXk)E
E.g. from E ≡ (∀X)p(X) ∨ ¬(∀X)p(X)
Rectifying expression: (∀X)p(X) ∨ ¬(∀Y )p(Y )
Moving Qs outside: (∀X)p(X) ∨ (∃Y )¬p(Y )
Dario Garigliotti Predicate Logic

28. Tautologies Involving Quantifiers
With those two steps, we can get an equivalent quantifier-free
expression (or prenex-form expression) of the shape
(Q1X1)(Q2X2)...(QkXk)E
E.g. from E ≡ (∀X)p(X) ∨ ¬(∀X)p(X)
Rectifying expression: (∀X)p(X) ∨ ¬(∀Y )p(Y )
Moving Qs outside: (∀X)p(X) ∨ (∃Y )¬p(Y )
Moving Qs outside: (∀X)(∃Y )(p(X) ∨ ¬p(Y ))
Dario Garigliotti Predicate Logic

29. Proofs in Predicate Logic: Basics
Given some hypotheses E1, E2, ..., Ek, a proof is a sequence of
expressions such that each of them
either is one if the Ei ’s
or follows from 0+ previous expressions by some rule of
inference
A rule of inference is of the shape (F1 ∧ F2 ∧ ... ∧ Fn) =⇒ F
Then, the law of variable substitution: given E asserted,
E =⇒ sub(E) is a tautology
E.g. p(X, Y ) =⇒ p(U, V )
E.g. p(X, Y ) =⇒ p(1, 2)
Dario Garigliotti Predicate Logic

30. Proofs in Predicate Logic: From Rules and Facts
Two main kind of hypotheses:
Facts: ground atomic formulas, e.g. p(1)
Rules: ”if-then” expressions, with a body of subgoals and a
head, e.g. p(X) ∧ q(Y ) =⇒ r(X)
Dario Garigliotti Predicate Logic

31. Proofs in Predicate Logic: From Rules and Facts
Two main kind of hypotheses:
Facts: ground atomic formulas, e.g. p(1)
Rules: ”if-then” expressions, with a body of subgoals and a
head, e.g. p(X) ∧ q(Y ) =⇒ r(X)
Then, a proof can be build in this way, e.g.
1 p(X) ∧ q(Y ) =⇒ r(X) (Hypothesis: rule)
2 p(1) (Hypothesis: fact)
3 q(3) (Hypothesis: fact)
Dario Garigliotti Predicate Logic

32. Proofs in Predicate Logic: From Rules and Facts
Two main kind of hypotheses:
Facts: ground atomic formulas, e.g. p(1)
Rules: ”if-then” expressions, with a body of subgoals and a
head, e.g. p(X) ∧ q(Y ) =⇒ r(X)
Then, a proof can be build in this way, e.g.
1 p(X) ∧ q(Y ) =⇒ r(X) (Hypothesis: rule)
2 p(1) (Hypothesis: fact)
3 q(3) (Hypothesis: fact)
4 p(1) ∧ q(3)
Dario Garigliotti Predicate Logic

33. Proofs in Predicate Logic: From Rules and Facts
Two main kind of hypotheses:
Facts: ground atomic formulas, e.g. p(1)
Rules: ”if-then” expressions, with a body of subgoals and a
head, e.g. p(X) ∧ q(Y ) =⇒ r(X)
Then, a proof can be build in this way, e.g.
1 p(X) ∧ q(Y ) =⇒ r(X) (Hypothesis: rule)
2 p(1) (Hypothesis: fact)
3 q(3) (Hypothesis: fact)
4 p(1) ∧ q(3)
5 p(1) ∧ q(3) =⇒ r(1)
Dario Garigliotti Predicate Logic

34. Proofs in Predicate Logic: From Rules and Facts
Two main kind of hypotheses:
Facts: ground atomic formulas, e.g. p(1)
Rules: ”if-then” expressions, with a body of subgoals and a
head, e.g. p(X) ∧ q(Y ) =⇒ r(X)
Then, a proof can be build in this way, e.g.
1 p(X) ∧ q(Y ) =⇒ r(X) (Hypothesis: rule)
2 p(1) (Hypothesis: fact)
3 q(3) (Hypothesis: fact)
4 p(1) ∧ q(3)
5 p(1) ∧ q(3) =⇒ r(1)
6 r(1)
Dario Garigliotti Predicate Logic

35. Provability and Truth
Idea of model: an interpretation which makes an expression
true
Similarly, a model of a collection of expressions {E1, E2, ..., En}
Entailment or satisfaction: E1, E2, ..., En |= E if every model
for the collection is also a model for E
Provability: E1, E2, ..., En E
Consistency of a proof system: provability =⇒ entailment
Completeness of a proof system: entailment =⇒ provability
Dario Garigliotti Predicate Logic

36. Predicate Logic
Thanks!
Questions?
Dario Garigliotti Predicate Logic