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Predicate Logic 16. Quantifiers
The Lecture
Jouko Väänänen: Predicate logic
First order (predicate logic) formulas
Quantifiers are the final elements that first order (i.e. predicate logic) formulas are built up from.First order formulas are built up from atomic formulas by means of logical operations: negation ¬, conjunction , disjunction , implication , equivalence , existential quantifier , and universal quantifier . Parentheses (,) are used for clarity.
Jouko Väänänen: Predicate logic
First order formulasare of the form
where A and B are first order formulas.
Parentheses (,) are used for clarity.
Jouko Väänänen: Predicate logic
First order formulasare of the form
where A and B are first order formulas.
Parentheses (,) are used for clarity.
atomic
Jouko Väänänen: Predicate logic
First order formulasare of the form
where A and B are first order formulas.
Parentheses (,) are used for clarity.
atomic
¬A
Jouko Väänänen: Predicate logic
First order formulasare of the form
where A and B are first order formulas.
Parentheses (,) are used for clarity.
atomic
¬A
A B
Jouko Väänänen: Predicate logic
First order formulasare of the form
where A and B are first order formulas.
Parentheses (,) are used for clarity.
atomic
¬A
A B
A B
Jouko Väänänen: Predicate logic
First order formulasare of the form
where A and B are first order formulas.
Parentheses (,) are used for clarity.
atomic
¬A
A B
A B
A B
Jouko Väänänen: Predicate logic
First order formulasare of the form
where A and B are first order formulas.
Parentheses (,) are used for clarity.
atomic
¬A
A B
A B
A BA B
Jouko Väänänen: Predicate logic
First order formulasare of the form
where A and B are first order formulas.
Parentheses (,) are used for clarity.
atomic
¬A
A B
A B
A BA B
xA
Jouko Väänänen: Predicate logic
First order formulasare of the form
where A and B are first order formulas.
Parentheses (,) are used for clarity.
atomic
¬A
A B
A B
A BA B
xA
xA
Jouko Väänänen: Predicate logic
Examples
P0(x) P1(x)
¬(x<y y<x)
x(xEy z(xEz ¬zEy))
x(B(x) z(Y(z) z<x))
Jouko Väänänen: Predicate logic
Universal quantifier explained
Jouko Väänänen: Predicate logic
Universal quantifier explained
xA: Every value of x satisfies A.
Jouko Väänänen: Predicate logic
Universal quantifier explained
xA: Every value of x satisfies A.Every tile is red.
Jouko Väänänen: Predicate logic
Universal quantifier explained
xA: Every value of x satisfies A.Every tile is red.Every x satisfies x2 0.
Jouko Väänänen: Predicate logic
Universal quantifier explained
xA: Every value of x satisfies A.Every tile is red.Every x satisfies x2 0.All vertices x and y are neighbors.
Jouko Väänänen: Predicate logic
Universal quantifier explained
xA: Every value of x satisfies A.Every tile is red.Every x satisfies x2 0.All vertices x and y are neighbors.All men are mortal.
Jouko Väänänen: Predicate logic
Universal quantifier explained
xA: Every value of x satisfies A.Every tile is red.Every x satisfies x2 0.All vertices x and y are neighbors.All men are mortal.Everybody loves her.
Jouko Väänänen: Predicate logic
Existential quantifier explained
Jouko Väänänen: Predicate logic
Existential quantifier explained
xA: Some value of x satisfies A.
Jouko Väänänen: Predicate logic
Existential quantifier explained
xA: Some value of x satisfies A.Some tiles are red.
Jouko Väänänen: Predicate logic
Existential quantifier explained
xA: Some value of x satisfies A.Some tiles are red.Some reals x satisfy x2 =2.
Jouko Väänänen: Predicate logic
Existential quantifier explained
xA: Some value of x satisfies A.Some tiles are red.Some reals x satisfy x2 =2.Some vertices x and y are neighbors.
Jouko Väänänen: Predicate logic
Existential quantifier explained
xA: Some value of x satisfies A.Some tiles are red.Some reals x satisfy x2 =2.Some vertices x and y are neighbors.There is a yellow tile.
Jouko Väänänen: Predicate logic
Existential quantifier explained
xA: Some value of x satisfies A.Some tiles are red.Some reals x satisfy x2 =2.Some vertices x and y are neighbors.There is a yellow tile.There is a vertex with two neighbors.
Jouko Väänänen: Predicate logic
Assignments and quantifiers
In order to define when an assignment satisfies a quantified formula, we need the concept of a modified assignment.
x y z
S 1 5 1
S(2/x) 2 5 1
S(8/z) 1 5 8
This row is a modified assignment
This row is another modified
assignment
Jouko Väänänen: Predicate logic
Modified assignments
Assignment s(a/x) is like assignment s except that the value of x is changed to a.
x y z
S 1 5 1
S(2/x) 2 5 1
S(8/z) 1 5 8
Jouko Väänänen: Predicate logic
Assignment satisfying a quantified formula
Jouko Väänänen: Predicate logic
Assignment satisfying a quantified formula
Assignment s satisfies xA in M if the modified assignment s(a/x) satisfies A in M for every a in M.
Jouko Väänänen: Predicate logic
Assignment satisfying a quantified formula
Assignment s satisfies xA in M if the modified assignment s(a/x) satisfies A in M for every a in M.Assignment s satisfies xA in M if the modified assignment s(a/x) satisfies A in M for some a in M.
Jouko Väänänen: Predicate logic
Satisfaction
We have defined when an assignment s satisfies a formula A in a structure M.When this is the case, we write M s A.
This is called the Tarski Truth Definition.
Tarski Truth Definition
11
Conjunction
Equivalence
Disjunction
Negation
Implication
Jouko Väänänen: Predicate logic
M s A B
if and only if
M s A and M s
B
M s AvB
if and only if
M s A or M s
B
M s A B
if and only if
M s A or M s
B
M s ¬A
if and only if
M s A
M s A B
if and only if
[M s A and M s
B] or
[M s A and M s
B]
Universal quantifier
M s
xA
if and only if
M s(a/x) A for all
a in M
Existential quantifier
M s
xA
if and only if
M s(a/x) A for
some a in M
Atomic
M s Pn(x)
if and only if s(x) Pn
M
M s R(x,y)
if and only if(s(x),s(y)) RM
AtomicAtomic
M s x=y
if and only if s(x)=s(y)
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