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)