Copyright © 2007 - 2015 Curt Hill Quantifiers. Copyright © 2007 - 2015 Curt Hill Introduction What we have seen is called propositional logic It includes.

Copyright © 2007 - 2015 Curt Hill

Quantifiers


Introduction

• What we have seen is called propositional logic

• It includes axioms and theorems concerning the operators of Boolean Algebra

• It lacks something• We want to strengthen this

into first order logic– AKA predicate calculus


Statement Categories• In mathematics, there are

three types of statements:– True– False– Open

• True statements are usually characterized by:– Statements with only constants

and a comparison such as = or > – A statement of fact – A tautology


True Statements

• 5=2+3• 5>2• The United States won its

independence from England• They may include variables in

limited ways:– x+5 = 5+x pq pq q


False statement

• False statements have a similar form but are false

• 5=4• 2>5• The United States won its

independence from Germany• They may also include variables:

– x (y z) x y x z– x = x + 1


Open statements• Open standards generally

cannot have a true or false value until a variable is given a value

• Almost all equations are open, the task of solving is finding a set of values that makes the statement true or determining that the set is empty

• 5x = 10 is neither true nor false• It is true if x = 2 and false

otherwise


Discussion

• Logic is two valued, we cannot allow true, false, and maybe

• Therefore, we have to find a way to make a statement that is open into one that is true or false

• One means of doing this is quantifiers

Predicates• Up to this point we have mostly dealt

with variables and operators• We may also use predicates• Predicate is a fancy name for a function

that returns a boolean value• It may have one or more arguments• The argument does not have to be a

boolean variable• This is not startling since any boolean

variable could hold the result of a predicate


An Example

• Let P(x) be the statement x>8• P(x) is still open• We may determine the truth value

of:– P(12)– P(2)

• If the predicate is arbitrary single letter names are sufficient

• We may have others as wellCopyright © 2007 - 2015 Curt Hill

Predicate Notation

• As always there are multiple ways to represent the same thing

• I prefer:predicate(parameter1, parameter2)– This matches programming notation

• Others use single letter predicates with subscript indicating parameter:Gx

• Thus P(x,y) or Px,y


Open Statements

• In a real sense we have skirted the Open statement problem

• P(x) is just as open as x>0• Now we get back to the topic• We next need quantifiers to say

something useful about the possible values



Quantifiers via familiar example

• You are probably familiar with summation notation– It gives us a compact way to

express an infinite sum or even a large finite sum

• This gives us a convenient way to denote the sum of a finite or infinite number of terms

• We also have a similar notation of products


Summation and Product

0

)12()1(4n

n n

n

i

in1

!


Generalization• This can be extended into

quantifiers with the following notation: (OP i:r:t)– This is not the only notation for

quantifiers

• OP is an operator which is symmetric, associative, and binary

•How many do we have?

• i is a list of dummies– Often just one but sometimes more

• r is the range• t is the term


Range• The range is a Boolean

expression that states the values that the dummies may assume– This corresponds to the summation

n

• A range of True indicates that the dummies may take on any or all values

• A range of False is an empty range


Term• The term corresponds to the

expression to the right of the summation

• It is evaluated for each acceptable dummy value

• These results are connected by the operator that starts the quantifier

• The term has the same type as the operator– This may be a Boolean or

arithmetic operator


Summation Example

• The summation example above would then become: = 4(+i : i i≥0 : (-1)i (2i+1))– + is the binary, symmetric,

associative operator– i is the dummy is the set of integers

• An alternative is: = 4(Σi : i i≥0 : (-1)i (2i+1))


Product Example

• The product example would then be:n! = (i : i 1 ≤ i ≤ n : i)– i is a dummy– n is a bound variable from

outside the quantifier

• An alternative is: n! = (Πi : i 1 ≤ i ≤ n : i)


Others• We can apply this notation to any

binary, symmetric, and associative operation such as– AND– OR– Equivalence– MAX– MIN

• MAX and MIN are the rare arithmetic functions that are idempotent

• They also have a unit and zero


Alternative notations• Two of these have their own

special symbols• And

– Pronounced For All – The universal quantifier

• Or – Pronounced There Exists – The existential quantifier

Open Questions Again• We now have a way to reduce an

open question into a true false proposition

• The open question:3x + 5 = 5x - 11

• Becomes( x : x : 3x + 5 = 5x - 11)

• This statement may be true only if a solution exists or false otherwise– No longer open



Predicates• Up to this point we have mostly dealt

with variables and operators• We may also use predicates• Predicate is a fancy name for a

function that returns a boolean value• It may have one or more arguments• The argument does not have to be a

boolean variable• This is not startling since any

boolean variable could hold the result of a predicate


Predicate Examples

• prime(n)– The argument n is prime or not

• student(p)– A person p is a student or not

• odd(n) or even(n)• Predicate names generally are

single word• This word should indicate the

purpose

Predicate Notation

• As always there are multiple ways to represent the same thing

• I prefer:predicate(parameter)

• Others use single letter predicates with subscript indicating parameter:Gx

• Thus P(x) or Px



More on quantifiers

• Quantifiers allow us to state things that are not directly expressible with just predicates, variables and the operators

• In practice this should allow us to state almost any real world fact or supposition

• A predicate may be defined in terms of a quantifier


Variables• There are two types of variables in

a quantified expression: bound and free

• A bound variable is part of the quantifier– It ranges over some set of values as

part of quantification– These are often known as dummies

• A free variable exists outside the quantified expression


Variable Example• Consider:

n! = (Πi : i 1 ≤ i ≤ n : i)• The variable i is a bound variable• It may only have a certain set of

values determined by the quantifier

• The variable n is free– It may take on any value– Although in this expression we

intend it only to be a positive integer

– This indicates our expression is not as good as we would like


Variable Scope

• The scope of a variable is that part of the expression where it is known

• In any quantified expression the bound variables have a scope contained by the parentheses

• This occasionally gets messy when a quantified expression contains another quantified expression


Scope Expression

• ( i:i : pred1(i)(k:k : pred2(i,k)))

• There are two bound variables• i is known in entire range• k is known only in second quantifier• Both may be used in the second

quantifier• This may cause a problem if they

use the same names


Resulting Types

• A quantifier produces the same type of result as its operator

• Boolean operators produce booleans

• Numeric operators produce numbers

• We may nest quantified expressions in other quantified expressions


Alternate Notation• Most textbooks use a looser notation• This usually omits the range or

incorporates it in the term• The form is something like this:

p (pp)– This is a statement of reflexivity of

equivalence– Somewhat more parallel to summation

• There is no specific mention that p must be boolean– This is just understood

• Quantifiers with a numeric dummy usually assume integer or real and hope it is obvious

Naming• Logicians seldom have to consider

large real world problems• Thus they tend to use single letter

predicates to keep the notation easy• This is an area where we want to

describe these, so many variables or predicates may exist

• More descriptive names are then needed

• This also explains using parenthesis for arguments



A Formula• A formula may be defined

inductively• A formula is:

– Any proposition or predicate– If F is a formula then F is a formula– If F and G are formulas then F G

is a formula•Also any of our other connectives

– If F is a formula then (x: … :F) and ( x: … :F) are also formulas


Another Definition: Sentence• A sentence is a formula with no

free variables• Like a proposition, a sentence does

not have to be true, but it will be true or false– It may be our job to prove it true or

false

• Recall Gödel’s statement that there exists a sentence which is both true and unprovable under any system– That uses this definition of sentence


Example Sentences• For each of the following:

– Is it true?– What does it mean?

• 3<6• 2+4<5• (x:x: (y:yR:x<y))• (x:x: (y:yR:x<y))• (i:i : (j:j :i=j2))• (i:i : ( j:j :j=i2))


Some quantifier examples

• Some apples are rotten– (a:a Apples: rotten(a))

• All women are beautiful– (w : w Females: beautiful(w))

• Some integers are positive• All squares are rectangles• Not all rectangles are squares• Prime numbers exist in the

integers


Prime Definition• Can we define a prime number?• prime(n) =

(i : i i > 1 i ≠ n : 0 ≠ n-floor(ni)*i )

• floor truncates a real number to an integer

• i is a dummy variable that can range over all integers greater than 1 and not equal to n


Prime Definition Again• This time with nested quantifiers• prime(n) =

(i : i 1 < i < n : (j:jI: j=ni ))

• Negation of an existential gives a universal so:

• prime(n) =(i : i 1 < i < n : (j:jI: jni ))


Summary

• What was open before (that is not possible to assign to truth value to) is now closed– We may not always know the truth

value– We do know that one exists

• We should now be able to state in a concise form any proposition

• Another presentation gives some rules for manipulating quantifiers

Copyright © 2007 - 2015 Curt Hill Quantifiers. Copyright © 2007 - 2015 Curt Hill Introduction What we have seen is called propositional logic It includes.

Documents

curt hillcopyright

curt hillintroductionwhat

curt hilldiscussionlogic

curt hillquantifierscopyright

curt hillpredicatesup

curt hillopen statementsin

curt hillpredicate notationas

curt hilltrue statements5