Copyright © Curt Hill Quantifiers

Copyright © Curt Hill 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

Copyright © Curt Hill 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

Copyright © Curt Hill 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 –  p  q  p  q  q

Copyright © Curt Hill 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

Copyright © Curt Hill 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

Copyright © Curt Hill 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 Copyright © Curt Hill

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 well Copyright © 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: G x Thus P(x,y) or P x,y Copyright © Curt Hill

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 Copyright © Curt Hill

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

Copyright © Curt Hill Summation and Product

Copyright © Curt Hill 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

Copyright © Curt Hill 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

Copyright © Curt Hill 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

Copyright © Curt Hill 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))

Copyright © Curt Hill 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)

Copyright © Curt Hill 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

Copyright © Curt Hill 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 Copyright © Curt Hill

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

Copyright © Curt Hill 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: G x Thus P(x) or P x Copyright © Curt Hill

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

Copyright © Curt Hill 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

Copyright © Curt Hill 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

Copyright © Curt Hill 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

Copyright © Curt Hill 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

Copyright © Curt Hill 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

Copyright © Curt Hill 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 (p  p) –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 Copyright © Curt Hill

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

Copyright © Curt Hill 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

Copyright © Curt Hill Example Sentences For each of the following: –Is it true? –What does it mean? 3<6 2+4<5 (  x:x   : (  y:y  R:x<y)) (  x:x   : (  y:y  R:x<y)) (  i:i   : (  j :j   :i=j 2 )) (  i:i   : (  j :j   :j=i 2 ))

Copyright © Curt Hill 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

Copyright © Curt Hill Prime Definition Can we define a prime number? prime(n) = (  i : i    i > 1  i ≠ n : 0 ≠ n-floor(n  i)*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

Copyright © Curt Hill Prime Definition Again This time with nested quantifiers prime(n) = (  i : i    1 < i < n :  (  j:j  I: j=n  i )) Negation of an existential gives a universal so: prime(n) = (  i : i    1 < i < n : (  j:j  I: j  n  i ))

Copyright © Curt Hill 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