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CS2013
Mathematics for Computing Science
Adam Wyner
University of Aberdeen
Computing Science
Slides adapted from
Michael P. Frank's course based on the text
Discrete Mathematics & Its Applications
(5th Edition)
by Kenneth H. Rosen
Predicate Logic
Continued
Agenda
•
•
•
•
Bound and free variables
Vacuous quantification
Empty domains
Complex expressions
– False antecedents
– Quantifiers with logical connectives
– Nested quantifiers
• Quantifier equivalences
• Defining (or not) other quantifiers
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Bound and Free Variables
• What do we say about the quantifiers and variables in the
following expressions (same point with )?
–
–
–
–
B(x)
x B(y,x)
x(B(x)  A(y))
xy (B(y)  A(x))
• A "relationship" between the variable associated with the
quantifier and variable associated with the predicate.
Cannot vary the variable unless it is bound.
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Bound and Free Variables
• An expression like P(x) is said to have a free
variable x (i.e., x is not “specified”). "She is
happy" does not have a truth value unless we
know whom "she" denotes.
• When we indicate whom "she" is, we can
determine if it is true with respect to a model.
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Bound and Free Variables
• A quantifier (either  or ) operates on an
expression having one or more free variables, and
it binds one or more of those variables to produce
an expression having one or more bound
variables.
• Expression with one free variable: P(x)
• Expression with the x binding the variable x
x P(x)
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Example of Binding
• P(x,y) has 2 free variables, x and y.
• x P(x,y) has 1 free variable and one bound
variable.
• An expression with zero free variables is a bonafide (actual) proposition.
P(jill',bill')
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Formal Definition of Free Variables
• The free-variable occurrences in an atomic
formula are all the variable occurrences in that
atomic formula.
• The free-variable occurrences in  are the freevariable occurrences in .
• The free-variable occurrences in
( connective ) are the free-variable occurrences
in  plus the free-variable occurrences in 
• The free-variable occurrences in x and x are
the free-variable occurrences in  except for any
occurrences of x.
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Examples
• Occurrences of variables that are not free are
bound. Start from atomic formula and work
outwards. Which (if any) variables are free in:
1.
2.
3.
4.
5.
6.
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x P(x)
x P(x)
y Q(x)
x P(b)
x(y R(x,y))
x(y R(x,z))
A. x x P(x)
B. x (P(x))  Q(x)
C. y Q(y)  x Q(x)
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Examples
1.
2.
3.
4.
5.
6.
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x P(x)
x P(x)
y Q(x)
x P(b)
x(y R(x,y))
x(y R(x,z))
(no free variable)
(no free variables)
(x is a free variable)
(no free variables)
(no free variables)
(z is a free variable)
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Exercise
Suppose (x:=a), where (x:=a) is the result of
substituting all free occurrences of the variable x
in  by the constant a.
What is ?
1.
2.
3.
4.
5.
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P(x)
R(x,y)
P(b)
x P(x)
yQ(x)
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Exercise
1.
2.
3.
4.
5.
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P(x)
R(x,y)
P(b)
x P(x)
yQ(x)
P(a)
R(a,y)
P(b)
x P(a)
y Q(a)
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Vacuous Quantification
• Recall definition: Let  be a formula.
Then x is true in D if every expression (x:=a)
is true in D, and false otherwise.
• xP(b) is true in D if every expression of the form
P(b)(x:=a) is true in D, and false otherwise.
• What is the set of all the expression of the form
P(b)(x:=a)?
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xP(b)
• What is the set of all expressions of the form
P(b)(x:=a)?
• That’s the singleton set {P(b)} !
• xP(b) is true in D if P(b) is true, and false
otherwise.
• So, xP(b) means the same as P(b)
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Empty Domains
• Let  be a formula. Then the proposition
x is true in D if every expression (x:=a) is true
in D, and false otherwise.
This is read as follows:
• Let  be a formula. Then the proposition
x is false in D if at least one expression (x:=a)
is false in D, and true otherwise.
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 could have been defined as
• Let  be a formula. Then the proposition
x is true in D if D is nonempty and every
expression (x:=a) is true in D, and false
otherwise.
– Under this definition, x P(x) would have been false
whenever D is empty. Every teddy_bear is happy is
false in a model where there is nothing. Sadness!
• But that’s not how it's done!
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Suppose D is Empty
Suppose D is empty.
x P(x) (e.g., P(x) means “x is occupied.”) is true
(sometimes called “vacuously true”).
For the same reason, x P(x) is also true.
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Consequences of the Standard
Position
Two logical equivalences in Predicate Logic:
x P(x)  x P(x)
x P(x)  x P(x)
(“no counterexample against P”)
So, one of the two quantifiers suffices (cf.,
functional completeness of a set of connectives in
propositional logic)
We’ll return to these equivalences later.
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False Antecedent
• Suppose M2: where D = {jill, bill, phil, will, mary},
is_happy' denotes {jill, bill, phil}, is_rich' denotes {}.
• x (is_rich'(x)  is_happy'(x))
• Is this formula T or F? Recall your T-tables for .
• It is clear that no constant for x will make is_rich'(x) true
since the denotation of is_rich'(x) is empty. In other
words, yQ(y).
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False Antecedent
•
•
•
•
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Then Q(a)  P(a) is true for every a
(since Q(a) is false for every a)
Consequently x (Q(x)  P(x)) is true because
Q(x) is false for every a.
A proposition  with a false antecedent is true!
We sometimes say the formula is vacuously true.
Yet, because the antecedent is always false, you
can never use the formula to conclude that P
holds of something.
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Vacuous truth
•
•
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Example 1: Think of a tax form: “Have you sent us
details about all your children?” You have no children,
so you’ve complied (without doing anything).
Example 2: Think of our definition of (x:=a) as “the
result of substituting all free occurrences of x in  by a”
No occurrences, so don't do anything (after which it’s
true that all occurrences have been substituted)
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Quantifiers with Connectives
Let the D be parking spaces at ABDN.
Let P(x) be "x is occupied."
Let Q(x) be "x is free of charge."
What do the following mean/paraphrase?
When are they T/F (construct models)?
1.
2.
3.
4.
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x (Q(x)  P(x))
x (Q(x)  P(x))
x (Q(x) P(x))
x (Q(x)  P(x))
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Construct English paraphrases
1.
2.
3.
4.
1.
2.
3.
4.
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x (Q(x)  P(x))
x (Q(x)  P(x))
x (Q(x)  P(x))
x (Q(x)  P(x))
Some places are free of charge and occupied
All places are free of charge and occupied
All places that are free of charge are occupied
For some places x, if x is free of charge then x is occupied
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Construct a Model where 1 and 4 are T,
while 2 and 3 are F
1.
2.
3.
4.
x (Q(x)  P(x)) (true for place a below)
x (Q(x)  P(x)) (false for places b below)
x (Q(x) P(x)) (false for place b below)
x (Q(x)  P(x)) (true for place a below)
M4: a model where D = {a, b}, I(Q) = {a, b}, I(P) = {a}.
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Construct a Model where 1 and 3 and 4
are T, but 2 is F
1.
2.
3.
4.
x (Q(x)  P(x))
x (Q(x)  P(x))
x (Q(x) P(x))
x (Q(x)  P(x))
M4: a model where D = {a, b}, I(Q) = {a}, I(P) = {a, b}.
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About x (Q(x)  P(x))
x (Q(x)  P(x))
For some x, if x is free of charge then x is occupied
x (Q(x)  P(x)) is true iff, for some place a,
Q(a)  P(a) is true.
Q(a)  P(a) is true iff Q(a) is false or P(a) is true.
Some place is either (not free of charge) or some
place (is occupied).
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Further
Remainder of Predicate Logic topics next week.
Then Proof.
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