Slides adapted fromMichael P. Frank's course based on the textDiscrete Mathematics & Its Applications

(5th Edition)by Kenneth H. Rosen

CS2013Mathematics for Computing Science

Adam WynerUniversity of Aberdeen

Computing Science

Predicate LogicContinued

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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))– xy (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 bona-

fide (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 free-variable 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. x P(x)2. x P(x)3. y Q(x)4. x P(b)5. x(y R(x,y))6. 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. x P(x) (no free variable)2. x P(x) (no free variables)3. y Q(x) (x is a free variable)4. x P(b) (no free variables)5. x(y R(x,y)) (no free variables)6. x(y R(x,z)) (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. P(x)

2. R(x,y)

3. P(b)

4. x P(x)

5. yQ(x)

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Exercise

1. P(x) P(a)

2. R(x,y) R(a,y)

3. P(b) P(b)

4. x P(x) x P(a)

5. yQ(x) 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) (“no counterexample against P”)

x P(x) x P(x)

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

• 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

• 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. x (Q(x) P(x))2. x (Q(x) P(x)) 3. x (Q(x) P(x))4. x (Q(x) P(x))

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Construct English paraphrases

1. x (Q(x) P(x))2. x (Q(x) P(x))3. x (Q(x) P(x))4. x (Q(x) P(x))

1. Some places are free of charge and occupied2. All places are free of charge and occupied 3. All places that are free of charge are occupied4. 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. x (Q(x) P(x)) (true for place a below)2. x (Q(x) P(x)) (false for places b below)3. x (Q(x) P(x)) (false for place b below)4. 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. x (Q(x) P(x)) 2. x (Q(x) P(x)) 3. x (Q(x) P(x)) 4. 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.