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Title: University of Aberdeen, Computing Science CS3511 Discrete Methods Kees van Deemter


1
University of Aberdeen, Computing
ScienceCS3511Discrete MethodsKees van Deemter
  • Slides adapted from Michael P. Franks Course
    Based on the TextDiscrete Mathematics Its
    Applications (5th Edition)by Kenneth H. Rosen

2
Module 1Part 2 Predicate Logic
  • Rosen 5th ed., 1.3-1.4 (but much extended)
  • 135 slides, 5 lectures

3
Predicate Logic (1.3)
Topic 3 Predicate Logic
  • We can use propositional logic to prove that
    certain real-life inferences are valid.
  • If its cold then it snows.
  • If it snows there are accidents
  • There are no accidents. Therefore
  • Its not cold
  • In propositional logic
  • ((c?s ?s?a ??a) ??c) is a tautology

4
Predicate Logic (1.3)
Topic 3 Predicate Logic
  • In propositional logic
  • (((c?s) ?(s?a) ??a) ??c) is a tautology
  • Saying this differently,It follows by
    propositional logic from(c?s) ?(s?a) ??a
    (premisse) that ?c (conclusion)

5
Predicate Logic (1.3)
Topic 3 Predicate Logic
  • But other valid inferences cannot be proven valid
    by propositional logic
  • Some girl is adored by everyone. Therefore
  • Everyone adores someone
  • For inferences like this, we need a more
    expressive logic
  • Needed treatment of some and every(a bit
    analogous to or and and)

6
Predicate Logic (1.3)
Topic 3 Predicate Logic
  • Predicate logic is an extension of propositional
    logic that permits quantification over classes of
    entities.
  • Propositional logic (recall) treats simple
    propositions (sentences) as atomic entities.
  • In contrast, predicate logic distinguishes the
    subject of a sentence from its predicate.

7
Applications of Predicate Logic
Topic 3 Predicate Logic
  • It is one of the most-used formal notations for
    writing mathematical definitions, axioms, and
    theorems.
  • For example, in linear algebra, a partial order
    is introduced saying that a relation R is
    reflexive and transitive and these notions are
    defined using predicate logic.

8
Practical Applications of Predicate Logic
Topic 3 Predicate Logic
  • Basis for many Artificial Intelligence systems.
  • E.g. automatic program verification systems.
  • Predicate-logic like statements are supported by
    some of the more sophisticated database query
    engines
  • There are also limitations associated with using
    predicate logic. More about that later

9
First A bit of grammar
Topic 3 Predicate Logic
  • In the sentence The dog is sleeping
  • The phrase the dog denotes the subject - which
    the sentence is about.
  • The phrase is sleeping denotes the predicate- a
    property that is true of the subject.
  • Predicate logic will follow the same pattern.

10
Formulas of predicate logic (informal)
Topic 3 Predicate Logic
  • We will use various kinds of individual constants
    that denote individuals/objects a,b,c,
  • Constants are a bit like names
  • Individual variables over objects x, y, z ,
  • The result of applying a predicate P to
    aconstant a is the proposition P(a)Meaning the
    object denoted by a has the property denoted by P.

11
Formulas of predicate logic (informal)
Topic 3 Predicate Logic
  • The result of applying a predicate P to
    avariable x is the propositional form P(x).
  • E.g. if P is a prime number, then P(x) is
    the propositional form x is a prime number.

12
Predicates/relations with n places
Topic 3 Predicate Logic
  • Predicate logic generalises the notion of a
    predicate to include propositional functions of
    any number of arguments. E.g.
  • R(x,y) x adores y
  • P(x,y,z) x gave y the grade z
  • Q(x,y,z,u) x(yz)u

13
Universes of Discourse (U.D.s)
Topic 3 Predicate Logic
  • Predicate Logic lets you state things about many
    objects at once.
  • E.g., let P(x) (x2) ? x . We can then
    say,For any number x, P(x) is true instead
    of(02 ? 0) ? (12 ? 1) ? (22 ? 2) ? ...
  • The collection of values that a variable x can
    take is called xs universe of discourse.(u.d.
    is something outside the formula that helps
    giving it its intended interpretation.)

14
Universes of Discourse (U.D.s)
Topic 3 Predicate Logic
  • E.g., let P(x)x2 ? x. We can then say,For
    any number x, P(x) is true instead of(02 ? 0)
    ? (12 ? 1) ? (22 ? 2) ? ...
  • Applying the notion of an u.d.
  • For any number x, P(x) is true is true when
    u.d. N
  • For any number x, P(x) is true is false when
    u.d. Z

15
Back to propositional logic
  • In propositional logic, we could not simply say
    whether a formula is TRUE what we could say is
    whether it is TRUE with respect to a given
    assignment of TRUE/FALSE to the Atoms in the
    formula
  • E.g., p?q is TRUE with respect to the assignment
    pTRUE, qTRUE

16
Predicate logic
  • In predicate logic, we say that a formula is TRUE
    (FALSE) with respect to a model
  • Model u.d. plus specification of the meanings
    of the predicates. This can be done e.g.
  • by giving an English equivalent of a predicate
  • by listing explicitly which objects the
    predicate is true of
  • ... as long the extension of the predicate is
    clear.(By definition, this is the set of objects
    in the u.d. for which the predicate holds.)

17
Quantifier Expressions
Topic 3 Predicate Logic
  • Quantifiers provide a notation that allows us to
    quantify (count) how many objects in the u.d.
    satisfy a given predicate.
  • ? is the FOR?LL or universal quantifier. ?
    is the ?XISTS or existential quantifier.
  • For example, ?x P(x) and ?x P(x) are propositions

18
Meaning of Quantified Expressions
Topic 3 Predicate Logic
  • First, informally
  • ?x P(x) means for all x in the u.d., P holds.
  • ?x P(x) means there exist x in the u.d. (that is,
    1 or more) such that P(x) is true.

19
Example ?
Topic 3 Predicate Logic
  • Let the u.d. be the parking spaces at UF.Let
    P(x) mean x is full.Then the existential
    quantification of P(x), ?x P(x), is the
    proposition saying that
  • Some parking spaces at UF are full.
  • There is a parking space at UF that is full.
  • At least one parking space at UF is full.

20
Example ?
Topic 3 Predicate Logic
  • Let the u.d. be parking spaces at UF.Let P(x)
    be the prop. form x is occupiedThen the
    universal quantification of P(x), ?x P(x), is the
    proposition
  • All parking spaces at UF are occupied.
  • For each parking space at UF, that space is
    full.

21
Syntax of predicate logic (for 1- and 2-place
predicates)
  • Variable x,y,z, Constants a,b,c,
  • 1-place predicates P,Q,
  • 2-place predicates R,S,
  • Atomic formulas If ? is a 1-pace predicate and
    ? a variable or constant then ?(?) is an atomic
    formula.
  • If ? is a 2-pace predicate and ? and ? are
    variables or constants then ?(?,?) is an atomic
    formula.

22
Syntax of predicate logic (for 1- and 2-place
predicates)
  • (Wellformed) Formulas
  • All atomic formulas are formulas
  • If ? and ? are formulas then ??, (? ??), (???),
    (? ??) are formulas.
  • If ? is a formula then ?x ? and ?x ? are
    formulas. (Likewise, ?y ? and ?y ?,and so on.)
  • This could be seen as overgenerating
  • wellformed formulas somewhat

23
Syntax of predicate logic (for 1- and 2-place
predicates)
  • Show that these are wellformed formulas
  • ?xP(x) ?yQ(x) ?x?y R(x,y) ?xP(b)

24
Syntax of predicate logic (for 1- and 2-place
predicates)
  • ?xP(x) ?yQ(x) ?x?y R(x,y) ?xP(b)
  • P(x) is a (atomic) formula, hence ?xP(x) is a
    formula
  • Q(x) is a (atomic) formula, hence ?yQ(x) is a
    formula
  • R(x,y) is a (atomic) formula, hence ?y R(x,y) is
    a formula, hence ?x?y R(x,y) is a formula
  • P(b) is a (atomic) formula, hence ?xP(b) is a
    formula

25
Syntax of predicate logic (for 1- and 2-place
predicates)
  • Examples ?xP(x) and ?yQ(x), ?x(?y R(x,y)),
    ?x(?x R(x,y)), ?xP(b) etc.
  • Lots of pathological cases. For example,
  • It will follow from the meaning of these formulas
    that ?xP(b) is true iff P(b) is true
  • Rule of thumb a quantifier that does not bind
    any variables can be ignored

26
Free and Bound Variables
Topic 3 Predicate Logic
  • An expression like P(x) is said to have a free
    variable x (i.e., x is not defined).
  • A quantifier (either ? or ?) operates on an
    expression having one or more free variables, and
    binds one or more of those variables, to produce
    an expression having one or more bound variables.

27
Example of Binding
Topic 3 Predicate Logic
  • P(x,y) has 2 free variables, x and y.
  • ?x P(x,y) has 1 free variable, and one bound
    variable. Which is which?
  • An expression with zero free variables is a
    bona-fide (actual) proposition.
  • An expression with one or more free variables is
    similar to a predicate e.g. let Q(y) ?x
    Adore(x,y)

y
x
28
Free variables, defined formally
Topic 3 Predicate Logic
  • The free-variable occurrences in an Atom are all
    the variable occurrences in that Atom
  • 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 all/any occurrences of x.

29
  • Occurrences of variables that are not free are
    bound.
  • Test your understanding Which (if any) variables
    are free in
  • ?x ?P(x)
  • ??x ?P(x)
  • ?yQ(x)
  • ?xP(b) (NB, b is a constant)
  • ?x(?y R(x,y))

30
  • Occurrences of variables that are not free are
    bound.
  • Check your understanding Which (if any)
    variables are free in
  • ?x ?P(x) no free variables
  • ??x ?P(x) no free variables
  • ?yQ(x) x is free
  • ?xP(b) (NB, b is a constant) no free var.
  • ?x(?y R(x,y)) no free variables

31
A more precise definition of the truth/falsity of
quantified formulas
Topic 3 Predicate Logic
  • (Formulation is simplified somewhat because we
    assume that every object in the u.d. D has a
    name (i.e., a constant referring to it)).
  • First some notation ?(xa) is the result of
    substituting all free occurrences of the variable
    x in ? by the constant a

32
Exercise
Topic 3 Predicate Logic
  • Say what ?(xa) is, if ?
  • P(x)
  • R(x,y)
  • P(b)
  • ?x ?P(x)
  • ?yQ(x)

33
Exercise
Topic 3 Predicate Logic
  • Say what ?(xa) is, if ?
  • P(x) ..... P(a)
  • R(x,y) .... R(a,y)
  • P(b) ..... P(b)
  • ?x ?P(x) ...... ?x ?P(x)
  • ?yQ(x) ...... ?yQ(a)

34
A more precise definition of the truth/falsity of
quantified formulas
Topic 3 Predicate Logic
  • (Formulation is simplified somewhat because we
    assume that every object in the u.d. has a
    name (i.e., a constant referring to it)).
  • Let ? be a formula. Then ?x? is true in D if at
    least one expression of the form ?(xa) is true
    in D, and false otherwise. (a can be any
    constant)

35
A more precise definition
Topic 3 Predicate Logic
  • Let ? be a formula. Then ?x? is true in D if at
    least one expression ?(xa) is true in D, and
    false otherwise.
  • A simple example ? P(x)
  • P(x) is a formula, hence ?x P(x) is true in D if
    at least one expression of the form P(a) is true
    in D, and false otherwise.

36
Similarly for ?
Topic 3 Predicate Logic
  • Let ? be a formula. Then the proposition ?x? is
    true in D if every expression of the form ?(xa)
    is true in D, and false otherwise.
  • A simple example ? P(x)
  • P(x) is a formula, hence ?x P(x) is true in D
    if every expression P(a) is true in D, and false
    otherwise.

37
Complex formulas
Topic 3 Predicate Logic
  • Example Let the u.d. of x and y be people.
  • Let L(x,y)x likes y (a predicate w. 2 f.v.s)
  • Then ?y L(x,y) There is someone whom x likes.
    (A predicate w. 1 free variable, x)
  • Then ?x (?y L(x,y)) Everyone has someone whom
    they like.
  • (a real proposition no free variables left)

38
Consequences of Binding (work out for yourself by
checking when each formula is true)
Topic 3 Predicate Logic
  • ?x ?x P(x) - x is not a free variable in ?x
    P(x), therefore the ?x binding isnt used,as it
    were.
  • (?x P(x)) ? Q(x) - The variable x is outside of
    the scope of the ?x quantifier, and is therefore
    free. Not a complete proposition!
  • (?x P(x)) ? (?x Q(x)) A complete proposition,
    and no superfluous quantifiers

39
Nested quantifiers
  • Assume S(x,y) means x sees y
  • u.d.all people
  • What does the following formula mean?
  • ?xS(x,a)

40
Nested quantifiers
  • Assume S(x,y) means x sees y.
  • u.d.all people
  • ?xS(x,a) means For every x, x sees a
  • In other words,Everyone sees a

41
Nested quantifiers
  • What does the following formula mean?
  • ?x(?y S(x,y))

42
Nested quantifiers
  • ?x(?y S(x,y)) means For every x, there exists a
    y such that x sees y
  • In other words Everyone sees someone

43
Quantifier Exercise
Topic 3 Predicate Logic
  • If R(x,y)x relies upon y, express the
    following in unambiguous English
  • ?x(?y R(x,y))
  • ?y(?x R(x,y))
  • ?x(?y R(x,y))
  • ?y(?x R(x,y))
  • ?x(?y R(x,y))

Everyone has someone to rely on.
Theres a poor overburdened soul whom everyone
relies upon (including himself)!
Theres some needy person who relies upon
everybody (including himself).
Everyone has someone who relies upon them.
Everyone relies upon everybody, (including
themselves)!
44
Quantifier Exercise
Topic 3 Predicate Logic
  • R(x,y)x relies upon y. Suppose the u.d. is
  • not empty. Now consider these formulas
  • ?x(?y R(x,y))
  • ?y(?x R(x,y))
  • ?x(?y R(x,y))
  • Which of them is most informative?
  • Which of them is least informative?

45
Quantifier Exercise
Topic 3 Predicate Logic
  • (Recall the u.d. is not empty. Empty u.d.s are
    discussed later.)
  • ?x(?y R(x,y)) Least informative
  • ?y(?x R(x,y))
  • ?x(?y R(x,y)) Most informative
  • If 3 is true then 2 must also be true.
  • If 2 is true then 1 must also be true.
  • We say 3 is logically stronger than 2 than 1

46
logically stronger than
  • General ? is logically stronger than ? iff
  • it is not possible for ? to be true and ? false
  • it is possible for ? to be true and ? false
  • E.g. ? John is older than 30, ? John
    is older than 20.
  • We write iff for if and only if

47
Natural language is ambiguous!
Topic 3 Predicate Logic
  • Everybody likes somebody.
  • For everybody, there is somebody they like,
  • ?x ?y Likes(x,y)
  • or, there is somebody (a popular person) whom
    everyone likes?
  • ?y ?x Likes(x,y)

Probably more likely.
48
Interactions between quantifiers and connectives
  • Let the u.d. be parking spaces at UF.
  • Let P(x) be x is occupied.
  • Let Q(x) be x is free of charge.
  • ?x (Q(x) ? P(x))
  • ?x (Q(x) ? P(x))
  • ?x (Q(x) ?P(x))
  • ?x (Q(x) ? P(x))

49
I. Construct English paraphrases
  • Let the u.d. be parking spaces at UF.
  • Let P(x) be x is occupied.
  • Let Q(x) be x is free of charge.
  • ?x (Q(x) ? P(x))
  • ?x (Q(x) ? P(x))
  • ?x (Q(x) ?P(x))
  • ?x (Q(x) ? P(x))

50
I. Construct English paraphrases
  1. ?x (Q(x) ? P(x)) Some places are free of charge
    and occupied
  2. ?x (Q(x) ? P(x)) All places are free of charge
    and occupied
  3. ?x (Q(x) ?P(x)) All places that are free of
    charge are occupied
  4. ?x (Q(x) ? P(x)) For some places x, if x is free
    of charge then x is occupied

51
About the last of these
  • 4. ?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
    and/or P(a) is true (conditional is only
    false in one row of table!) Some places are
    either (not free of charge) and/or occupied

52
About the last of these
  1. ?x (Q(x) ? P(x)) When confused by a conditional
    re-write it using negation and disjunction ?x
    (?Q(x) ? P(x)) ?? (p. 67) ?x ?Q(x) ? ?x P(x)
    Some places are not free of charge or some
    places are occupied

53
  • Combinations to remember
  • ?x (Q(x) ? P(x))
  • ?x (Q(x) ?P(x))

54
II. Construct a model where 1 and 4 are true,
while 2 and 3 are false
  • Let the u.d. be parking spaces at UF.
  • Let P(x) be x is occupied.
  • Let Q(x) be x is free of charge.
  • ?x (Q(x) ? P(x))
  • ?x (Q(x) ? P(x))
  • ?x (Q(x) ?P(x))
  • ?x (Q(x) ? P(x))

55
II. Construct a model where 1 and 4 are true,
while 2 and 3 are false
  • ?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)
  • One solution a model with exactly two objects in
    it. One object has the property Q and the
    property P the other object has the property Q
    but not the property P. In a diagram
  • a Q P b Q not-P

56
III. Construct a model where 1 and 3 and 4 are
true, but 2 is false
  • ?x (Q(x) ? P(x))
  • ?x (Q(x) ? P(x))
  • ?x (Q(x) ?P(x))
  • ?x (Q(x) ? P(x))
  • Here is such a model (using a diagram). It has
    just two objects in its u.d., called a and b
  • a Q P b not-Q P

57
Quantifier Equivalence Laws
Topic 3 Predicate Logic
  • Expanding quantifiers If u.d.a,b,c, ?x P(x) ?
    P(a) ? P(b) ? P(c) ? ?x P(x) ? P(a) ? P(b) ?
    P(c) ?
  • From those, we can prove the laws?x P(x) ?
    ??x ?P(x)?x P(x) ? ??x ?P(x)
  • Which propositional equivalence laws can be used
    to prove this?

DeMorgan's
58
Remember
  • In propositional logic, we can strictly speaking
    only build formulas of finite size.
  • E.g., we can write P(a) ? P(b) P(a) ? P(b) ?
    P(c) P(a) ? P(b) ? P(c) ? P(d) , etc.
  • But this way, we could never say that all natural
    numbers have P

59
Formulas of infinite length?
  • In predicate logic, you can say this easily
    ?xP(x)
  • Its sometimes useful to pretend that
    propositional logic allows infinitely long
    formulas, but in the official version this is
    not possible.

60
More Equivalence Laws
Topic 3 Predicate Logic
  • ?x ?y P(x,y) ? ?y ?x P(x,y)?x ?y P(x,y) ? ?y ?x
    P(x,y)
  • ?x (P(x) ? Q(x)) ? (?x P(x)) ? (?x Q(x))?x (P(x)
    ? Q(x)) ? (?x P(x)) ? (?x Q(x))

61
More Equivalence Laws
Topic 3 Predicate Logic
  • ?x ?y P(x,y) ? ?y ?x P(x,y)?x ?y P(x,y) ? ?y ?x
    P(x,y)
  • ?x (P(x) ? Q(x)) ? (?x P(x)) ? (?x Q(x))?x (P(x)
    ? Q(x)) ? (?x P(x)) ? (?x Q(x))
  • Lets prove the last of these equivalences using
    the definition of the truth of a formula of the
    form ?x f

62
A Pred. Log. equivalence proven
Topic 3 Predicate Logic
  • ?x (P(x) ? Q(x)) ? (?x P(x)) ? (?x Q(x))Proof
    ? Suppose ?x (P(x) ? Q(x)) is true. So, there
    is a constant a such that (P(x) ? Q(x)) (xa) is
    true. So, for that a, the formula P(a) ? Q(a) is
    true. One possibility is that P(a) is true. In
    this case, P(x)(xa) is true. So, ?x P(x) is
    true, so (?x P(x)) ? (?x Q(x)) is true. The other
    possibility is that Q(a) is true. In this case,
    Q(x)(xa) is true. So, ?x Q(x) is true, so (?x
    P(x)) ? (?x Q(x))is true.

63
A Pred. Log. equivalence proven
Topic 3 Predicate Logic
  • ?x (P(x) ? Q(x)) ? (?x P(x)) ? (?x Q(x))Proof
    ? Suppose (?x P(x)) ? (?x Q(x)) is true. One
    possibility is that ?x P(x) is true. This would
    mean that there is an a such that P(x) (xa) is
    true. So, for that constant a, P(a) is
    true.Therefore, P(a) ? Q(a) would also be true.
    Hence, (P(x) ? Q(x))(xa) would be true. Hence,
    ?x (P(x) ? Q(x)) would be true. The other
    possibility is that ?x Q(x). From this, ?x (P(x)
    ? Q(x)) is proven in the same way QED

64
More Equivalence Laws
Topic 3 Predicate Logic
  • ?x ?y P(x,y) ? ?y ?x P(x,y)?x ?y P(x,y) ? ?y ?x
    P(x,y)
  • ?x (P(x) ? Q(x)) ? (?x P(x)) ? (?x Q(x))?x (P(x)
    ? Q(x)) ? (?x P(x)) ? (?x Q(x))
  • How about this one?
  • ?x (P(x) ? Q(x)) ? (?x P(x)) ? (?x Q(x))?

65
More Equivalence Laws
Topic 3 Predicate Logic
  • How about this one?
  • ?x (P(x) ? Q(x)) ? (?x P(x)) ? (?x Q(x)) ?
  • This equivalence statement is false.
    Counterexample (i.e. model making this false)
  • P(x) xs birthday is on 30 AprilQ(x) xs
    birthday is on 20 December

66
Cases to think carefully about
  • Lets discuss some specific cases
  • These might seem unusual, but its
    important not to loose your way when they occur
  • Quantifiers that dont bind any variables
  • Quantification in an empty u.d.
  • ?x (Q(x) ?P(x)) when ??yQ(y)

67
1. Quantifiers that do not bind ?xP(b)
  • Recall definition Let ? be a formula. Then ?x?
    is true in D if every expression ?(xa) is true
    in D, and false otherwise.
  • ?xP(b) is true in D if every expression of the
    form P(b)(xa) is true in D, and false
    otherwise.
  • What is the set of all the expression of the form
    P(b)(xa)?

68
Consider ?xP(b)
  • What is the set of all the expression of the form
    P(b)(xa)?
  • Thats the singleton set P(b) ! So,
  • ?xP(b) is true in D if P(b) is true, and false
    otherwise.
  • So, ?xP(b) means just P(b)

69
2. Empty u.d. We defined
  • Let ? be a formula. Then the proposition ?x? is
    true in D if every expression ?(xa) 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 ?(xa) is
    false in D, and true otherwise.

70
? could have been defined differently. For
example,
  • Let ? be a formula. Then the proposition ?x? is
    true in D if D is nonempty and every expression
    ?(xa) is true in D, and false otherwise.
  • Under this definition, ?x P(x) would have been
    false whenever D is empty (e.g., when there are
    no parking spaces at U.F., and P occupied)
  • But thats not how its done!

71
Suppose D is empty
  • Suppose the u.d. is empty. Consider
  • ?x P(x) (e.g., P(x) means x is occupied.)
  • This counts as true. (Sometimes called vacuously
    true)
  • For the same reason,
  • ?x ?P(x) is also true

72
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)
  • Well return to these equivalences later.

73
3. ?x (Q(x) ?P(x)) when ??yQ(y)
74
When many implications are combined
  • Consider the formula r ?x (Q(x) ?P(x)) with
    respect to a nonempty u.d.
  • Suppose ??yQ(y)
  • (For example, Q might mean being more than 4
    meters tall)
  • Can you work out whether r is true?

75
When many implications are combined
  • Consider ?x (Q(x) ?P(x)) in a nonempty u.d.
  • Suppose, however, ??yQ(y)
  • 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
  • Once again, we sometimes say it is vacuously
    true (because the antecedent is always false, so
    you can never use the formula to conclude that P
    holds of something).

76
Vacuous truth
  • Example 1 Think of a tax form Have you sent us
    details about all your children? No children ?
    Youve complied (without doing anything)
  • Example 2 Think of our definition of ?(xa) as
    the result of substituting all free occurrences
    of x in ? by aNo occurrences ? dont do
    anything (after which its true that all
    occurrences have been substituted)

77
Some consequences of these definitions
  • Sometimes, predicate logic is taught very
    informally
  • This makes it easy to understand simple formulas
  • But each more complex case has to be explained
    separately, as if it was an exception
  • By defining things properly once, complex
    formulas fall out as special cases
  • One example quantifier nesting

78
Back to Quantifier Exercise
Topic 3 Predicate Logic
  • R(x,y)x relies upon y. Suppose the u.d. is
  • empty. Now consider these formulas
  • ?x(?y R(x,y))
  • ?y(?x R(x,y))
  • ?x(?y R(x,y))
  • Which of them is most informative?
  • Which of them is least informative?

79
Back to our Quantifier Exercise
Topic 3 Predicate Logic
  • Now that the u.d. is empty,
  • ?x(?y R(x,y)) is uninformative
  • ?y(?x R(x,y)) is false!
  • ?x(?y R(x,y)) is uninformative
  • ?x(P(x)) is true if P(a) is true for all a.
  • If no x exist then this makes ?x(P(x))
    (vacuously) true
  • ?x(P(x)) is true if P(a) is true for some a.
  • If no x exist then this makes ?x(P(x)) false.

80
Back to models Look at the old formula 3 again
  • 3. ?x (Q(x) ?P(x)) Consider this model
  • a ?Q P b ?Q?P
  • c?Q P d ?Q?P
  • Is the proposition true or false in the model?

81
Look at formula number 3 again
  • 3. ?x (Q(x) ?P(x)) is true in this model
  • a ?Q P b ?Q?P
  • c?Q P d ?Q?P
  • (3) Is true iff each of these is true ?x (Q(a)
    ?P(a)) ?x (Q(b) ?P(b)) ?x (Q(c) ?P(c)) ?x
    (Q(d) ?P(d))

82
Look at formula number 3 again
  • 3. ?x (Q(x) ?P(x)) is true in this model
  • a ?Q P b ?Q?P
  • c?Q P d ?Q?P
  • (3) Is true because each of these is true Q(a)
    ?P(a) F T, hence T Q(b) ?P(b) F F, hence T
    Q(c) ?P(c) F T, hence T Q(d) ?P(d) F F,
    hence T

83
Predicate Logic essentials
  • Predicates of different arity (one-place,
    two-place, etc.)
  • Quantifiers ?x P(x) For all x, P(x). ?x
    P(x) There is/are x such that P(x).
  • Universes of discourse, bound free vars.
  • The rest follows empty domains, quantified
    implications, quantifier nesting

84
Some common shorthands
Topic 3 Predicate Logic
  • Sometimes the universe of discourse is restricted
    within the quantification, e.g.,
  • ?xgt0 P(x) is shorthand forFor all x that are
    greater than zero, P(x).
  • How would you write this in formal notation?

85
Some common shorthands
Topic 3 Predicate Logic
  • Sometimes the universe of discourse is restricted
    within the quantification, e.g.,
  • ?xgt0 P(x) is shorthand forFor all x that are
    greater than zero, P(x).?x (xgt0 ? P(x))

86
Some common shorthands
Topic 3 Predicate Logic
  • Sometimes the universe of discourse is restricted
    within the quantification, e.g.,
  • ?xgt0 P(x) is shorthand forFor all x that are
    greater than zero, P(x).?x (xgt0 ? P(x))
  • ?xgt0 P(x) is shorthand forThere is an x greater
    than zero such that P(x).?x (xgt0 ? P(x))

87
Some common shorthands
Topic 3 Predicate Logic
  • Consecutive quantifiers of the same type can be
    combined ?xyz P(x,y,z) ?def ?x ?y ?z P(x,y,z)
    ?xyz P(x,y,z) ?def ? x ? y ? z P(x,y,z)

88
Theorems about logic
  • We are studying logical languages/calculi to
    allow you to use them (better)
  • Logicians study logical languages/calculi to
    understand their limitations
  • Meta-theorems can, e.g., say things like
    cannot be expressed in predicate logic

89
Theorems about logic
  • About propositional logic, we asked What types
    of things can we express? How many connectives
    do we need?
  • About predicate logic, logicians ask similar
    questions. For example, are these two quantifiers
    enough to be able to say everything?
  • This is a question about the expressive power of
    predicate logic

90
Example one
  • As per their name, quantifiers can be used to
    express that a predicate is true of a given
    quantity (number) of objects.
  • Example Can predicate logic say there exists at
    most one object with property P?

91
Example at most one
  • Example Can predicate logic say there exists at
    most one object with property P?
  • Yes (provided we have equality)
  • ?x?y ((P(x) ? P(y)) ? x y)

92
Example one
  • Can predicate logic say there exists exactly one
    object with property P?

93
Example one
  • Can predicate logic say there exists exactly one
    object with property P? ?xP(x) ? ?x?y((P(x)
    ?P(y))?x y)
  • There exist x such that P(x) andThere exists
    at most one x such that P(x)
  • Abbreviation ?!x P(x)(there exists exactly one
    x such that P(x))

94
Example one
  • Another way to write this
  • ?x (P(x) ? ??y (P(y) ? y? x))There is an x such
    that P(x), such that there is no y such that P(y)
    and y? x.
  • ?x binds x throughout the conjunction?x (P(x) ?
    ??y (P(y) ? y? x))

95
At least two
  • Can predicate logic say there exist at least two
    objects with property P?

96
At least two
  • Can predicate logic say there exist at least two
    objects with property P?
  • Yes?x ?y ((P(x) ? P(y)) ? x? y)
  • Incorrect would be ?xP(x) ? ?y(P(y) ? x?
    y),(where x occurs free, and whichtherefore
    does not express a proposition)

97
Exactly two
  • Can predicate logic say there exist exactly two
    objects with property P?

98
Exactly wo
  • Can predicate logic say there exist exactly two
    objects with property P?
  • Yes
  • ?x ?y (P(x) ? P(y) ? x? y ? ?z
    (P(z) ? (z x ? z y) ))

99
Whats wrong with
  • ?x ?y (P(x) ? P(y) ? x? y) ? ?z (P(z) ? (z x ?
    z y ))as a formalisation of exactly two?

100
Whats wrong with
  • ?x ?y (P(x) ? P(y) ? x? y) ? ?z (P(z) ? (z x ?
    z y ))as a formalisation of exactly two?
  • This is a conjunction of two separate
    propositions. As a result, x and y are not bound,
    so this is not even a proposition

101
infinitely many
  • Can predicate logic say there exist infinitely
    many objects with property P?
  • No! This follows from the so-called Compactness
    Theorem An infinite set S of formulas has a
    model iff every finite subset of S has a model
  • How about finitely many?

102
finitely many
  • How about finitely many?
  • Suppose there existed a formula ? there exist
    only finitely many x such that so and so
  • Then ?? there exist infinitely many x such
    that so and so
  • We know that such a formula does not exist
  • So, also ? does not exist

103
  • (Of course if we allow infinite conjunctions
    then all this can be expressed?!x P(x) ? ?2!x
    P(x) ? ?3!x P(x) ? )

104
  • Can predicate logic say most objects have
    property P?
  • No! This follows from the Compactness Theorem as
    well. Again, this is unless we allow infinitely
    long disjunctions.
  • Can predicate logic say many objects have
    property P?
  • No, only precisely defined quantities

105
Decidability
  • Weve shown you two ways of checking
    propositional-logic equivalencies
  • Checking truth tables
  • Using equivalence laws
  • Youve seen how (1) can be done algorithmically.
  • This shows that checking propositional logic
    equivalence is decidable

106
Decidability
  • Checking proplog equivalence is decidable
  • Checking predlog equivalence is not
    decidableTherefore, theorem proving will always
    remain an art (for both computers and humans)
  • Some fragments of predlog are decidable. One
    application PROLOG

107
Bonus Topic Logic Programming
  • Some programming languages are based entirely on
    (a part of) predicate logic
  • The most famous one is called Prolog.
  • A Prolog program is a set of propositions
    (facts) and (rules) in predicate logic.
  • The input to the program is a query
    proposition.
  • Want to know if it is true or false.
  • The Prolog interpreter does some automated
    deduction to determine whether the query follows
    from the facts.

108
Facts in Prolog
  • A fact in Prolog represents a simple,
    non-compound proposition in predicate logic.
  • E.g., likes(john,mary)
  • Lowercase symbols are used for constants and
    predicates, uppercase is used for variables.

109
Rules in Prolog
  • A rule in Prolog represents a universally
    quantified proposition of the general form ?x?y
    (P(x,y)?Q(x)),where x and y are variables, P a
    possibly compound predicate, and Q an atomic
    proposition.
  • In Prolog q(X) - p(X,Y).
  • i.e., the ?,? quantifiers are implicit.
  • Example likable(X) - likes(Y,X).

110
Rules in Prolog
  • Note that ?x?y (P(x,y)?Q(x))is equivalent to
    ?x((?y P(x,y)?Q(x))
  • In other words, Q(x) holds if you can find a y
    such that P(x,y)

111
Conjunction and Disjunction
  • Logical conjunction is encoded using multiple
    comma-separated terms in a rule.
  • Logical disjunction is encoded using multiple
    rules.
  • E.g., ?x ((P(x)?Q(x))?R(x))?S(x) can be rendered
    in Prolog as
  • s(X) - p(X),q(X)
  • s(X) - r(X)

112
Deduction in Prolog
  • When a query is input to the Prolog interpreter,
  • it searches its database to determine if the
    query can be proven true from the available
    facts.
  • if so, it returns yes, if not, no (!)
  • If the query contains any variables, all values
    that make the query true are printed.

113
Simple Prolog Example
  • An example input program
  • likes(john,mary).
  • likes(mary,fred).
  • likes(fred,mary).
  • likable(X) - likes(Y,X).
  • An example query ? likable(Z)returns ...

114
Simple Prolog Example
  • An example input program
  • likes(john,mary).
  • likes(mary,fred).
  • likes(fred,mary).
  • likable(X) - likes(Y,X).
  • An example query ? likable(Z)returns
    mary fred

115
Relation between PROLOG and Predicate Logic
  • PROLOG cannot use all predicate logic formulas.
    (It covers a decidable fragment of predicate
    logic.)
  • It uses negation as failure
  • Based on these limitations, PROLOG-based
    deduction is decidable.
  • PROLOG is more than just logic (I/O, cut!), but
    logic is its hard core.

116
Deduction Example (as appetiser for the next
lecture)
Topic 3 Predicate Logic
  • Premises
  • ?x H(x) ? M(x) (e.g. Humans are mortal)
  • ?x G(x) ? ?M(x) (Gods are immortal).
  • Suppose you need to prove the Conclusion ??x
    (H(x) ? G(x)) (No human is a god.)

117
A semantic proof in informal style
Topic 3 Predicate Logic
  • ?x (H(x) ? M(x)) (Humans are mortal) and
  • ?x (G(x) ? ?M(x)) (Gods are immortal).
  • Suppose ?x (H(x) ? G(x)). For example,
  • H(a) ? G(a). Then
  • By the first premisse, we have M(a).
  • By the second premisse, we have ?M(a).
  • Contradiction! Therefore, it follows that
  • ??x (H(x) ? G(x)) (No human is a god.)

118
Alternative proof, using equivalences
Topic 3 Predicate Logic
  • ?x (H(x)?M(x)) and ?x (G(x)??M(x)).
  • ?x (?M(x)??H(x)) Contrapositive.
  • ?x (G(x)??M(x) ? ?M(x)??H(x))
  • ?x (G(x)??H(x)) Transitivity of ?.
  • ?x (?G(x) ? ?H(x)) Definition of ?.
  • ?x ?(G(x) ? H(x)) DeMorgans law.
  • ??x (G(x) ? H(x)) An equivalence law.

119
Bonus Number Theory Examples
Topic 3 Predicate Logic
  • Let u.d. the natural numbers 0, 1, 2,
  • What do the following mean?
  • ?x (E(x) ? (?y x2y))
  • ?x (P(x) ? (xgt1 ? ??yz xyz ? y?1 ? z?1))

120
Bonus more Number Theory Examples
Topic 3 Predicate Logic
  • Let u.d. the natural numbers 0, 1, 2,
  • A number x has the property E if and only if it
    is equal to 2 times some other number.
    (even!)?x (E(x) ? (?y x2y))
  • A number has P, iff its greater than 1 and it
    isnt the product of any non-unity numbers.
    (prime!)
  • ?x (P(x) ? (xgt1 ? ??yz xyz ? y?1 ? z?1))

121
Goldbachs Conjecture (unproven)
Topic 3 Predicate Logic
  • Using E(x) and P(x) from previous slide,
  • ?x( xgt2 ? E(x) ?
  • ?p ?q P(p) ? P(q) ? pq x).

122
Goldbachs Conjecture (unproven)
Topic 3 Predicate Logic
  • Using E(x) and P(x) from previous slide,
  • ?x( xgt2 ? E(x) ?
  • ?p ?q P(p) ? P(q) ? pq x).
  • Every even number greater than 2 is the sum of
    two primes.

123
Game Theoretic Semantics
Topic 3 Predicate Logic
  • Thinking in terms of a competitive game can help
    you tell whether a proposition with nested
    quantifiers is true.
  • The game has two players, both with the same
    knowledge
  • Verifier Wants to demonstrate that the
    proposition is true.
  • Falsifier Wants to demonstrate that the
    proposition is false.
  • The Rules of the Game Verify or Falsify
  • Read the quantifiers from left to right, picking
    values of variables.
  • When you see ?, the falsifier gets to select
    the value.
  • When you see ?, the verifier gets to select the
    value.
  • If the verifier can always win, then the
    proposition is true.
  • Otherwise it is false.

124
Lets Play, Verify or Falsify!
Topic 3 Predicate Logic

Let B(x,y) xs birthday is followed within
14 days
by ys birthday.
Suppose I claim that among you ?x ?y B(x,y)
Your turn, as falsifier You pick any x ?
(so-and-so)
?y B(so-and-so,y)
My turn, as verifier I pick any y ?
(such-and-such)
B(so-and-so,such-and-such)
125
Summing up strict Predicate Logic without
extensions and simplified notations
Topic 3 Predicate Logic
  • Syntax definition
  • Variable x,y,z, Constants a,b,c,
  • 1-place predicates P,Q,
  • 2-place predicates R,S,
  • Atomic formulas If ? is a 1-pace predicate and
    ? a variable or constant then ?(?) is an atomic
    formula.
  • If ? is a 2-pace predicate and ? and ? are
    variables or constants then ?(?,?) is an atomic
    formula. and so on for 3-place, etc.

126
Syntax of predicate logic (for 1- and 2-place
predicates)
  • (Wellformed) Formulas
  • All atomic formulas are formulas
  • If ? and ? are formulas then ??, (? ??), (???),
    (? ??) are formulas.
  • If ? is a formula then ?x ? and ?y ? are
    formulas.

127
Semantics of predicate logic(this time using
models)
  • The truth conditions for propositional logic
    connectives are as specified in their truth-table
    definitions
  • ?x? is true with respect to a model if at least
    one expression ?(xa) is true with respect to
    the model, and false otherwise.
  • ?x? is true with respect to a model if every
    expression ?(xa) is true with respect to the
    model, and false otherwise

128
Semantics of predicate logic(using models in a
more precise way)
  • Observe Effectively, the meaning of formulas is
    defined via rules of the form ... is true iff
    .... This is called a truth definition.
  • For those of you who like a formal treatment of
    truth definitions, in which the role of models is
    made mathematically more explicit, here goes ...

129
Formally what is a model?
  • A model M is a pair ltD,Igt where D is a set (an
    u.d.), and I is a suitable interpretation
    function. Informally speaking, a given M defines
    one way of giving a concrete meaning to the
    symbols (other than ??????) in your formulas.
  • If a is an individual constant then I(a)?D
  • If P is a 1-place predicate then I(P)?D
  • If R is a 2-place predicate then I(P) is a set of
    pairs (?,?) such that ? ?D, and ? ?D

130
  • (This way to specify the meaning of a
    2-place predicate will become clearer after
    the lectures about relations. An example
    might help )

131
An example of a model
  • Suppose M (D,I) where DJohn,Mary,Bill and
    I(j)John, I(m)Mary, I(b)Bill,
    I(B)John,BillI(G)MaryI(A)(John,Mary),(Bi
    ll,Mary)
  • Note John is a person (i.e., a part of the
    world)j is an individual constant (i.e., a part
    of the language of predicate logic). j names
    John.

132
Formally (where M ltD,Igt) atomic formulas
  • Here is the start of the new truth definition
  • A formula of the form P(a) is true with respect
    to M iff I(a) ? I(P).
  • A formula of the form R(a,b) is true with
    respect to M iff (I(a),I(b)) ? I(R).

133
Formally (where M ltD,Igt) formulas with
propositional connectives
  • A formula of the form ?? is true wrt M iff ? is
    false wrt M
  • A formula of the form ??? is true wrt M iff ? is
    true wrt M or ? is true wrt M or both
  • A formula of the form ??? is true wrt M iff ? is
    true wrt M and ? is true wrt M
  • A formula of the form ? ? ? is true wrt M iff ?
    is false wrt M or ? is true wrt M or both

134
Formally (where M ltD,Igt) quantified formulas
  • A formula of the form ?x? is true wrt M iff at
    least one expression ?(xa) is true wrt M.
    (More precisely there is an a such that ...)
  • A formula of the form ?x? is true wrt M iff all
    expressions ?(xa) are true wrt M. (More
    precisely for all a, ...)
  • (End of truth definition.)

135
models some illustrations
  • Suppose M (D,I) , where DJohn,Mary,Bill, and
    I(j)John, I(m)Mary, I(b)Bill,
    andI(B)John,BillI(G)MaryI(A)(John,Mary)
    ,(Bill,Mary)
  • Is B(j) true wrt M?

136
models some illustrations
  • Suppose M( D,I) , where DJohn,Mary,Bill, and
    I(j)John, I(m)Mary, I(b)Bill,
    andI(B)John,BillI(G)MaryI(A)(John,Mary)
    ,(Bill,Mary)
  • Is B(j) true wrt M? Yes

137
models some illustrations
  • Suppose M(D,I), where DJohn,Mary,Bill, and
    I(j)John, I(m)Mary, I(b)Bill,
    andI(B)John,BillI(G)MaryI(A)(John,Mary)
    ,(Bill,Mary)
  • Is B(j) true wrt M? Yes
  • Is B(j)?B(b)?A(m,m) true wrt M?

138
models some illustrations
  • Suppose M(D,I), where DJohn,Mary,Bill, and
    I(j)John, I(m)Mary, I(b)Bill,
    andI(B)John,BillI(G)MaryI(A)(John,Mary)
    ,(Bill,Mary)
  • Is B(j) true wrt M? Yes
  • Is B(j)?B(b)?A(m,m) true wrt M? No

139
models some illustrations
  • Suppose M(D,I), where DJohn,Mary,Bill, and
    I(j)John, I(m)Mary, I(b)Bill,
    andI(B)John,BillI(G)MaryI(A)(John,Mary)
    ,(Bill,Mary)
  • Is B(j) true wrt M? Yes
  • Is B(j)?B(b)?A(m,m) true wrt M? No
  • Is ?x(B(x)?A(x,m)) true wrt M?

140
models some illustrations
  • Suppose M(D,I), where DJohn,Mary,Bill, and
    I(j)John, I(m)Mary, I(b)Bill,
    andI(B)John,BillI(G)MaryI(A)(John,Mary)
    ,(Bill,Mary)
  • Is B(j) true wrt M? Yes
  • Is B(j)?B(b)?A(m,m) true wrt M? No
  • Is ?x(B(x)?A(x,m)) true wrt M? Yes

141
models some illustrations
  • ?x(B(x)?A(x,m)) true wrt M iff
  • all expressions B(a) ?A(a,m) are true wrt M.
  • These are the following
  • B(j) ?A(j,m)
  • B(b) ?A(b,m)
  • B(m) ?A(m,m)

142
why is the latter formula true?
  • ?x(B(x)?A(x,m)) true wrt M iff
  • all expressions B(a) ?A(a,m) are true wrt M.
  • These are the following
  • B(j) ?A(j,m) true wrt M
  • B(b) ?A(b,m) true wrt M
  • B(m) ?A(m,m)

143
why is the latter formula true?
  • ?x(B(x)?A(x,m)) true wrt M iff
  • all expressions B(a) ?A(a,m) are true wrt M.
  • These are the following
  • B(j) ?A(j,m) true wrt M
  • B(b) ?A(b,m) true wrt M
  • B(m) ?A(m,m) true wrt M (antecedent false!)
  • Consequently ?x(B(x)?A(x,m)) is true wrt M

144
Predicate logic optional extensions
  • This completes the strict version of predicate
    logic
  • In addition, we have introduced a number of
    simplifications and extensions
  • Brackets
  • all outermost brackets can be omitted
  • brackets that are superfluous because of
    associativity can be omitted, e.g., p?q?r instead
    of (p?q)?r
  • we sometimes add extra brackets for clarity, or
    .. rather than (..), to show which brackets
    match

145
Predicate logic optional extensions
  • Quantifiers
  • we sometimes write ?xyz instead of ?x?y?z, or
    ?xyz instead of ?x?y?z
  • ?!x P(x), ?2!x P(x), etc. for exact numbers
  • restricted quantification e.g., ?xgt0? instead
    of ?x(xgt0??) or ?xgt0? instead of ?x(xgt0??)
  • Other
  • we sometimes write ??? instead of ???
  • we sometimes allow ourselves to suggest
    infinitely long formulas using dots, e.g. p1 ?
    p2 ? ...
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