# Predicate Logic (PowerPoint) by yurtgc548

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Predicate Logic

Colin Campbell
A Formal Language
   Predicate Logic provides a way to
formalize natural language so that
ambiguity is removed.
   Mathematical arguments (proofs) are
usually written in pseudo logic.
   Many programming languages involve
elements of predicate logic. e.g. if..then
Predicate vs Propositional
Logic
   Predicate logic uses the same connectives
as propositional logic but allows you to
refer to different elements of the
universe.
   It also introduces quantifiers…so you can
say things like..
   It always rains in England.
   Sometimes it is sunny in Bristol.
Propositional Logic Revised
   Propositional Logic is a formal approach to
sentences.

If David likes Mary then Mary likes David
David likes Mary
Mary likes David
Semantics vs Proof Theory
   There are two approaches to reasoning in
propositional logic.

Semantics = truth tables and valuations

Proof theory    = formal notion of proof from
axioms and inference rule
Truth-Tables
   The truth value of compound sentences can then be
determined by application of truth tables defined for
each connective
A    A     A    B    AB     A    B    AB
t     f     t    t     t       t   t     t
f     t     f    t     f       f   t     t     etc
t    f     f       t   f     f
f    f     f       f   f     t
A sentence B truth-functionally follows from sentence A if
for every row of the truth-table in which A is true B is also
true. Denoted A|=B
Formal Proof
T=set of assumptions (sentences) then T|-A if we can
generate a sequence of sentences
Axioms for propositional logic
S1,S2,…,Si-1,Si,…,Sn=A               1. A(B A)
2. (A (B C)) ((A B)
Where either                            (A C))
Si=axiom                             3. (B  A) ((B A)
= a sentence in T (an assumption)      B)

or                                   Inference Rule: Modus
S1,…,Si-1 (inference rule) Si       Ponens
From A,AB infer B
Relating Semantics and Proof
Soundness

Proof theory  Semantics

Completeness

Semantics  Proof theory

Propositional logic is sound and complete

Semantics = Proof theory
Decidability
   A formal logic is decidable if there is a mechanical
(algorithmic) way of determining whether A follows
from T.
T|=A ?
Propositional logic is
decidable: at worst we
need only consider
yes   every row of the truth-
table for which T is
true, and there are only
finitely many.
No
Predicate Logic: Introducing
Quantifiers
    Predicate logic allows for quantified statements by
introducing the quantifiers  “for all” and  “there
exists”.
     Why do we need quantifiers ? Consider
All birds have wings
Tweety is a bird
Therefore, Tweety has wings
Propositional logic: All birds have wings= (Robins have
wings)(Crows have wings)(Eagles have wings)
..........etc.
This is impractical.
Quantifiers 2:
   In predicate logic this translates to

xBx   W x 
• Unlike propositional logic,
predicate logic has variables- in
this case x.
BTweety                  • It also has constants – in this
case Tweety
W Tweety                • Tweety is an instantiation of
the predicates B and W.

B(x)= x is a Bird, W(x)=x has wings,
x (B(x)W(x)) = For all x, if x is a bird then x has wings
B(Tweety)=Tweety is a bird.
W(Tweety)=Tweety has wings
Functions
   It is also desirable to be able to represent functional
relationships in predicate logic. Consider

Blue eyed people have blue eyed fathers
Some people have blue eyes
Therefore, some people’s fathers have blue eyes
Translates to            x is a variable ranging across all people
              
x BEx   BE F x     BE(x) -means “person x has blue eyes”
xBEx 
F is a function mapping from one
 
xBE F x              person x to another person F(x), where
F(x) is the father of x
Syntax: Building Blocks for
Predicate Logic
   Formula in predicate are built up from the
following elements
   (1) Predicates: PL is a set of n-ary
predicates. e.g.
   B(x)=x is a bird –unary
   M(x,y)=x is married to y –binary
   BT(x,y,z)=y is between x and z –tertiary
Syntax (2)
   (2) Functions: FNL is a set of n-ary
functions.
   F(x)=Father of x, D(x,y,z)=Day of the
weeks falling on the date x/y/z.
   (3) Constants: CL is a set of constants
e.g. Tweety, numbers, Aunty Ethel
   (4) Variables: a set of variables, x,y,z etc
   (5) Connectives, Quantifiers and
Brackets: ,,,,, (,),[,]
Terms
    Terms are possible instantiations of predicates.
     The set of terms TL is defined recursively in the
following way:
(i) All variables and constants are terms
(ii) If f is an n - ary function symbols and t1....tn are
terms then f(t1....tn) is a term.
FNL={father, mother, first-child}, CL={bill}, VL={x} -
ranging across parents
x, bill, father(x), mother(x), first-child(x), father(bill), mother(bill),
first-child(bill), father(father(x)), father(mother(x)), first-child(father(x))
father(father(bill)), father(mother(bill)),.......etc
Formula in Predicate Logic
   Atomic Formula: The atomic formula ATL, have the
form P(t1,...., tn) where P is an n-ary relation in PL
and ti are terms for i=1,...,n
   E.g. PNL={Rich,Happy},
FNL={father,mother},CL={jim, fred, mary},
VL={x,y}
   Atomic formula include…Rich(x), Happy(y), Rich(bill),
Happy(father(y)), Rich(mother(mary)),
Happy(father(mother(x))) etc
Formula 2
   Well-Formed Formula: The set of well-formed
formula WFL is defined recursively as follows:

(i) all atomic formula are well-formed formula
(ii) If F and G are well-formed formula so are (F),
(FG), (FG), (FG), (FG)
(iii) If F is a well-formed formula and xVL then
(xF) and (xF) are well-formed formula.

E.g. Rich(x)Happy(x), Rich(x)Happy(y),
Rich(bill)Happy(father(fred)),
(x Happy(x)) yRich(y)), x(Rich(x) Happy(x))
Translating from Logic to
English
   Example: PL ={married, younger-than, equal} all binary
VL={x,y,z} ranging over all parents
CL={bill, john, george, mary}
FNL={father, mother, first-child} -all unary
Connectives ,,,,
Quantifiers ,, Brackets
married(mary,john) translates to ”Mary is married to John”
equal(father(bill),george) translates to ”George is the father of Bill”
equal(mother(mother(bill)),mary) translates to ”Mary is the maternal
grandmother of Bill”
(equal(mother(x),y)equal(father(x),y)) translates to ”y is a parent of x”
xy(married(x,y) married(y,x)) translates to ”for any parents x and y
if x is married to y then y is married to x”
xy(equal(mother(x),y) translates to ”everyone has a mother”
Translating from English to
Logic
   “Anyone who is persistent can learn logic” translates
to x(persistent(x)learn(x, logic))
   “All blocks which are on top of or attached to blocks
which have been moved, have also been moved”
translates to xy((block(x)block(y)(ontop(x,y)
attached(x,y))moved(y))moved(x))
   “You can fool some of the people all of the time, and
you can fool all the people some of the time, but you
can’t fool all the people all the time” translates to
xy(person(x)time(y)fool-at(y,x)) 
yx(person(x)time(y)fool-at(y,x)) 
(x y (person(x)time(y)fool-at(y,x)))
Removing Ambiguity
   Bertrand Russell gave the following example of how
translating to logic can remove ambiguity.
   Consider the sentence “The current king of France is
bald”. Is this sentence true?
   Translating to logic gives:
   x(KF(x)B(x))
   KF(x)= x is the current king of France.
   B(x)= x is bald.
   This is clearly false! Why?
Whoops-Missed a bit
Its only me!
So the correct translation is
xKF(x)
x(KF(x)y(KF(y)equals(x,y)))
x(KF(x)B(x))

Still false though!

What about “the current king of France is not bald” – is it
true or false?
Declarative Formula
   Some formula have a truth-value while others don’t.
   xy(equal(mother(x),y) has a truth-value
   So does x(Rich(x)Happy(x))
   But neither Rich(x) nor married(x,y) married(y,x)
have truth values.
   In predicate logic declarative formula are closed.
Non-declarative form are open.
Open and Closed Formula
   Scope of a Quantifier: The scope of x (resp. x) in x
F (resp x F) is F
   E.g. In x(Rich(x)Happy(x)) the scope of x is
(Rich(x)Happy(x))
   In xy(equal(mother(x),y) the scope of x is
y(equal(mother(x),y) and the scope of y is
(equal(mother(x),y)
   Free and Bound Occurrence of a Variable: A bound
occurrence of a variable in a formula is an occurrence
immediately following a quantifier or an occurrence within
the scope of a quantifier, which has the same variable
immediately after the quantifier.
   Any other variable is free.
Open and Closed:2
   Example: In the formula (xP(x,y))Q(x) the
occurrence of x in P(x,y) is bound but the occurrence
of x in Q(x) is free.
    All occurrences of x in x(P(x,y)Q(x)) are bound.
   Open and Closed Formula: A closed formula is a
well-formed formula with no free occurrences of any
variable. An open formula is a formula that is not
closed
   Example: xy(equal(mother(x),y) is closed since
all occurrences of x fall within the cope of x and all
occurrences of y fall within the sope of y.
Interpretations
   Really a language of predicate logic is only a set of
symbols which can be interpreted in different ways.
   Consider the formula: yxP(x,y)
   Quantification over natural numbers, P(x,y) means
xy then interpretation is `for all natural numbers x
there is some natural y for which xy’
   Quantification over people, P(x,y) means `y is the
father of x’ then interpretation is `there is some who
is the father of everyone’
Interpretations:2
U1                                           U2
Formal Logic
I                   J   PL J={PJ,QJ,RJ,..}
PLI={PI,QI,RI,..}       PL={P,Q,R,..}
FL J={fJ,gJ,hJ,..}
FLI={fI,gI,hI,..}       FL={f,g,h,..}
CL J={aJ,bJ,cJ,..}
CLI={aI,bI,cI,..}       CL={a,b,c,..}
Theories and Models
   Theory: A theory of a language of
predicate logic L is any finite set of
closed well-formed formula of L.
   Model: A model of a theory T is an
interpretation of L under which all the
formula in T are true.
Satisfiability
   A theory of L, is satisfiable if it has at
least one model.
   Any theory that is not satisfiable is said
to be inconsistent.
   Any theory for which every possible
interpretation is a model is said to be
valid.
Satisfiability: 2
   The theory consisting of the single
formula x(P(x) P(x)) is inconsistent.
   The theory {x (P(x) P(x))} is valid.
Example
   Consider the following theories:
T1  xyPx, y   Pc, f c 
T2  xPx, f x   yPc, y 

   Also consider the following two interpretations
Interpretation I: Let U1 be the set of all integers, cI=0,
fI is the successor function, PI is >

Interpretation J: Let U2 be the set of all integers,
cJ=0, fJ is the predecessor function, PI is 
Example: 2
Under I
T1 means
If for every integer x there is some integer y such that x>y
then 0>0+1
Clearly this is false since the antecedent is true and the
consequence is false - I is not a model of T1

T2 means
If for every integer x x>x+1 then there exists an integer y
such that 0>y
This is true since the antecedent is false - I is a model of T2
Example: 3
Under J
T1 means
If for every integer x there is some integer y such that xy
then 0  0-1
Clearly this is true since both the antecedent and
consequence are true - J is a model of T1

T2 means
If for every integer x, x x-1 then there exists an integer y
such that 0y
This is true since, again, both antecedent and consequence
are true - J is a model of T2
Example: 4
   We see that T1 is not a valid theory since I
is not a model of T1.
   Both I and J are models of T2. Is T2 valid?
   Let K be any interpretation of L then T2
means
   If for every element xU PK (x, fK(x)) then
there exists some element yU such that
PK(cK, y ).
Example: 5
   Two possibilities
   Antecedent false -in this case the
conditional is automatically true.
   Antecedent true -in this case since cKU
we have PK (cK, fK(cK)) and hence the
consequence of the conditional must
also be true since we can select y to be
fK(cK).
Semantic Entailment
   A theory T is said to entail a closed formula F if F is true
in every model of T.
T F                            T F

models of F      models of F       models of F      models of F

models of T                                          models of T

models of F    models of F         T F
models of T                 T F
Decidability
T F                 T F

Proof                Proof
Procedure             Procedure        Predicate logic is only
Query: does F        Query: does F      semi-decidable