PREDICATE LOGIC

1
PREDICATE LOGIC
Models, counterexamples
2
Another Tree
v
\ a
b
Some planets contain water, and some planets are
suitable for life. Therefore, there is a planet
containing water and suitable for life.
v
a
v
b
Ca
Lb
Cx x is a planet containing waterLx x is a
planet suitable for life
(Ca La)
v
Ca
La
X
(Cb Lb)
v
Cb
Lb
X
?
IS THE ARGUMENT NON-VALID?
3
Model
( x)Cx ( y)Ly( x)(Cx Lx)
E
v
E
\ a
b
E
Alpha ... planet containing water, but not
suitable for living
v
a
v
b
Beta ..... planet not containing water, but
suitable for living
Ca
Some planets contain water, and some planets are
suitable for life.
Lb
YES
(Ca La)
v
There is a planet containing water and suitable
for life.
NO!
Ca
La
X
(Cb Lb)
v
THE ARGUMENT IS NOT VALID!
Cb
Lb
X
?
4
Model
A
Domain Alpha, Beta
Name interpretation
Alpha ... planet containing water, but not
suitable for living
I(a) Alpha
I(b) Beta
Beta ..... planet not containing water, but
suitable for living
Predicate interpretation
Some planets contain water, and some planets are
suitable for life.
YES
There is a planet containing water and suitable
for life.
NO!
THE ARGUMENT IS NOT VALID!
true in A
false in A
5
Models of theory ( x)Cx ( y)Ly
E
E
B

Domain Alpha, Beta, Earth
Domain Mercury, Mars, Venus, Pluto
Name interpretation
Name interpretation
I(a) Alpha
I(a) Mercury
I(b) Beta
I(b) Mars
Predicate interpretation
Predicate interpretation
...
true in B
true in B
6
Model

• A model consists of
• Domain, i.e. non-empty set of objects
• Interpretation I of names and predicates such
  that
• Every name a is interpreted by an object I(a)
  from the domain
• Each n-ary predicate P is interpreted by an
  assignment I(P) of truth values to all ordered
  n-tuples from the domain

When is a formula satisfied (true) in a model?

• Only formulas created using following rules are
  well formed formulas
• If F is an n-ary predicate, and a1,..., an are
  names, then Fa1...an is a formula
• If A and B are formulas, then also A, (AB),
  (AvB), (A?B), (A?B) are formulas
• If A is a formula, a is a name and x is a
  variable, then also ( x)A(ax) and (
  x)A(ax) are formulas, where A(aß) is a result
  of replacing all occurrences of a name or
  variable a with a name or variable ß.

7
Semantic Entailment
For a set of formulas X and a formula A, X A (X
semantically entails A) if and only if EVERY
model satisfying (making true) each formula in X
also satisfies A.
So, if there is a model satisfying all formulas
in X, but not satisfying formula A, then X A.
In that case, argument X, therefore A is not
valid. The model in question is a counterexample
for this argument.
8
Another Argument
Everybody likes somebody. Therefore, everybody
like me.
E
Lxy x likes yj me
E
9
Another Argument
Domain Juraj, Ann, Betty, Colin, Dan, ...
( x)( y)Lxy ( x)Lxj
A
E
\ j
b
c
...
a
I(j) Juraj I(b) Betty I(d) DanI(a)
Ann I(c) Colin ...
A
a
v
Laj
( y)Ljy
b
v
E
Ljb
( y)Lay
c
E
v
Lac
d
v
Lbd
10
Another Argument
Domain Juraj, Ann, Betty, Colin, Dan, ...
( x)( y)Lxy ( x)Lxj
A
E
\ j
b
c
...
a
I(j) Juraj I(b) Betty I(d) DanI(a)
Ann I(c) Colin ...
A
a
v
Laj
( y)Ljy
b
v
E
Ljb
( y)Lay
c
E
v
Lac
d
v
Lbd
11
Another Argument
A
Domain Juraj, Ann, Betty, Colin, Dan, ...
( x)( y)Lxy ( x)Lxj
A
E
\ j
b
c
...
a
I(j) Juraj I(b) Betty I(d) DanI(a)
Ann I(c) Colin ...
A
a
v
Laj
( y)Ljy
b
v
E
Ljb
( y)Lay
c
E
v
Lac
B
Domain Juraj, Ann
I(j) Juraj I(a) Ann
d
v
TRUE
Lbd
FALSE
12
How to deal with a predicate logic argument

• Translate the argument into predicate logic
  formulas, premises X and conclusion A.
• Try to analyze tree for X, A.
• If the tree closes, then fine X A, so the
  argument is valid.
• If the tree doesnt close but it completes,
  construct a counterexample a model satisfying
  all the premises X and not satisfying the
  conclusion A then fine X A, so the argument is
  not valid.
• If the tree doesnt seem to close, neither is
  seems to get completed, then think!
• Step back and try to see a pattern which allows
  you to construct a counterexample a model
  satisfying all the premises X and not satisfying
  the conclusion A.
• If youre successful, then fine X A, so the
  argument is not valid.
• If youre not, try harder!

13
Completeness Theorem
X A a tree for set of formulas X,A closes
X A every model satisfying each formula in X
also satisfies A
Let X be a set of (predicate logic) formulas, and
let A be a (predicate logic) formula. Then X A
if and only iff X A

• If X A, then some tree for X,A closes. Can
  some (other) tree complete but remain open?
• NO! Why?
• Because, for a complete tree with an open branch
  we could construct a model A satisfying all
  premises X and A.
• But it would mean, that X A. And thus
  (Completeness Theorem) X A.
• And it cannot, because we supposed that X A!

14
Arguments
Every logician is a smart person. Some people
(persons) are smart. Therefore, there is a smart
logician.
For every two philosophers its true, that either
one inspired the other or the other way round.
Every philosopher is inspired by him-/her- self.
There are some philosophers. Therefore, there is
a philosopher which inspired every philosopher.
For every two philosophers its true, that either
one inspired the other or the other way round.
Every philosopher is inspired by him-/her- self.
(There are some philosophers.) (If a philosopher
inspired another philosopher which in turn
inspired yet another one, then the first
philosopher also inspired the last
one.) Therefore, there is a philosopher which
inspired every philosopher.