PREDICATE LOGIC

1
PREDICATE LOGIC
Trees, proofs
2
Argument
All male philosophers have beards.Socrates is a
male philosopher. Therefore, Socrates has a beard.
s Socrates B... ... is bearded M... ...
is a male P... ... is a philosopher
3
Tree
\ s
MsPs
(Ms Ps) ? Bs
v
(Ms Ps)
Bs
X
Ms
Ps
X
X
THE ARGUMENT IS VALID
4
Another Tree
If nobody contributes to Oxfam then there is
someone who dies of hunger. Therefore, there is
a person who dies of hunger if he/she doesnt
contribute to Oxfam.
v
\ a
b
a
b
v
v
Ca
Hb
Cx x contributes to OxfamHx x dies of hunger
(Ca ? Ha)
(Cb ? Hb)
CaHa
CbHb
X
X
THE ARGUMENT IS VALID
5
Well formed formulas

• 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.

Every variable has to be quantified, i.e. it has
to be within a scope of its quantifier. So, Px,
Lxb, ( x)Lxy, are not formulas! Strings like
this (containing non-quantified variables) are
called expressions. Propositional logic rules for
well formed formulas apply for predicate logic
formulas and expressions. When do we skip
brackets? (See the rules for well-formed formulas
above)
E
6
Another Tree
An action is not bad only if its not selfish.
Not every action is bad. Therefore, some
actions are not selfish.
Ax x is an actionBx x is badSx x is
selfish
X
X
X
X
THE ARGUMENT IS VALID
7
Tree rules and proof
To complete a tree these rules have to
applied for all existing names
If some tree for set of formulas X,A closes,
then we write X A. The tree is called a proof of
formula A from axioms X (in theory X).
8
Argument
If x is similar to y, then y is similar to x. If
x is similar to y and y is similar to z, then x
is similar to z. Everything is similar to
something. Therefore, everything is similar to
itself.
Sxy x is similar to y
9
Proof
X
(Sab Sba)
Saa
X
THE ARGUMENT IS VALID
Sab
Sba
X
X
10
Argument
If x is similar to y, then y is similar to x. If
x is similar to y and y is similar to z, then x
is similar to z. Everything is similar to
something. Therefore, if x is similar to y but
not similar to z, then y is not similar to z.
11
Syntactical entailment

• For a valid argument does the tree always close?
• Yes!
• What if the argument is not valid?