12/4/2007
1
QUANTIFICATION
Chapter 9
PROPOSITIONAL LOGIC
What we have been doing over the last couple of
weeks is called
propositional logic
.
Propositional logic deals with the logic of simple
atomic sentences with named objects.
Cube(a)
refers to a property of an object
a
. We
know which object that is.
We are now going to extend our understanding of
logic to include more complex cases.
WHAT IS WRONG WITH PROPOSITIONAL
LOGIC?
Suppose we want to say “All objects are cubes.”
What we can claim with propositional logic.
1 atomic sentence:
1 property of 1 object:
Cube(a)
1 relationship of a fixed number of objects:
Between(a,b,c)
Connectives
Finite number of relationships (properties), each
with a fixed number of objects
Conjunction, disjunction, conditional symbol, …
Cube(b)
∧
((Tet(a)
∨
(Dodec
(c)) →
LeftOf(a,b))
PROPERTIES OF PROPOSITIONAL CALCULUS
In a propositional calculus (a formal system):
Objects must be named.
“Finite” claims
We have a limited number of objects and relations
Can we use propositional logic to claim
something like the following?
All objects are small.
No objects are tetrahedron.
Some object is large.
We can express sentences like “all objects are
small” in a limited domain like Tarski‟s
World
simply by listing them all individually:
Small(a)
∧
Small(b)
∧
Small(c)
∧
…
This becomes impractical and impossible if we
expand the domain to the real world. How would
we express a sentence like “all actors are rich” in
propositional logic?
You would have to list all actors individually,
which is clearly impossible to do.
If we don‟t have a limited domain, then we will
need a new mechanism, one which allows us to
say things like “all actors are rich” without
having to list them all.
This new mechanism is called
predicate logic
and
it involves the notion of
quantification
.
Predicate logic is also called
firstorder logic
.
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CREATING QUANTIFIED SENTENCES
Redefining terms
Defining wellformed formulas
(wff)
Redefining sentences
“
ATOMIC SENTENCES
” (
THUS FAR)
Formed by a single predicate followed by one or
more terms:
e is larger than b
Larger(e,b)
e is identical to a
e = a
A sentence expresses a claim that is either
true
or
false
.
QUANTIFICATION
To be able to work with ideas such as “all”,
“most” or „none” we need to be able to talk about
objects as a group (set), rather than as individual
objects.
We do that by introducing the notion of a
variable
, which is a reference to an object without
the need of specifying what that object is.
It is similar to the use of a variable in algebra.
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 Spring '08
 DeHaan
 Logic, Predicate logic, Claire, Firstorder logic

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