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Negation, Conjunction, Disjunction
In FOL, atomic sentences have a limited expressive power.
Consider the following set of
statements:
Claire is not taller than Max.
Max is at home and Claire fed Scruffy.
Either Claire is a student or Max is a student.
In order to form more complex statements in FOL, such as those above, we need to
combine atomic sentences with
connectives
.
The first set of connectives we will consider
are called Boolean connectives and they correspond to the English expressions
it is not the
case that
,
and
, and
or
.
The Boolean connectives, along with several other we will discuss later, are called “truth
functional” connectives.
The reason for this is that the truth value of a complex sentence
using Boolean connectives is determined by the truth values of the simple sentences
involved.
Perhaps the best way to illustrate is to construct a
truth table
for each
connective, showing that the overall truth value of Boolean statements depends on the
atomic sentences of which the Boolean statement is composed.
For each new connective
we introduce into our system, its truthfunctional definition will be given by way of a truth
table.
Negation:
¬
Our first connective
¬
will be used to express negation.
In English we might use the
expressions
not
,
is not the case that
,
non
, or
un
to express such negation.
In FOL, all of
these Englishlanguage ways of expressing negation can be handled by the
¬
symbol.
Suppose that we want to express our first example from above in FOL.
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 Fall '06
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