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“mcsftl” — 2010/9/8 — 0:40 — page 5 — #11
Propositions
Definition.
A
proposition
is a statement that is either true or false.
For example, both of the following statements are propositions. The first is true
and the second is false.
Proposition 1.0.1.
2 + 3 = 5.
Proposition 1.0.2.
1 + 1 = 3.
Being true or false doesn’t sound like much of a limitation, but it does exclude
statements such as, “Wherefore art thou Romeo?” and “Give me an
A
!”.
Unfortunately, it is not always easy to decide if a proposition is true or false, or
even what the proposition means. In part, this is because the English language is
riddled with ambiguities. For example, consider the following statements:
1. “You may have cake, or you may have ice cream.”
2. “If pigs can ﬂy, then you can understand the Chebyshev bound.”
3. “If you can solve any problem we come up with, then you get an
A
for the
course.”
4. “Every American has a dream.”
What
precisely
do these sentences mean? Can you have both cake and ice cream
or must you choose just one dessert? If the second sentence is true, then is the
Chebyshev bound incomprehensible? If you can solve some problems we come up
with but not all, then do you get an
A
for the course? And can you still get an
A
even if you can’t solve any of the problems? Does the last sentence imply that all
Americans have the same dream or might some of them have different dreams?
Some uncertainty is tolerable in normal conversation. But when we need to
formulate ideas precisely—as in mathematics and programming—the ambiguities
inherent in everyday language can be a real problem. We can’t hope to make an
exact argument if we’re not sure exactly what the statements mean. So before we
start into mathematics, we need to investigate the problem of how to talk about
mathematics.
To get around the ambiguity of English, mathematicians have devised a special
minilanguage for talking about logical relationships. This language mostly uses
ordinary English words and phrases such as “or”, “implies”, and “for all”. But
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“mcsftl” — 2010/9/8 — 0:40 — page 6 — #12
Chapter 1
Propositions
mathematicians endow these words with definitions more precise than those found
in an ordinary dictionary. Without knowing these definitions, you might sometimes
get the gist of statements in this language, but you would regularly get misled about
what they really meant.
Surprisingly, in the midst of learning the language of mathematics, we’ll come
across the most important open problem in computer science—a problem whose
solution could change the world.
1.1
Compound Propositions
In English, we can modify, combine, and relate propositions with words such as
“not”, “and”, “or”, “implies”, and “ifthen”. For example, we can combine three
propositions into one like this:
If
all humans are mortal
and
all Greeks are human,
then
all Greeks are mortal.
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 Fall '10
 TomLeighton,Dr.MartenvanDijk
 Computer Science, Logic, Proposition, Quantification, Universal quantification

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