epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

Introduction and Basic Terminology 7
introduced. Indeed, all the results already established and all the definitions
already stated as parts of a context can be used in the construction of
the proofs of other results in that same context. As this book focuses more
on the "nuts and bolts" of proof design than on the development of a
mathematical theory, it does not include the construction of a mathematical
setting for the material presented. This approach is supposed to provide the
reader with the basic tools to use for the construction of proofs in a variety
of mathematical settings.
At this point we want to emphasize the difference between the vaHdity of
an argument and the truth or falsity of the results of an argument. An
argument is valid if its hypothesis suppHes sufficient and certain basis for the
conclusion to be reached. An argument can be vaHd and reach a false
conclusion, as in the following example, in which one of the hypotheses
is false.
All birds are able to fly.
Penguins are birds.
Therefore, penguins are able to fly.
An argument can be invahd and reach a true conclusion. Consider the
following argument:
Cows have four legs.
Giraffes have four legs.
Therefore, giraffes are taller than cows.
In the example just given, it is clear that the information we have (cows have
four legs; giraffes have four legs) does not imply that "giraffes are taller that
cows," which is nonetheless a true fact. The only conclusion we could
legitimately reach is that giraffes and cows have the same number of legs.
In other cases the possible flaws in the reasoning process are more subtle.
Consider the following argument.
// Joe wins the state lottery, he can afford a new car.
Joe did not win the state lottery.
Therefore, Joe cannot afford a new car.
The hypotheses for this argument are: If Joe wins the state lottery, he can
afford a new car. Joe did not win the state lottery. The conclusion reached is:
Joe cannot afford a new car.
This is an example of incorrect (nonvahd) reasoning. Indeed, Joe did not
win the state lottery, so he might not be able to afford a new car (the